Personality

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Personality

Personality is a structure gathering interrelated behavioral, cognitive and emotional patterns that biological and environmental factors influence; these interrelated patterns are relatively stable over time periods, but they change over the entire lifetime. While there is no generally agreed-upon definition of personality, most theories focus on motivation and psychological interactions with the environment one is surrounded by. Trait-based personality theories, such as those defined by Raymond Cattell, define personality as traits that predict an individual's behavior. On the other hand, more behaviorally-based approaches define personality through learning and habits. Nevertheless, most theories view personality as relatively stable.

The study of the psychology of personality, called personality psychology, attempts to explain the tendencies that underlie differences in behavior. Psychologists have taken many different approaches to the study of personality, including biological, cognitive, learning, and trait-based theories, as well as psychodynamic, and humanistic approaches. The various approaches used to study personality today reflect the influence of the first theorists in the field, a group that includes Sigmund Freud, Alfred Adler, Gordon Allport, Hans Eysenck, Abraham Maslow, and Carl Rogers.

Measuring

Personality can be determined through a variety of tests. Due to the fact that personality is a complex idea, the dimensions of personality and scales of such tests vary and often are poorly defined. Two main tools to measure personality are objective tests and projective measures. Examples of such tests are the: Big Five Inventory (BFI), Minnesota Multiphasic Personality Inventory (MMPI-2), Rorschach Inkblot test, Neurotic Personality Questionnaire KON-2006, or Eysenck's Personality Questionnaire (EPQ-R). All of these tests are beneficial because they have both reliability and validity, two factors that make a test accurate. "Each item should be influenced to a degree by the underlying trait construct, giving rise to a pattern of positive intercorrelations so long as all items are oriented (worded) in the same direction." A recent, but not well-known, measuring tool that psychologists use is the 16PF. It measures personality based on Cattell's 16-factor theory of personality. Psychologists also use it as a clinical measuring tool to diagnose psychiatric disorders and help with prognosis and therapy planning.

Personality is frequently broken into factors or dimensions, statistically extracted from large questionnaires through factor analysis. When brought back to two dimensions, often the dimensions of introvert-extrovert and neuroticism (emotionally unstable-stable) are used as first proposed by Eysenck in the 1960s.

Five-factor inventory

The Big Five personality traits

Many factor analyses found what is called the Big Five, which are openness to experience, conscientiousness, extraversion, agreeableness, and neuroticism (or emotional stability), known as "OCEAN". These components are generally stable over time, and about half of the variance appears to be attributable to a person's genetics rather than the effects of one's environment.

Some research has investigated whether the relationship between happiness and extraversion seen in adults also can be seen in children. The implications of these findings can help identify children who are more likely to experience episodes of depression and develop types of treatment that such children are likely to respond to. In both children and adults, research shows that genetics, as opposed to environmental factors, exert a greater influence on happiness levels. Personality is not stable over the course of a lifetime, but it changes much more quickly during childhood, so personality constructs in children are referred to as temperament. Temperament is regarded as the precursor to personality.

Another interesting finding has been the link found between acting extraverted and positive affect. Extraverted behaviors include acting talkative, assertive, adventurous, and outgoing. For the purposes of this study, positive affect is defined as experiences of happy and enjoyable emotions. This study investigated the effects of acting in a way that is counter to a person's dispositional nature. In other words, the study focused on the benefits and drawbacks of introverts (people who are shy, socially inhibited, and non-aggressive) acting extraverted, and of extraverts acting introverted. After acting extraverted, introverts' experience of positive affect increased whereas extraverts seemed to experience lower levels of positive affect and suffered from the phenomenon of ego depletion. Ego depletion, or cognitive fatigue, is the use of one's energy to overtly act in a way that is contrary to one's inner disposition. When people act in a contrary fashion, they divert most, if not all, (cognitive) energy toward regulating this foreign style of behavior and attitudes. Because all available energy is being used to maintain this contrary behavior, the result is an inability to use any energy to make important or difficult decisions, plan for the future, control or regulate emotions, or perform effectively on other cognitive tasks.

One question that has been posed is why extroverts tend to be happier than introverts. The two types of explanations that attempt to account for this difference are instrumental theories and temperamental theories. The instrumental theory suggests that extraverts end up making choices that place them in more positive situations and they also react more strongly than introverts to positive situations. The temperamental theory suggests that extroverts have a disposition that generally leads them to experience a higher degree of positive affect. In their study of extraversion, Lucas and Baird found no statistically significant support for the instrumental theory but did, however, find that extraverts generally experience a higher level of positive affect.

Research has been done to uncover some of the mediators that are responsible for the correlation between extraversion and happiness. Self-esteem and self-efficacy are two such mediators.

Self-efficacy is one's belief about abilities to perform up to personal standards, the ability to produce desired results, and the feeling of having some ability to make important life decisions. Self-efficacy has been found to be related to the personality traits of extraversion and subjective well-being.

Self-efficacy, however, only partially mediates the relationship between extraversion (and neuroticism) and subjective happiness. This implies that there are most likely other factors that mediate the relationship between subjective happiness and personality traits. Self-esteem maybe another similar factor. Individuals with a greater degree of confidence about themselves and their abilities seem to have both higher degrees of subjective well-being and higher levels of extraversion.

Other research has examined the phenomenon of mood maintenance as another possible mediator. Mood maintenance is the ability to maintain one's average level of happiness in the face of an ambiguous situation – meaning a situation that has the potential to engender either positive or negative emotions in different individuals. It has been found to be a stronger force in extroverts. This means that the happiness levels of extraverted individuals are less susceptible to the influence of external events. This finding implies that extraverts' positive moods last longer than those of introverts.

Developmental biological model

Modern conceptions of personality, such as the Temperament and Character Inventory have suggested four basic temperaments that are thought to reflect basic and automatic responses to danger and reward that rely on associative learning. The four temperaments, harm avoidance, reward dependence, novelty-seeking and persistence, are somewhat analogous to ancient conceptions of melancholic, sanguine, choleric, phlegmatic personality types, although the temperaments reflect dimensions rather than distance categories.

The harm avoidance trait has been associated with increased reactivity in insular and amygdala salience networks, as well as reduced 5-HT2 receptor binding peripherally, and reduced GABA concentrations. Novelty seeking has been associated with reduced activity in insular salience networks increased striatal connectivity. Novelty seeking correlates with dopamine synthesis capacity in the striatum and reduced auto receptor availability in the midbrain. Reward dependence has been linked with the oxytocin system, with increased concentration of plasma oxytocin being observed, as well as increased volume in oxytocin-related regions of the hypothalamus. Persistence has been associated with increased striatal-mPFC connectivity, increased activation of ventral striatal-orbitofrontal-anterior cingulate circuits, as well as increased salivary amylase levels indicative of increased noradrenergic tone.

Environmental influences

It has been shown that personality traits are more malleable by environmental influences than researchers originally believed. Personality differences predict the occurrence of life experiences.

One study has shown how the home environment, specifically the types of parents a person has, can affect and shape their personality. Mary Ainsworth's strange situation experiment showcased how babies reacted to having their mother leave them alone in a room with a stranger. The different styles of attachment, labeled by Ainsworth, were Secure, Ambivalent, avoidant, and disorganized. Children who were securely attached tend to be more trusting, sociable, and are confident in their day-to-day life. Children who were disorganized were reported to have higher levels of anxiety, anger, and risk-taking behavior.

Judith Rich Harris's group socialization theory postulates that an individual's peer groups, rather than parental figures, are the primary influence of personality and behavior in adulthood. Intra- and intergroup processes, not dyadic relationships such as parent-child relationships, are responsible for the transmission of culture and for environmental modification of children's personality characteristics. Thus, this theory points at the peer group representing the environmental influence on a child's personality rather than the parental style or home environment.

Tessuya Kawamoto's Personality Change from Life Experiences: Moderation Effect of Attachment Security talked about some significant laboratory tests. The study mainly focused on the effects of life experiences on change in personality and life experiences. The assessments suggested that "the accumulation of small daily experiences may work for the personality development of university students and that environmental influences may vary by individual susceptibility to experiences, like attachment security".

Some studies suggest that a shared family environment between siblings has less influence on personality than individual experiences of each child. Identical twins have similar personalities largely because they share the same genetic makeup rather than their shared environment.

Cross-cultural studies

There has been some recent debate over the subject of studying personality in a different culture. Some people think that personality comes entirely from culture and therefore there can be no meaningful study in cross-culture study. On the other hand, many believe that some elements are shared by all cultures and an effort is being made to demonstrate the cross-cultural applicability of "the Big Five".

Cross-cultural assessment depends on the universality of personality traits, which is whether there are common traits among humans regardless of culture or other factors. If there is a common foundation of personality, then it can be studied on the basis of human traits rather than within certain cultures. This can be measured by comparing whether assessment tools are measuring similar constructs across countries or cultures. Two approaches to researching personality are looking at emic and etic traits. Emic traits are constructs unique to each culture, which are determined by local customs, thoughts, beliefs, and characteristics. Etic traits are considered universal constructs, which establish traits that are evident across cultures that represent a biological basis of human personality. If personality traits are unique to the individual culture, then different traits should be apparent in different cultures. However, the idea that personality traits are universal across cultures is supported by establishing the Five-Factor Model of personality across multiple translations of the NEO-PI-R, which is one of the most widely used personality measures. When administering the NEO-PI-R to 7,134 people across six languages, the results show a similar pattern of the same five underlying constructs that are found in the American factor structure.

Similar results were found using the Big Five Inventory (BFI), as it was administered in 56 nations across 28 languages. The five factors continued to be supported both conceptually and statistically across major regions of the world, suggesting that these underlying factors are common across cultures. There are some differences across culture, but they may be a consequence of using a lexical approach to study personality structures, as language has limitations in translation and different cultures have unique words to describe emotion or situations. Differences across cultures could be due to real cultural differences, but they could also be consequences of poor translations, biased sampling, or differences in response styles across cultures. Examining personality questionnaires developed within a culture can also be useful evidence for the universality of traits across cultures, as the same underlying factors can still be found. Results from several European and Asian studies have found overlapping dimensions with the Five-Factor Model as well as additional culture-unique dimensions. Finding similar factors across cultures provides support for the universality of personality trait structure, but more research is necessary to gain stronger support.

Historical development of concept

The modern sense of individual personality is a result of the shifts in culture originating in the Renaissance, an essential element in modernity. In contrast, the Medieval European's sense of self was linked to a network of social roles: "the household, the Kinship network, the guild, the corporation – these were the building blocks of personhood". Stephen Greenblatt observes, in recounting the recovery (1417) and career of Lucretius' poem De rerum natura: "at the core of the poem lay key principles of a modern understanding of the world." "Dependent on the family, the individual alone was nothing," Jacques Gélis observes. "The characteristic mark of the modern man has two parts: one internal, the other external; one dealing with his environment, the other with his attitudes, values, and feelings." Rather than being linked to a network of social roles, the modern man is largely influenced by the environmental factors such as: "urbanization, education, mass communication, industrialization, and politicization." In 2006, for example, scientists reported a relationship between personality and political views as follows: "Preschool children who 20 years later were relatively liberal were characterized as: developing close relationships, self-reliant, energetic, somewhat dominating, relatively under-controlled, and resilient. Preschool children subsequently relatively conservative at age 23 were described as: feeling easily victimized, easily offended, indecisive, fearful, rigid, inhibited, and relatively over-controlled and vulnerable."

Temperament and philosophy

William James (1842–1910)

William James (1842–1910) argued that temperament explains a great deal of the controversies in the history of philosophy by arguing that it is a very influential premise in the arguments of philosophers. Despite seeking only impersonal reasons for their conclusions, James argued, the temperament of philosophers influenced their philosophy. Temperament thus conceived is tantamount to a bias. Such bias, James explained, was a consequence of the trust philosophers place in their own temperament. James thought the significance of his observation lay on the premise that in philosophy an objective measure of success is whether philosophy is peculiar to its philosopher or not, and whether a philosopher is dissatisfied with any other way of seeing things or not.

Mental make-up

James argued that temperament may be the basis of several divisions in academia, but focused on philosophy in his 1907 lectures on Pragmatism. In fact, James' lecture of 1907 fashioned a sort of trait theory of the empiricist and rationalist camps of philosophy. As in most modern trait theories, the traits of each camp are described by James as distinct and opposite, and maybe possessed in different proportions on a continuum, and thus characterize the personality of philosophers of each camp. The "mental make-up" (i.e. personality) of rationalist philosophers is described as "tender-minded" and "going by "principles", and that of empiricist philosophers is described as "tough-minded" and "going by "facts." James distinguishes each not only in terms of the philosophical claims they made in 1907, but by arguing that such claims are made primarily on the basis of temperament. Furthermore, such categorization was only incidental to James' purpose of explaining his pragmatist philosophy and is not exhaustive.

Empiricists and rationalists

John Locke (1632–1704)

According to James, the temperament of rationalist philosophers differed fundamentally from the temperament of empiricist philosophers of his day. The tendency of rationalist philosophers toward refinement and superficiality never satisfied an empiricist temper of mind. Rationalism leads to the creation of closed systems, and such optimism is considered shallow by the fact-loving mind, for whom perfection is far off. Rationalism is regarded as pretension, and a temperament most inclined to abstraction.

Empiricists, on the other hand, stick with the external senses rather than logic. British empiricist John Locke's (1632–1704) explanation of personal identity provides an example of what James referred to. Locke explains the identity of a person, i.e. personality, on the basis of a precise definition of identity, by which the meaning of identity differs according to what it is being applied to. The identity of a person is quite distinct from the identity of a man, woman, or substance according to Locke. Locke concludes that consciousness is personality because it "always accompanies thinking, it is that which makes everyone to be what he calls self," and remains constant in different places at different times.

Benedictus Spinoza (1632–1677)

Rationalists conceived of the identity of persons differently than empiricists such as Locke who distinguished identity of substance, person, and life. According to Locke, Rene Descartes (1596–1650) agreed only insofar as he did not argue that one immaterial spirit is the basis of the person "for fear of making brutes thinking things too." According to James, Locke tolerated arguments that a soul was behind the consciousness of any person. However, Locke's successor David Hume (1711–1776), and empirical psychologists after him denied the soul except for being a term to describe the cohesion of inner lives. However, some research suggests Hume excluded personal identity from his opus An Inquiry Concerning Human Understanding because he thought his argument was sufficient but not compelling. Descartes himself distinguished active and passive faculties of mind, each contributing to thinking and consciousness in different ways. The passive faculty, Descartes argued, simply receives, whereas the active faculty produces and forms ideas, but does not presuppose thought, and thus cannot be within the thinking thing. The active faculty mustn't be within self because ideas are produced without any awareness of them, and are sometimes produced against one's will.

Rationalist philosopher Benedictus Spinoza (1632–1677) argued that ideas are the first element constituting the human mind, but existed only for actually existing things. In other words, ideas of non-existent things are without meaning for Spinoza, because an idea of a non-existent thing cannot exist. Further, Spinoza's rationalism argued that the mind does not know itself, except insofar as it perceives the "ideas of the modifications of body", in describing its external perceptions, or perceptions from without. On the contrary, from within, Spinoza argued, perceptions connect various ideas clearly and distinctly. The mind is not the free cause of its actions for Spinoza. Spinoza equates the will with the understanding and explains the common distinction of these things as being two different things as an error which results from the individual's misunderstanding of the nature of thinking.

Biology

The biological basis of personality is the theory that anatomical structures located in the brain contribute to personality traits. This stems from neuropsychology, which studies how the structure of the brain relates to various psychological processes and behaviors. For instance, in human beings, the frontal lobes are responsible for foresight and anticipation, and the occipital lobes are responsible for processing visual information. In addition, certain physiological functions such as hormone secretion also affect personality. For example, the hormone testosterone is important for sociability, affectivity, aggressiveness, and sexuality. Additionally, studies show that the expression of a personality trait depends on the volume of the brain cortex it is associated with.

Personology

Personology confers a multidimensional, complex, and comprehensive approach to personality. According to Henry A. Murray, personology is:

The branch of psychology which concerns itself with the study of human lives and the factors that influence their course which investigates individual differences and types of personality ... the science of men, taken as gross units ... encompassing "psychoanalysis" (Freud), "analytical psychology" (Jung), "individual psychology" (Adler) and other terms that stand for methods of inquiry or doctrines rather than realms of knowledge.

From a holistic perspective, personology studies personality as a whole, as a system, but at the same time through all its components, levels, and spheres.

Growth hormone

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Growth_hormone

 

Growth hormone 1 (pituitary)
Growth hormone

Growth hormone (GH) or somatotropin, also known as human growth hormone (hGH or HGH) in its human form, is a peptide hormone that stimulates growth, cell reproduction, and cell regeneration in humans and other animals. It is thus important in human development. GH also stimulates production of IGF-1 and increases the concentration of glucose and free fatty acids. It is a type of mitogen which is specific only to the receptors on certain types of cells. GH is a 191-amino acid, single-chain polypeptide that is synthesized, stored and secreted by somatotropic cells within the lateral wings of the anterior pituitary gland.

A recombinant form of HGH called somatropin (INN) is used as a prescription drug to treat children's growth disorders and adult growth hormone deficiency. In the United States, it is only available legally from pharmacies by prescription from a licensed health care provider. In recent years in the United States, some health care providers are prescribing growth hormone in the elderly to increase vitality. While legal, the efficacy and safety of this use for HGH has not been tested in a clinical trial. Many of the functions of HGH remain unknown.

In its role as an anabolic agent, HGH has been used by competitors in sports since at least 1982, and has been banned by the IOC and NCAA. Traditional urine analysis does not detect doping with HGH, so the ban was not enforced until the early 2000s, when blood tests that could distinguish between natural and artificial HGH were starting to be developed. Blood tests conducted by WADA at the 2004 Olympic Games in Athens, Greece, targeted primarily HGH. Use of the drug for performance enhancement is not currently approved by the FDA.

GH has been studied for use in raising livestock more efficiently in industrial agriculture and several efforts have been made to obtain governmental approval to use GH in livestock production. These uses have been controversial. In the United States, the only FDA-approved use of GH for livestock is the use of a cow-specific form of GH called bovine somatotropin for increasing milk production in dairy cows. Retailers are permitted to label containers of milk as produced with or without bovine somatotropin.

Nomenclature

The names somatotropin (STH) or somatotropic hormone refer to the growth hormone produced naturally in animals and extracted from carcasses. Hormone extracted from human cadavers is abbreviated hGH. The main growth hormone produced by recombinant DNA technology has the approved generic name (INN) somatropin and the brand name Humatrope, and is properly abbreviated rhGH in the scientific literature. Since its introduction in 1992 Humatrope has been a banned sports doping agent, and in this context is referred to as HGH.

The term growth hormone has been incorrectly applied to refer to anabolic sex hormones in the European beef hormone controversy, which initially restricts the use of estradiol, progesterone, testosterone, zeranol, melengestrol acetate and trenbolone acetate.

Biology

Gene

Main articles: Growth hormone 1 and Growth hormone 2

Genes for human growth hormone, known as growth hormone 1 (somatotropin; pituitary growth hormone) and growth hormone 2 (placental growth hormone; growth hormone variant), are localized in the q22-24 region of chromosome 17 and are closely related to human chorionic somatomammotropin (also known as placental lactogen) genes. GH, human chorionic somatomammotropin, and prolactin belong to a group of homologous hormones with growth-promoting and lactogenic activity.

Structure

The major isoform of the human growth hormone is a protein of 191 amino acids and a molecular weight of 22,124 daltons. The structure includes four helices necessary for functional interaction with the GH receptor. It appears that, in structure, GH is evolutionarily homologous to prolactin and chorionic somatomammotropin. Despite marked structural similarities between growth hormone from different species, only human and Old World monkey growth hormones have significant effects on the human growth hormone receptor.

Several molecular isoforms of GH exist in the pituitary gland and are released to blood. In particular, a variant of approximately 20 kDa originated by an alternative splicing is present in a rather constant 1:9 ratio, while recently an additional variant of ~ 23-24 kDa has also been reported in post-exercise states at higher proportions. This variant has not been identified, but it has been suggested to coincide with a 22 kDa glycosylated variant of 23 kDa identified in the pituitary gland. Furthermore, these variants circulate partially bound to a protein (growth hormone-binding protein, GHBP), which is the truncated part of the growth hormone receptor, and an acid-labile subunit (ALS).

Regulation

See also: Hypothalamic–pituitary–somatotropic axis

Secretion of growth hormone (GH) in the pituitary is regulated by the neurosecretory nuclei of the hypothalamus. These cells release the peptides growth hormone-releasing hormone (GHRH or somatocrinin) and growth hormone-inhibiting hormone (GHIH or somatostatin) into the hypophyseal portal venous blood surrounding the pituitary. GH release in the pituitary is primarily determined by the balance of these two peptides, which in turn is affected by many physiological stimulators (e.g., exercise, nutrition, sleep) and inhibitors (e.g., free fatty acids) of GH secretion.

Somatotropic cells in the anterior pituitary gland then synthesize and secrete GH in a pulsatile manner, in response to these stimuli by the hypothalamus. The largest and most predictable of these GH peaks occurs about an hour after onset of sleep with plasma levels of 13 to 72 ng/mL. Maximal secretion of GH may occur within minutes of the onset of slow-wave (SW) sleep (stage III or IV). Otherwise there is wide variation between days and individuals. Nearly fifty percent of GH secretion occurs during the third and fourth NREM sleep stages. Surges of secretion during the day occur at 3- to 5-hour intervals. The plasma concentration of GH during these peaks may range from 5 to even 45 ng/mL. Between the peaks, basal GH levels are low, usually less than 5 ng/mL for most of the day and night. Additional analysis of the pulsatile profile of GH described in all cases less than 1 ng/ml for basal levels while maximum peaks were situated around 10-20 ng/mL.

A number of factors are known to affect GH secretion, such as age, sex, diet, exercise, stress, and other hormones. Young adolescents secrete GH at the rate of about 700 μg/day, while healthy adults secrete GH at the rate of about 400 μg/day. Sleep deprivation generally suppresses GH release, particularly after early adulthood.

Stimulators of growth hormone (GH) secretion include:

• Peptide hormones
  • GHRH (somatocrinin) through binding to the growth hormone-releasing hormone receptor (GHRHR)
  • Ghrelin through binding to growth hormone secretagogue receptors (GHSR)
• Sex hormones
  • Increased androgen secretion during puberty (in males from testes and in females from adrenal cortex)
  • Testosterone and DHEA
  • Estrogen
• Clonidine and L-DOPA by stimulating GHRH release
• α4β2 nicotinic agonists, including nicotine, which also act synergistically with clonidine
• Hypoglycemia, arginine and propranolol by inhibiting somatostatin release
• Deep sleep
• Glucagon
• Niacin as nicotinic acid (vitamin B₃)
• Fasting
• Vigorous exercise

Inhibitors of GH secretion include:

• GHIH (somatostatin) from the periventricular nucleus 
• circulating concentrations of GH and IGF-1 (negative feedback on the pituitary and hypothalamus)
• Hyperglycemia
• Glucocorticoids
• Dihydrotestosterone
• Insulin
• Phenothiazines

In addition to control by endogenous and stimulus processes, a number of foreign compounds (xenobiotics such as drugs and endocrine disruptors) are known to influence GH secretion and function.

Function

Main pathways in endocrine regulation of growth

Effects of growth hormone on the tissues of the body can generally be described as anabolic (building up). Like most other peptide hormones, GH acts by interacting with a specific receptor on the surface of cells.

Increased height during childhood is the most widely known effect of GH. Height appears to be stimulated by at least two mechanisms:

1. Because polypeptide hormones are not fat-soluble, they cannot penetrate cell membranes. Thus, GH exerts some of its effects by binding to receptors on target cells, where it activates the MAPK/ERK pathway. Through this mechanism GH directly stimulates division and multiplication of chondrocytes of cartilage.
2. GH also stimulates, through the JAK-STAT signaling pathway, the production of insulin-like growth factor 1 (IGF-1, formerly known as somatomedin C), a hormone homologous to proinsulin. The liver is a major target organ of GH for this process and is the principal site of IGF-1 production. IGF-1 has growth-stimulating effects on a wide variety of tissues. Additional IGF-1 is generated within target tissues, making it what appears to be both an endocrine and an autocrine/paracrine hormone. IGF-1 also has stimulatory effects on osteoblast and chondrocyte activity to promote bone growth.

In addition to increasing height in children and adolescents, growth hormone has many other effects on the body:

• Increases calcium retention, and strengthens and increases the mineralization of bone
• Increases muscle mass through sarcomere hypertrophy
• Promotes lipolysis
• Increases protein synthesis
• Stimulates the growth of all internal organs excluding the brain
• Plays a role in homeostasis
• Reduces liver uptake of glucose
• Promotes gluconeogenesis in the liver
• Contributes to the maintenance and function of pancreatic islets
• Stimulates the immune system
• Increases deiodination of T4 to T3

Biochemistry

GH has a short biological half-life of about 10 to 20 minutes.

Clinical significance

Excess

The most common disease of GH excess is a pituitary tumor composed of somatotroph cells of the anterior pituitary. These somatotroph adenomas are benign and grow slowly, gradually producing more and more GH. For years, the principal clinical problems are those of GH excess. Eventually, the adenoma may become large enough to cause headaches, impair vision by pressure on the optic nerves, or cause deficiency of other pituitary hormones by displacement.

Prolonged GH excess thickens the bones of the jaw, fingers and toes, resulting in heaviness of the jaw and increased size of digits, referred to as acromegaly. Accompanying problems can include sweating, pressure on nerves (e.g. carpal tunnel syndrome), muscle weakness, excess sex hormone-binding globulin (SHBG), insulin resistance or even a rare form of type 2 diabetes, and reduced sexual function.

GH-secreting tumors are typically recognized in the fifth decade of life. It is extremely rare for such a tumor to occur in childhood, but, when it does, the excessive GH can cause excessive growth, traditionally referred to as pituitary gigantism.

Surgical removal is the usual treatment for GH-producing tumors. In some circumstances, focused radiation or a GH antagonist such as pegvisomant may be employed to shrink the tumor or block function. Other drugs like octreotide (somatostatin agonist) and bromocriptine (dopamine agonist) can be used to block GH secretion because both somatostatin and dopamine negatively inhibit GHRH-mediated GH release from the anterior pituitary.

Deficiency

Main article: Growth hormone deficiency

The effects of growth hormone (GH) deficiency vary depending on the age at which they occur. Alterations in somatomedin can result in growth hormone deficiency with two known mechanisms; failure of tissues to respond to somatomedin, or failure of the liver to produce somatomedin. Major manifestations of GH deficiency in children are growth failure, the development of a short stature, and delayed sexual maturity. In adults, somatomedin alteration contributes to increased osteoclast activity, resulting in weaker bones that are more prone to pathologic fracture and osteoporosis. However, deficiency is rare in adults, with the most common cause being a pituitary adenoma. Other adult causes include a continuation of a childhood problem, other structural lesions or trauma, and very rarely idiopathic GHD.

Adults with GHD "tend to have a relative increase in fat mass and a relative decrease in muscle mass and, in many instances, decreased energy and quality of life".

Diagnosis of GH deficiency involves a multiple-step diagnostic process, usually culminating in GH stimulation tests to see if the patient's pituitary gland will release a pulse of GH when provoked by various stimuli.

Psychological effects

Quality of life

Several studies, primarily involving patients with GH deficiency, have suggested a crucial role of GH in both mental and emotional well-being and maintaining a high energy level. Adults with GH deficiency often have higher rates of depression than those without. While GH replacement therapy has been proposed to treat depression as a result of GH deficiency, the long-term effects of such therapy are unknown.

Cognitive function

GH has also been studied in the context of cognitive function, including learning and memory. GH in humans appears to improve cognitive function and may be useful in the treatment of patients with cognitive impairment that is a result of GH deficiency.

Medical uses

Main article: Growth hormone therapy

Replacement therapy

GH is used as replacement therapy in adults with GH deficiency of either childhood-onset or adult-onset (usually as a result of an acquired pituitary tumor). In these patients, benefits have variably included reduced fat mass, increased lean mass, increased bone density, improved lipid profile, reduced cardiovascular risk factors, and improved psychosocial well-being. Long acting growth hormone (LAGH) analogues are now available for treating growth hormone deficiency both in children and adults. These are once weekly injections as compared to conventional growth hormone which has to be taken as daily injections. LAGH injection 4 times a month has been found to be as safe and effective as daily growth hormone injections.

Other approved uses

GH can be used to treat conditions that produce short stature but are not related to deficiencies in GH. However, results are not as dramatic when compared to short stature that is solely attributable to deficiency of GH. Examples of other causes of shortness often treated with GH are Turner syndrome, Growth failure secondary to chronic kidney disease in children, Prader–Willi syndrome, intrauterine growth restriction, and severe idiopathic short stature. Higher ("pharmacologic") doses are required to produce significant acceleration of growth in these conditions, producing blood levels well above normal ("physiologic").

One version of rHGH has also been FDA approved for maintaining muscle mass in wasting due to AIDS.

Off-label use

Main article: HGH controversies

Off-label prescription of HGH is controversial and may be illegal.

Claims for GH as an anti-aging treatment date back to 1990 when the New England Journal of Medicine published a study wherein GH was used to treat 12 men over 60. At the conclusion of the study, all the men showed statistically significant increases in lean body mass and bone mineral density, while the control group did not. The authors of the study noted that these improvements were the opposite of the changes that would normally occur over a 10- to 20-year aging period. Despite the fact the authors at no time claimed that GH had reversed the aging process itself, their results were misinterpreted as indicating that GH is an effective anti-aging agent. This has led to organizations such as the controversial American Academy of Anti-Aging Medicine promoting the use of this hormone as an "anti-aging agent".

A Stanford University School of Medicine meta-analysis of clinical studies on the subject published in early 2007 showed that the application of GH on healthy elderly patients increased muscle by about 2 kg and decreased body fat by the same amount. However, these were the only positive effects from taking GH. No other critical factors were affected, such as bone density, cholesterol levels, lipid measurements, maximal oxygen consumption, or any other factor that would indicate increased fitness. Researchers also did not discover any gain in muscle strength, which led them to believe that GH merely let the body store more water in the muscles rather than increase muscle growth. This would explain the increase in lean body mass.

GH has also been used experimentally to treat multiple sclerosis, to enhance weight loss in obesity, as well as in fibromyalgia, heart failure, Crohn's disease and ulcerative colitis, and burns. GH has also been used experimentally in patients with short bowel syndrome to lessen the requirement for intravenous total parenteral nutrition.

In 1990, the US Congress passed an omnibus crime bill, the Crime Control Act of 1990, that amended the Federal Food, Drug, and Cosmetic Act, that classified anabolic steroids as controlled substances and added a new section that stated that a person who "knowingly distributes, or possesses with intent to distribute, human growth hormone for any use in humans other than the treatment of a disease or other recognized medical condition, where such use has been authorized by the Secretary of Health and Human Services" has committed a felony.

The Drug Enforcement Administration of the US Department of Justice considers off-label prescribing of HGH to be illegal, and to be a key path for illicit distribution of HGH. This section has also been interpreted by some doctors, most notably the authors of a commentary article published in the Journal of the American Medical Association in 2005, as meaning that prescribing HGH off-label may be considered illegal. And some articles in the popular press, such as those criticizing the pharmaceutical industry for marketing drugs for off-label use (with concern of ethics violations) have made strong statements about whether doctors can prescribe HGH off-label: "Unlike other prescription drugs, HGH may be prescribed only for specific uses. U.S. sales are limited by law to treat a rare growth defect in children and a handful of uncommon conditions like short bowel syndrome or Prader-Willi syndrome, a congenital disease that causes reduced muscle tone and a lack of hormones in sex glands." At the same time, anti-aging clinics where doctors prescribe, administer, and sell HGH to people are big business. In a 2012 article in Vanity Fair, when asked how HGH prescriptions far exceed the number of adult patients estimated to have HGH-deficiency, Dragos Roman, who leads a team at the FDA that reviews drugs in endocrinology, said "The F.D.A. doesn't regulate off-label uses of H.G.H. Sometimes it's used appropriately. Sometimes it's not."

Side effects

Injection-site reaction is common. More rarely, patients can experience joint swelling, joint pain, carpal tunnel syndrome, and an increased risk of diabetes. In some cases, the patient can produce an immune response against GH. GH may also be a risk factor for Hodgkin's lymphoma.

One survey of adults that had been treated with replacement cadaver GH (which has not been used anywhere in the world since 1985) during childhood showed a mildly increased incidence of colon cancer and prostate cancer, but linkage with the GH treatment was not established.

Performance enhancement

Main article: Growth hormone in sports

The first description of the use of GH as a doping agent was Dan Duchaine's "Underground Steroid handbook" which emerged from California in 1982; it is not known where and when GH was first used this way.

Athletes in many sports have used human growth hormone in order to attempt to enhance their athletic performance. Some recent studies have not been able to support claims that human growth hormone can improve the athletic performance of professional male athletes. Many athletic societies ban the use of GH and will issue sanctions against athletes who are caught using it. However, because GH is a potent endogenous protein, it is very difficult to detect GH doping. In the United States, GH is legally available only by prescription from a medical doctor.

Dietary supplements

To capitalize on the idea that GH might be useful to combat aging, companies selling dietary supplements have websites selling products linked to GH in the advertising text, with medical-sounding names described as "HGH Releasers". Typical ingredients include amino acids, minerals, vitamins, and/or herbal extracts, the combination of which are described as causing the body to make more GH with corresponding beneficial effects. In the United States, because these products are marketed as dietary supplements, it is illegal for them to contain GH, which is a drug. Also, under United States law, products sold as dietary supplements cannot have claims that the supplement treats or prevents any disease or condition, and the advertising material must contain a statement that the health claims are not approved by the FDA. The FTC and the FDA do enforce the law when they become aware of violations.

Agricultural use

In the United States, it is legal to give a bovine GH to dairy cows to increase milk production, and is legal to use GH in raising cows for beef; see article on Bovine somatotropin, cattle feeding, dairy farming and the beef hormone controversy.

The use of GH in poultry farming is illegal in the United States. Similarly, no chicken meat for sale in Australia is administered hormones.

Several companies have attempted to have a version of GH for use in pigs (porcine somatotropin) approved by the FDA but all applications have been withdrawn.

Drug development history

Main article: Growth hormone treatment § History

Genentech pioneered the use of recombinant human growth hormone for human therapy, which was approved by the FDA in 1985.

Prior to its production by recombinant DNA technology, growth hormone used to treat deficiencies was extracted from the pituitary glands of cadavers. Attempts to create a wholly synthetic HGH failed. Limited supplies of HGH resulted in the restriction of HGH therapy to the treatment of idiopathic short stature. Very limited clinical studies of growth hormone derived from an Old World monkey, the rhesus macaque, were conducted by John C. Beck and colleagues in Montreal, in the late 1950s. The study published in 1957, which was conducted on "a 13-year-old male with well-documented hypopituitarism secondary to a crainiophyaryngioma," found that: "Human and monkey growth hormone resulted in a significant enhancement of nitrogen storage ... (and) there was a retention of potassium, phosphorus, calcium, and sodium. ... There was a gain in body weight during both periods. ... There was a significant increase in urinary excretion of aldosterone during both periods of administration of growth hormone. This was most marked with the human growth hormone. ... Impairment of the glucose tolerance curve was evident after 10 days of administration of the human growth hormone. No change in glucose tolerance was demonstrable on the fifth day of administration of monkey growth hormone." The other study, published in 1958, was conducted on six people: the same subject as the Science paper; an 18-year-old male with statural and sexual retardation and a skeletal age of between 13 and 14 years; a 15-year-old female with well-documented hypopituitarism secondary to a craniopharyngioma; a 53-year-old female with carcinoma of the breast and widespread skeletal metastases; a 68-year-old female with advanced postmenopausal osteoporosis; and a healthy 24-year-old medical student without any clinical or laboratory evidence of systemic disease.

In 1985, unusual cases of Creutzfeldt–Jakob disease were found in individuals that had received cadaver-derived HGH ten to fifteen years previously. Based on the assumption that infectious prions causing the disease were transferred along with the cadaver-derived HGH, cadaver-derived HGH was removed from the market.

In 1985, biosynthetic human growth hormone replaced pituitary-derived human growth hormone for therapeutic use in the U.S. and elsewhere.

As of 2005, recombinant growth hormones available in the United States (and their manufacturers) included Nutropin (Genentech), Humatrope (Lilly), Genotropin (Pfizer), Norditropin (Novo), and Saizen (Merck Serono). In 2006, the U.S. Food and Drug Administration (FDA) approved a version of rHGH called Omnitrope (Sandoz). A sustained-release form of growth hormone, Nutropin Depot (Genentech and Alkermes) was approved by the FDA in 1999, allowing for fewer injections (every 2 or 4 weeks instead of daily); however, the product was discontinued by Genentech/Alkermes in 2004 for financial reasons (Nutropin Depot required significantly more resources to produce than the rest of the Nutropin line).

Polyketide synthase

From Wikipedia, the free encyclopedia

Polyketide synthases (PKSs) are a family of multi-domain enzymes or enzyme complexes that produce polyketides, a large class of secondary metabolites, in bacteria, fungi, plants, and a few animal lineages. The biosyntheses of polyketides share striking similarities with fatty acid biosynthesis.

The PKS genes for a certain polyketide are usually organized in one operon or in gene clusters. Type I and type II PKSs form either large modular protein complexes or dissociable molecular assemblies; type III PKSs exist as smaller homodimeric proteins.

Classification

Reaction mechanisms of type I, II and III PKSs. Decarboxylation of malonyl unit followed by thio-Claisen condensation. a) (cis-AT) type I PKS with acyl carrier protein (ACP), keto synthase (KS) and acyl transferase (AT) domains covalently bound to another . b) Type II PKS with KSα-KSβ heterodimer and ACP as separate proteins. c) ACP-independent Type III PKS.

PKSs can be classified into three types:

• Type I PKSs are large, complex protein structures with multiple modules which in turn consist of several domains that are usually covalently connected to each other and fulfill different catalytic steps. The minimal composition of a type I PKS module consists of an acyltransferase (AT) domain, which is responsible for choosing the building block to be used, a keto synthase (KS) domain, which catalyzes the C-C bond formation and an acyl carrier protein (ACP) domain, also known as thiolation domain. The latter contains a conserved Ser residue, post-translationally modified with a phosphopantetheine at the end of which the polyketide chain is covalently bound during biosynthesis as a thioester. Moreover, multiple other optional domains can also exist within a module like ketoreductase or dehydratase domains which alter the default 1,3-dicarbonyl functionality of the installed ketide by sequential reduction to an alcohol and double bond, respectively. These domains work together like an assembly line. This type of type I PKSs is also referred to as cis-acyltransferase polyketide synthases (cis-AT PKSs). In contrast to that, so called trans-AT PKSs evolved independently and lack AT domains in their modules. This activity is provided by free-standing AT domains instead. Moreover, they often contain uncommon domains with unique catalytic activities.
• Type II PKSs behave very similarly to type I PKS but with one key difference: Instead of one large megaenzyme, type II PKSs are separate, monofunctional enzymes. The smallest possible type II PKS consists of an ACP, as well as two heterodimeric KS units (KSα, which catalyzes the C-C bond formation and KSβ, also known as 'chain length factor' — CLF, since it can determine the carbon chain length), which fulfill a similar function as the AT, KS and ACP domains in type I PKSs, even though type II PKSs are lacking a separate AT domain. Additionally, type II PKSs often work iteratively where multiple chain elongation steps are carried out by the same enzyme, similar to type III PKSs.
• Type III PKSs are small homodimers of 40 kDa proteins that combine all the activities from the essential type I and II PKS domains. However, in contrast to type I and II PKSs they do not require an ACP-bound substrate. Instead, they can use a free acyl-CoA substrate for chain elongation. Moreover, type III PKSs contain a Cys-His-Asn catalytic triad in their active center, with the cysteine residue acting as the attacking nucleophile, whereas type I and II PKSs are characterized by a Cys-His-His catalytic triad. Typical products of type III PKSs include phenolic lipids like alkylresorcinols

• In addition to these three types of PKSs, they can be further classified as iterative or noniterative. Iterative Type I PKSs reuse domains in a cyclic fashion. Other classifications include the degree of reduction performed during the synthesis of the growing polyketide chain.
  • NR-PKSs — non-reducing PKSs, the products of which are true polyketides
  • PR-PKSs — partially reducing PKSs
  • FR-PKSs — fully reducing PKSs, the products of which are fatty acid derivatives

Modules and domains

Biosynthesis of the doxorubicin precursor, є-rhodomycinone. The polyketide synthase reactions are shown on top.

Each type I polyketide-synthase module consists of several domains with defined functions, separated by short spacer regions. The order of modules and domains of a complete polyketide-synthase is as follows (in the order N-terminus to C-terminus):

• Starting or loading module: AT-ACP-
• Elongation or extending modules: -KS-AT-[DH-ER-KR]-ACP-
• Termination or releasing domain: -TE

Domains:

• AT: Acyltransferase
• ACP: Acyl carrier protein with an SH group on the cofactor, a serine-attached 4'-phosphopantetheine
• KS: Keto-synthase with an SH group on a cysteine side-chain
• KR: Ketoreductase
• DH: Dehydratase
• ER: Enoylreductase
• MT: Methyltransferase O- or C- (α or β)
• SH: PLP-dependent cysteine lyase
• TE: Thioesterase

The polyketide chain and the starter groups are bound with their carboxy functional group to the SH groups of the ACP and the KS domain through a thioester linkage: R-C(=O)OH + HS-protein <=> R-C(=O)S-protein + H₂O.

The ACP carrier domains are similar to the PCP carrier domains of nonribosomal peptide synthetases, and some proteins combine both types of modules.

Stages

The growing chain is handed over from one thiol group to the next by trans-acylations and is released at the end by hydrolysis or by cyclization (alcoholysis or aminolysis).

Starting stage:

• The starter group, usually acetyl-CoA or its analogues, is loaded onto the ACP domain of the starter module catalyzed by the starter module's AT domain.

Elongation stages:

• The polyketide chain is handed over from the ACP domain of the previous module to the KS domain of the current module, catalyzed by the KS domain.
• The elongation group, usually malonyl-CoA or methylmalonyl-CoA, is loaded onto the current ACP domain catalyzed by the current AT domain.
• The ACP-bound elongation group reacts in a Claisen condensation with the KS-bound polyketide chain under CO₂ evolution, leaving a free KS domain and an ACP-bound elongated polyketide chain. The reaction takes place at the KS_n-bound end of the chain, so that the chain moves out one position and the elongation group becomes the new bound group.
• Optionally, the fragment of the polyketide chain can be altered stepwise by additional domains. The KR (keto-reductase) domain reduces the β-keto group to a β-hydroxy group, the DH (dehydratase) domain splits off H₂O, resulting in the α-β-unsaturated alkene, and the ER (enoyl-reductase) domain reduces the α-β-double-bond to a single-bond. It is important to note that these modification domains actually affect the previous addition to the chain (i.e. the group added in the previous module), not the component recruited to the ACP domain of the module containing the modification domain.
• This cycle is repeated for each elongation module.

Termination stage:

• The TE domain hydrolyzes the completed polyketide chain from the ACP-domain of the previous module.

Pharmacological relevance

Polyketide synthases are an important source of naturally occurring small molecules used for chemotherapy. For example, many of the commonly used antibiotics, such as tetracycline and macrolides, are produced by polyketide synthases. Other industrially important polyketides are sirolimus (immunosuppressant), erythromycin (antibiotic), lovastatin (anticholesterol drug), and epothilone B (anticancer drug).

Polyketides are a large family of natural products widely used as drugs, pesticides, herbicides, and biological probes.

There are antifungal and antibacterial polyketide compounds, namely ophiocordin and ophiosetin.

And are researched for the synthesis of biofuels and industrial chemicals.

Ecological significance

Only about 1% of all known molecules are natural products, yet it has been recognized that almost two thirds of all drugs currently in use are at least in part derived from a natural source. This bias is commonly explained with the argument that natural products have co-evolved in the environment for long time periods and have therefore been pre-selected for active structures. Polyketide synthase products include lipids with antibiotic, antifungal, antitumor, and predator-defense properties; however, many of the polyketide synthase pathways that bacteria, fungi and plants commonly use have not yet been characterized. Methods for the detection of novel polyketide synthase pathways in the environment have therefore been developed. Molecular evidence supports the notion that many novel polyketides remain to be discovered from bacterial sources.

Pythagorean theorem

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Pythagorean_theorem 

 

Pythagorean theorem

Type	Theorem
Field	Euclidean geometry
Statement	The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Symbolic statement	$a^2+b^2=c^2$
Generalizations	Law of cosines Solid geometry Non-Euclidean geometry Differential geometry
Consequences	Pythagorean triple Reciprocal Pythagorean theorem Complex number Euclidean distance Pythagorean trigonometric identity

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:

${\displaystyle a^2+b^2=c^2.}$

The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.

When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.

The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound.

Proofs using constructed squares

Rearrangement proof of the Pythagorean theorem.
(The area of the white space remains constant throughout the translation rearrangement of the triangles. At all moments in time, the area is always c². And likewise, at all moments in time, the area is always a²+b².)

Rearrangement proofs

In one rearrangement proof, two squares are used whose sides have a measure of $a+b$ and which contain four right triangles whose sides are a, b and c, with the hypotenuse being c. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length c. Each outer square has an area of ${\displaystyle (a+b{)}^2}$ as well as ${\displaystyle 2ab+c^2}$, with $2ab$ representing the total area of the four triangles. Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of length a and b. These rectangles in their new position have now delineated two new squares, one having side length a is formed in the bottom-left corner, and another square of side length b formed in the top-right corner. In this new position, this left side now has a square of area ${\displaystyle (a+b{)}^2}$ as well as ${\displaystyle 2ab+a^2+b^2}$. Since both squares have the area of ${\displaystyle (a+b{)}^2}$ it follows that the other measure of the square area also equal each other such that ${\displaystyle 2ab+c^2}$ = ${\displaystyle 2ab+a^2+b^2}$. With the area of the four triangles removed from both side of the equation what remains is ${\displaystyle a^2+b^2=c^2.}$ 

In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. In this way they also form two boxes, this time in consecutive corners, with areas ${\displaystyle a^2}$ and ${\displaystyle b^2}$which will again lead to a second square of with the area ${\displaystyle 2ab+a^2+b^2}$.

English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, and mentions the proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him." Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues. 

Algebraic proofs

Diagram of the two algebraic proofs

The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side c, as shown in the lower part of the diagram. This results in a larger square, with side a + b and area (a + b)². The four triangles and the square side c must have the same area as the larger square,

${\displaystyle (b+a{)}^2=c^2+4\frac{ab}{2}=c^2+2ab,}$

giving

${\displaystyle c^2=(b+a{)}^2-2ab=b^2+2ab+a^2-2ab=a^2+b^2.}$

A similar proof uses four copies of a right triangle with sides a, b and c, arranged inside a square with side c as in the top half of the diagram. The triangles are similar with area $\frac{1}{2}ab$, while the small square has side b − a and area (b − a)². The area of the large square is therefore

${\displaystyle (b-a{)}^2+4\frac{ab}{2}=(b-a{)}^2+2ab=b^2-2ab+a^2+2ab=a^2+b^2.}$

But this is a square with side c and area c², so

${\displaystyle c^2=a^2+b^2.}$

Other proofs of the theorem

This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs.

Proof using similar triangles

In this section, and as usual in geometry, a "word" of two capital letters, such as AB denotes the length of the line segment defined by the points labeled with the letters, and not a multiplication. So, AB² denotes the square of the length AB and not the product ${\displaystyle A\times B^2.}$

Proof using similar triangles

This proof is based on the proportionality of the sides of three similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle, ACH, is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the triangle postulate: The sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:

${\displaystyle \frac{BC}{AB}=\frac{BH}{BC}\text{ and }\frac{AC}{AB}=\frac{AH}{AC}.}$

The first result equates the cosines of the angles θ, whereas the second result equates their sines.

These ratios can be written as

${\displaystyle B{C}^2=AB\times BH\text{ and }A{C}^2=AB\times AH.}$

Summing these two equalities results in

${\displaystyle B{C}^2+A{C}^2=AB\times BH+AB\times AH=AB(AH+BH)=A{B}^2,}$

which, after simplification, demonstrates the Pythagorean theorem:

${\displaystyle B{C}^2+A{C}^2=A{B}^2.}$

The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements, and that the theory of proportions needed further development at that time.

Einstein's proof by dissection without rearrangement

Albert Einstein gave a proof by dissection in which the pieces do not need to be moved. Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and two similar shapes that each include one of two legs instead of the hypotenuse (see Similar figures on the three sides). In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts. Those two parts have the same shape as the original right triangle, and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle. Because the ratio of the area of a right triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well.

Trigonometric proof using Einstein's construction

Right triangle on the hypotenuse dissected into two similar right triangles on the legs, according to Einstein's proof.

Both the proof using similar triangles and Einstein's proof rely on constructing the height to the hypotenuse of the right triangle $\mathrm{△}ABC$. The same construction provides a trigonometric proof of the Pythagorean theorem using the definition of the sine as a ratio inside a right triangle:

${\displaystyle \sin \alpha =\frac{a}{c},}$

${\displaystyle \sin \beta =\frac{b}{c},}$

${\displaystyle c=b\sin \beta +a\sin \alpha =\frac{b^2}{c}+\frac{a^2}{c},}$

and thus

${\displaystyle c^2=a^2+b^2.}$

This proof is essentially the same as the above proof using similar triangles, where some ratios of lengths are replaced by sines.

Euclid's proof

Proof in Euclid's Elements

In outline, here is how the proof in Euclid's Elements proceeds. The large square is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are shown to be congruent, proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow.

Let A, B, C be the vertices of a right triangle, with a right angle at A. Drop a perpendicular from A to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs.

For the formal proof, we require four elementary lemmata:

1. If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (side-angle-side).
2. The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
3. The area of a rectangle is equal to the product of two adjacent sides.
4. The area of a square is equal to the product of two of its sides (follows from 3).

Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square.

Illustration including the new lines

Showing the two congruent triangles of half the area of rectangle BDLK and square BAGF

The proof is as follows:

1. Let ACB be a right-angled triangle with right angle CAB.
2. On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order. The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate.
3. From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively.
4. Join CF and AD, to form the triangles BCF and BDA.
5. Angles CAB and BAG are both right angles; therefore C, A, and G are collinear.
6. Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC, since both are the sum of a right angle and angle ABC.
7. Since AB is equal to FB, BD is equal to BC and angle ABD equals angle FBC, triangle ABD must be congruent to triangle FBC.
8. Since A-K-L is a straight line, parallel to BD, then rectangle BDLK has twice the area of triangle ABD because they share the base BD and have the same altitude BK, i.e., a line normal to their common base, connecting the parallel lines BD and AL. (lemma 2)
9. Since C is collinear with A and G, and this line is parallel to FB, then square BAGF must be twice in area to triangle FBC.
10. Therefore, rectangle BDLK must have the same area as square BAGF = AB².
11. By applying steps 3 to 10 to the other side of the figure, it can be similarly shown that rectangle CKLE must have the same area as square ACIH = AC².
12. Adding these two results, AB² + AC² = BD × BK + KL × KC
13. Since BD = KL, BD × BK + KL × KC = BD(BK + KC) = BD × BC
14. Therefore, AB² + AC² = BC², since CBDE is a square.

This proof, which appears in Euclid's Elements as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares. This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used.

Proofs by dissection and rearrangement

Another by rearrangement is given by the middle animation. A large square is formed with area c², from four identical right triangles with sides a, b and c, fitted around a small central square. Then two rectangles are formed with sides a and b by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas a² and b², which must have the same area as the initial large square.

The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse – or conversely the large square can be divided as shown into pieces that fill the other two. This way of cutting one figure into pieces and rearranging them to get another figure is called dissection. This shows the area of the large square equals that of the two smaller ones.

Animation showing proof by rearrangement of four identical right triangles

Animation showing another proof by rearrangement

Proof using an elaborate rearrangement

Proof by area-preserving shearing

Visual proof of the Pythagorean theorem by area-preserving shearing

As shown in the accompanying animation, area-preserving shear mappings and translations can transform the squares on the sides adjacent to the right-angle onto the square on the hypotenuse, together covering it exactly. Each shear leaves the base and height unchanged, thus leaving the area unchanged too. The translations also leave the area unchanged, as they do not alter the shapes at all. Each square is first sheared into a parallelogram, and then into a rectangle which can be translated onto one section of the square on the hypotenuse.

Other algebraic proofs

A related proof was published by future U.S. President James A. Garfield (then a U.S. Representative) (see diagram). Instead of a square it uses a trapezoid, which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram. The area of the trapezoid can be calculated to be half the area of the square, that is

$\frac{1}{2}(b+a{)}^2.$

The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of $\frac{1}{2}$, which is removed by multiplying by two to give the result.

Proof using differentials

One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus.

The triangle ABC is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y, the side AC of length x and the side AB of length a, as seen in the lower diagram part.

Diagram for differential proof

If x is increased by a small amount dx by extending the side AC slightly to D, then y also increases by dy. These form two sides of a triangle, CDE, which (with E chosen so CE is perpendicular to the hypotenuse) is a right triangle approximately similar to ABC. Therefore, the ratios of their sides must be the same, that is:

${\displaystyle \frac{dy}{dx}=\frac{x}{y}.}$

This can be rewritten as ${\displaystyle ydy=xdx}$ , which is a differential equation that can be solved by direct integration:

${\displaystyle \int ydy=\int xdx,}$

giving

${\displaystyle y^2=x^2+C.}$

The constant can be deduced from x = 0, y = a to give the equation

${\displaystyle y^2=x^2+a^2.}$

This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of dx and dy.

Converse

The converse of the theorem is also true:

Given a triangle with sides of length a, b, and c, if a² + b² = c², then the angle between sides a and b is a right angle.

For any three positive real numbers a, b, and c such that a² + b² = c², there exists a triangle with sides a, b and c as a consequence of the converse of the triangle inequality.

This converse appears in Euclid's Elements (Book I, Proposition 48): "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."

It can be proved using the law of cosines or as follows:

Let ABC be a triangle with side lengths a, b, and c, with a² + b² = c². Construct a second triangle with sides of length a and b containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length c = √a² + b², the same as the hypotenuse of the first triangle. Since both triangles' sides are the same lengths a, b and c, the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths a and b in the original triangle is a right angle.

The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proved without assuming the Pythagorean theorem.

A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let c be chosen to be the longest of the three sides and a + b > c (otherwise there is no triangle according to the triangle inequality). The following statements apply:

• If a² + b² = c², then the triangle is right.
• If a² + b² > c², then the triangle is acute.
• If a² + b² < c², then the triangle is obtuse.

Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:

sgn(α + β − γ) = sgn(a² + b² − c²),

where α is the angle opposite to side a, β is the angle opposite to side b, γ is the angle opposite to side c, and sgn is the sign function.

Consequences and uses of the theorem

Pythagorean triples

Main article: Pythagorean triple

A Pythagorean triple has three positive integers a, b, and c, such that a² + b² = c². In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13).

A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a, b and c is 1).

The following is a list of primitive Pythagorean triples with values less than 100:

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Inverse Pythagorean theorem

Given a right triangle with sides $a,b,c$ and altitude $d$ (a line from the right angle and perpendicular to the hypotenuse $c$). The Pythagorean theorem has,

${\displaystyle a^2+b^2=c^2}$

while the inverse Pythagorean theorem relates the two legs $a,b$ to the altitude $d$,

${\displaystyle \frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{d^2}}$

The equation can be transformed to,

${\displaystyle \frac{1}{(xz{)}^2}+\frac{1}{(yz{)}^2}=\frac{1}{(xy{)}^2}}$

where $x^2+y^2=z^2$ for any non-zero real $x,y,z$. If the ${\displaystyle a,b,d}$ are to be integers, the smallest solution ${\displaystyle a>b>d}$ is then

${\displaystyle \frac{1}{{20}^2}+\frac{1}{{15}^2}=\frac{1}{{12}^2}}$

using the smallest Pythagorean triple ${\displaystyle 3,4,5}$. The reciprocal Pythagorean theorem is a special case of the optic equation

${\displaystyle \frac{1}{p}+\frac{1}{q}=\frac{1}{r}}$

where the denominators are squares and also for a heptagonal triangle whose sides $p,q,r$ are square numbers.

Incommensurable lengths

The spiral of Theodorus: A construction for line segments with lengths whose ratios are the square root of a positive integer

One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable (so the ratio of which is not a rational number) can be constructed using a straightedge and compass. Pythagoras' theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation.

The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras' theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is not a perfect square, it is a realization of a length incommensurable with the unit, such as √2, √3, √5 . For more detail, see Quadratic irrational.

Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit. According to one legend, Hippasus of Metapontum (ca. 470 B.C.) was drowned at sea for making known the existence of the irrational or incommensurable.

Complex numbers

The absolute value of a complex number z is the distance r from z to the origin.

For any complex number

$z=x+iy,$

the absolute value or modulus is given by

${\displaystyle r=|z|=\sqrt{x^2+y^2}.}$

So the three quantities, r, x and y are related by the Pythagorean equation,

${\displaystyle r^2=x^2+y^2.}$

Note that r is defined to be a positive number or zero but x and y can be negative as well as positive. Geometrically r is the distance of the z from zero or the origin O in the complex plane.

This can be generalised to find the distance between two points, z₁ and z₂ say. The required distance is given by

${\displaystyle |z_1-z_2|=\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2},}$

so again they are related by a version of the Pythagorean equation,

${\displaystyle |z_1-z_2{|}^2=(x_1-x_2{)}^2+(y_1-y_2{)}^2.}$

Euclidean distance

Main article: Euclidean distance

The distance formula in Cartesian coordinates is derived from the Pythagorean theorem. If (x₁, y₁) and (x₂, y₂) are points in the plane, then the distance between them, also called the Euclidean distance, is given by

$\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2}.$

More generally, in Euclidean n-space, the Euclidean distance between two points, $A=(a_1,a_2,\dots ,a_n)$ and $B=(b_1,b_2,\dots ,b_n)$, is defined, by generalization of the Pythagorean theorem, as:

$\sqrt{(a_1-b_1{)}^2+(a_2-b_2{)}^2+\cdots +(a_n-b_n{)}^2}=\sqrt{\sum _{i=1}^n(a_i-b_i{)}^2}.$

If instead of Euclidean distance, the square of this value (the squared Euclidean distance, or SED) is used, the resulting equation avoids square roots and is simply a sum of the SED of the coordinates:

${\displaystyle (a_1-b_1{)}^2+(a_2-b_2{)}^2+\cdots +(a_n-b_n{)}^2=\sum _{i=1}^n(a_i-b_i{)}^2.}$

The squared form is a smooth, convex function of both points, and is widely used in optimization theory and statistics, forming the basis of least squares.

Euclidean distance in other coordinate systems

If Cartesian coordinates are not used, for example, if polar coordinates are used in two dimensions or, in more general terms, if curvilinear coordinates are used, the formulas expressing the Euclidean distance are more complicated than the Pythagorean theorem, but can be derived from it. A typical example where the straight-line distance between two points is converted to curvilinear coordinates can be found in the applications of Legendre polynomials in physics. The formulas can be discovered by using Pythagoras' theorem with the equations relating the curvilinear coordinates to Cartesian coordinates. For example, the polar coordinates (r, θ) can be introduced as:

${\displaystyle x=r\cos \theta ,y=r\sin \theta .}$

Then two points with locations (r₁, θ₁) and (r₂, θ₂) are separated by a distance s:

${\displaystyle s^2=(x_1-x_2{)}^2+(y_1-y_2{)}^2=(r_1\cos {\theta }_1-r_2\cos {\theta }_2{)}^2+(r_1\sin {\theta }_1-r_2\sin {\theta }_2{)}^2.}$

Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as:

${\displaystyle \begin{array}{rl}{s}^2& =r_1^2+r_2^2-2r_1{r}_2\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\right)\\ & =r_1^2+r_2^2-2r_1{r}_2\cos \left({\theta }_1-{\theta }_2\right)\\ & =r_1^2+r_2^2-2r_1{r}_2\cos \mathrm{\Delta }\theta ,\end{array}}$

using the trigonometric product-to-sum formulas. This formula is the law of cosines, sometimes called the generalized Pythagorean theorem. From this result, for the case where the radii to the two locations are at right angles, the enclosed angle Δθ = π/2, and the form corresponding to Pythagoras' theorem is regained: ${\displaystyle s^2=r_1^2+r_2^2.}$ The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles.

Pythagorean trigonometric identity

Main article: Pythagorean trigonometric identity

Similar right triangles showing sine and cosine of angle θ

In a right triangle with sides a, b and hypotenuse c, trigonometry determines the sine and cosine of the angle θ between side a and the hypotenuse as:

$\sin \theta =\frac{b}{c},\cos \theta =\frac{a}{c}.$

From that it follows:

${\cos }^2\theta +{\sin }^2\theta =\frac{a^2+b^2}{c^2}=1,$

where the last step applies Pythagoras' theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. In similar triangles, the ratios of the sides are the same regardless of the size of the triangles, and depend upon the angles. Consequently, in the figure, the triangle with hypotenuse of unit size has opposite side of size sin θ and adjacent side of size cos θ in units of the hypotenuse.

Relation to the cross product

The area of a parallelogram as a cross product; vectors a and b identify a plane and a × b is normal to this plane.

The Pythagorean theorem relates the cross product and dot product in a similar way:

${\displaystyle \Vert \mathbf{a}\times \mathbf{b}{\Vert }^2+(\mathbf{a}\cdot \mathbf{b}{)}^2=\Vert \mathbf{a}{\Vert }^2\Vert \mathbf{b}{\Vert }^2.}$

This can be seen from the definitions of the cross product and dot product, as

$\begin{array}{rl}\mathbf{a}\times \mathbf{b}& =ab\mathbf{n}\sin \theta \\ \mathbf{a}\cdot \mathbf{b}& =ab\cos \theta ,\end{array}$

with n a unit vector normal to both a and b. The relationship follows from these definitions and the Pythagorean trigonometric identity.

This can also be used to define the cross product. By rearranging the following equation is obtained

${\displaystyle \Vert \mathbf{a}\times \mathbf{b}{\Vert }^2=\Vert \mathbf{a}{\Vert }^2\Vert \mathbf{b}{\Vert }^2-(\mathbf{a}\cdot \mathbf{b}{)}^2.}$

This can be considered as a condition on the cross product and so part of its definition, for example in seven dimensions.

Generalizations

Similar figures on the three sides

The Pythagorean theorem generalizes beyond the areas of squares on the three sides to any similar figures. This was known by Hippocrates of Chios in the 5th century BC, and was included by Euclid in his Elements:

If one erects similar figures (see Euclidean geometry) with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side.

This extension assumes that the sides of the original triangle are the corresponding sides of the three congruent figures (so the common ratios of sides between the similar figures are a:b:c). While Euclid's proof only applied to convex polygons, the theorem also applies to concave polygons and even to similar figures that have curved boundaries (but still with part of a figure's boundary being the side of the original triangle).

The basic idea behind this generalization is that the area of a plane figure is proportional to the square of any linear dimension, and in particular is proportional to the square of the length of any side. Thus, if similar figures with areas A, B and C are erected on sides with corresponding lengths a, b and c then:

$\frac{A}{a^2}=\frac{B}{b^2}=\frac{C}{c^2},$

$\Rightarrow A+B=\frac{a^2}{c^2}C+\frac{b^2}{c^2}C.$

But, by the Pythagorean theorem, a² + b² = c², so A + B = C.

Conversely, if we can prove that A + B = C for three similar figures without using the Pythagorean theorem, then we can work backwards to construct a proof of the theorem. For example, the starting center triangle can be replicated and used as a triangle C on its hypotenuse, and two similar right triangles (A and B ) constructed on the other two sides, formed by dividing the central triangle by its altitude. The sum of the areas of the two smaller triangles therefore is that of the third, thus A + B = C and reversing the above logic leads to the Pythagorean theorem a² + b² = c². (See also Einstein's proof by dissection without rearrangement)

Generalization for similar triangles, green area A + B = blue area C

Pythagoras' theorem using similar right triangles

Generalization for regular pentagons

Law of cosines

Main article: Law of cosines

The separation s of two points (r₁, θ₁) and (r₂, θ₂) in polar coordinates is given by the law of cosines. Interior angle Δθ = θ₁−θ₂.

The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines, which states that

$${\displaystyle a^2+b^2-2ab\cos \theta =c^2}$$

where $\theta$ is the angle between sides $a$ and $b$.

When $\theta$ is $\frac{\pi }{2}$ radians or 90°, then ${\displaystyle \cos \theta =0}$, and the formula reduces to the usual Pythagorean theorem.

Arbitrary triangle

Generalization of Pythagoras' theorem by Tâbit ibn Qorra. Lower panel: reflection of triangle CAD (top) to form triangle DAC, similar to triangle ABC (top).

At any selected angle of a general triangle of sides a, b, c, inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled c. Inscribing the isosceles triangle forms triangle CAD with angle θ opposite side b and with side r along c. A second triangle is formed with angle θ opposite side a and a side with length s along c, as shown in the figure. Thābit ibn Qurra stated that the sides of the three triangles were related as:

$a^2+b^2=c(r+s).$

As the angle θ approaches π/2, the base of the isosceles triangle narrows, and lengths r and s overlap less and less. When θ = π/2, ADB becomes a right triangle, r + s = c, and the original Pythagorean theorem is regained.

One proof observes that triangle ABC has the same angles as triangle CAD, but in opposite order. (The two triangles share the angle at vertex A, both contain the angle θ, and so also have the same third angle by the triangle postulate.) Consequently, ABC is similar to the reflection of CAD, the triangle DAC in the lower panel. Taking the ratio of sides opposite and adjacent to θ,

${\displaystyle \frac{c}{b}=\frac{b}{r}.}$

Likewise, for the reflection of the other triangle,

${\displaystyle \frac{c}{a}=\frac{a}{s}.}$

Clearing fractions and adding these two relations:

${\displaystyle cs+cr=a^2+b^2,}$

the required result.

The theorem remains valid if the angle $\theta$ is obtuse so the lengths r and s are non-overlapping.

General triangles using parallelograms

Generalization for arbitrary triangles, green area = blue area

Construction for proof of parallelogram generalization

Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras' theorem, and was considered a generalization by Pappus of Alexandria in 4 AD

The lower figure shows the elements of the proof. Focus on the left side of the figure. The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same base b and height h. However, the left green parallelogram also has the same area as the left green parallelogram of the upper figure, because they have the same base (the upper left side of the triangle) and the same height normal to that side of the triangle. Repeating the argument for the right side of the figure, the bottom parallelogram has the same area as the sum of the two green parallelograms.

Solid geometry

Pythagoras' theorem in three dimensions relates the diagonal AD to the three sides.

A tetrahedron with outward facing right-angle corner

In terms of solid geometry, Pythagoras' theorem can be applied to three dimensions as follows. Consider a rectangular solid as shown in the figure. The length of diagonal BD is found from Pythagoras' theorem as:

${\overline{BD}}^2={\overline{BC}}^2+{\overline{CD}}^2,$

where these three sides form a right triangle. Using horizontal diagonal BD and the vertical edge AB, the length of diagonal AD then is found by a second application of Pythagoras' theorem as:

${\overline{AD}}^2={\overline{AB}}^2+{\overline{BD}}^2,$

or, doing it all in one step:

${\overline{AD}}^2={\overline{AB}}^2+{\overline{BC}}^2+{\overline{CD}}^2.$

This result is the three-dimensional expression for the magnitude of a vector v (the diagonal AD) in terms of its orthogonal components {v_k} (the three mutually perpendicular sides):

$\Vert \mathbf{v}{\Vert }^2=\sum _{k=1}^3\Vert {\mathbf{v}}_k{\Vert }^2.$

This one-step formulation may be viewed as a generalization of Pythagoras' theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras' theorem to a succession of right triangles in a sequence of orthogonal planes.

A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem, named for Jean Paul de Gua de Malves: If a tetrahedron has a right angle corner (like a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This result can be generalized as in the "n-dimensional Pythagorean theorem":

Let ${\displaystyle x_1,x_2,\dots ,x_n}$ be orthogonal vectors in Rⁿ. Consider the n-dimensional simplex S with vertices ${\displaystyle 0,x_1,\dots ,x_n}$. (Think of the (n − 1)-dimensional simplex with vertices $x_1,\dots ,x_n$ not including the origin as the "hypotenuse" of S and the remaining (n − 1)-dimensional faces of S as its "legs".) Then the square of the volume of the hypotenuse of S is the sum of the squares of the volumes of the n legs.

This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras' theorem applies. In a different wording:

Given an n-rectangular n-dimensional simplex, the square of the (n − 1)-content of the facet opposing the right vertex will equal the sum of the squares of the (n − 1)-contents of the remaining facets.

Inner product spaces

Vectors involved in the parallelogram law

The Pythagorean theorem can be generalized to inner product spaces, which are generalizations of the familiar 2-dimensional and 3-dimensional Euclidean spaces. For example, a function may be considered as a vector with infinitely many components in an inner product space, as in functional analysis.

In an inner product space, the concept of perpendicularity is replaced by the concept of orthogonality: two vectors v and w are orthogonal if their inner product $⟨\mathbf{v},\mathbf{w}⟩$ is zero. The inner product is a generalization of the dot product of vectors. The dot product is called the standard inner product or the Euclidean inner product. However, other inner products are possible.

The concept of length is replaced by the concept of the norm ‖v‖ of a vector v, defined as:

$\Vert \mathbf{v}\Vert \equiv \sqrt{⟨\mathbf{v},\mathbf{v}⟩}.$

In an inner-product space, the Pythagorean theorem states that for any two orthogonal vectors v and w we have

${\Vert \mathbf{v}+\mathbf{w}\Vert }^2={\Vert \mathbf{v}\Vert }^2+{\Vert \mathbf{w}\Vert }^2.$

Here the vectors v and w are akin to the sides of a right triangle with hypotenuse given by the vector sum v + w. This form of the Pythagorean theorem is a consequence of the properties of the inner product:

${\displaystyle \begin{array}{rl}{\Vert \mathbf{v}+\mathbf{w}\Vert }^2& =⟨\mathbf{v}\mathbf{+}\mathbf{w},\mathbf{v}\mathbf{+}\mathbf{w}⟩\\ & =⟨\mathbf{v},\mathbf{v}⟩+⟨\mathbf{w},\mathbf{w}⟩+⟨\mathbf{v}\mathbf{,}\mathbf{w}⟩+⟨\mathbf{w}\mathbf{,}\mathbf{v}⟩\\ & ={\Vert \mathbf{v}\Vert }^2+{\Vert \mathbf{w}\Vert }^2,\end{array}}$

where ${\displaystyle ⟨\mathbf{v}\mathbf{,}\mathbf{w}⟩=⟨\mathbf{w}\mathbf{,}\mathbf{v}⟩=0}$ because of orthogonality.

A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is the parallelogram law :

$2\Vert \mathbf{v}{\Vert }^2+2\Vert \mathbf{w}{\Vert }^2=\Vert \mathbf{v}\mathbf{+}\mathbf{w}{\Vert }^2+\Vert \mathbf{v}\mathbf{-}\mathbf{w}{\Vert }^2,$

which says that twice the sum of the squares of the lengths of the sides of a parallelogram is the sum of the squares of the lengths of the diagonals. Any norm that satisfies this equality is ipso facto a norm corresponding to an inner product.

The Pythagorean identity can be extended to sums of more than two orthogonal vectors. If v₁, v₂, ..., v_n are pairwise-orthogonal vectors in an inner-product space, then application of the Pythagorean theorem to successive pairs of these vectors (as described for 3-dimensions in the section on solid geometry) results in the equation

${\displaystyle {\Vert \sum _{k=1}^n{\mathbf{v}}_k\Vert }^2=\sum _{k=1}^n\Vert {\mathbf{v}}_k{\Vert }^2}$

Sets of m-dimensional objects in n-dimensional space

Another generalization of the Pythagorean theorem applies to Lebesgue-measurable sets of objects in any number of dimensions. Specifically, the square of the measure of an m-dimensional set of objects in one or more parallel m-dimensional flats in n-dimensional Euclidean space is equal to the sum of the squares of the measures of the orthogonal projections of the object(s) onto all m-dimensional coordinate subspaces.

In mathematical terms:

${\mu }_{ms}^2=\sum _{i=1}^x{{\mu }^{\mathbf{2}}}_{m{p}_i}$

where:

• ${\mu }_m$ is a measure in m-dimensions (a length in one dimension, an area in two dimensions, a volume in three dimensions, etc.).
• $s$ is a set of one or more non-overlapping m-dimensional objects in one or more parallel m-dimensional flats in n-dimensional Euclidean space.
• ${\mu }_{ms}$ is the total measure (sum) of the set of m-dimensional objects.
• $p$ represents an m-dimensional projection of the original set onto an orthogonal coordinate subspace.
• ${\mu }_{m{p}_i}$ is the measure of the m-dimensional set projection onto m-dimensional coordinate subspace $i$. Because object projections can overlap on a coordinate subspace, the measure of each object projection in the set must be calculated individually, then measures of all projections added together to provide the total measure for the set of projections on the given coordinate subspace.
• $x$ is the number of orthogonal, m-dimensional coordinate subspaces in n-dimensional space (Rⁿ) onto which the m-dimensional objects are projected (m ≤ n):
  $${\displaystyle x=(\genfrac{}{}{0ex}{}{n}{m})=\frac{n!}{m!(n-m)!}}$$

  

Non-Euclidean geometry

The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate. Thus, right triangles in a non-Euclidean geometry do not satisfy the Pythagorean theorem. For example, in spherical geometry, all three sides of the right triangle (say a, b, and c) bounding an octant of the unit sphere have length equal to π/2, and all its angles are right angles, which violates the Pythagorean theorem because ${\displaystyle a^2+b^2=2c^2>c^2}$.

Here two cases of non-Euclidean geometry are considered—spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines.

However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the condition that two of the angles sum to the third, say A+B = C. The sides are then related as follows: the sum of the areas of the circles with diameters a and b equals the area of the circle with diameter c.

Spherical geometry

Spherical triangle

For any right triangle on a sphere of radius R (for example, if γ in the figure is a right angle), with sides a, b, c, the relation between the sides takes the form:

${\displaystyle \cos \frac{c}{R}=\cos \frac{a}{R}\cos \frac{b}{R}.}$

This equation can be derived as a special case of the spherical law of cosines that applies to all spherical triangles:

${\displaystyle \cos \frac{c}{R}=\cos \frac{a}{R}\cos \frac{b}{R}+\sin \frac{a}{R}\sin \frac{b}{R}\cos \gamma .}$

For infinitesimal triangles on the sphere (or equivalently, for finite spherical triangles on a sphere of infinite radius), the spherical relation between the sides of a right triangle reduces to the Euclidean form of the Pythagorean theorem. To see how, assume we have a spherical triangle of fixed side lengths a, b, and c on a sphere with expanding radius R. As R approaches infinity the quantities a/R, b/R, and c/R tend to zero and the spherical Pythagorean identity reduces to ${\displaystyle 1=1,}$ so we must look at its asymptotic expansion.

The Maclaurin series for the cosine function can be written as $\cos x=1-\frac{1}{2}{x}^2+O\left(x^4\right)$ with the remainder term in big O notation. Letting ${\displaystyle x=c/R}$ be a side of the triangle, and treating the expression as an asymptotic expansion in terms of R for a fixed c,

${\displaystyle \begin{array}{r}\cos \frac{c}{R}=1-\frac{c^2}{2R^2}+O\left(R^{-4}\right)\end{array}}$

and likewise for a and b. Substituting the asymptotic expansion for each of the cosines into the spherical relation for a right triangle yields

${\displaystyle \begin{array}{rl}1-\frac{c^2}{2R^2}+O\left(R^{-4}\right)& =\left(1-\frac{a^2}{2R^2}+O\left(R^{-4}\right)\right)\left(1-\frac{b^2}{2R^2}+O\left(R^{-4}\right)\right)\\ & =1-\frac{a^2}{2R^2}-\frac{b^2}{2R^2}+O\left(R^{-4}\right).\end{array}}$

Subtracting 1 and then negating each side,

${\displaystyle \frac{c^2}{2R^2}=\frac{a^2}{2R^2}+\frac{b^2}{2R^2}+O\left(R^{-4}\right).}$

Multiplying through by 2R², the asymptotic expansion for c in terms of fixed a, b and variable R is

${\displaystyle c^2=a^2+b^2+O\left(R^{-2}\right).}$

The Euclidean Pythagorean relationship $c^2=a^2+b^2$ is recovered in the limit, as the remainder vanishes when the radius R approaches infinity.

For practical computation in spherical trigonometry with small right triangles, cosines can be replaced with sines using the double-angle identity ${\displaystyle \cos 2\theta =1-2{\sin }^2\theta }$ to avoid loss of significance. Then the spherical Pythagorean theorem can alternately be written as

${\displaystyle {\sin }^2\frac{c}{2R}={\sin }^2\frac{a}{2R}+{\sin }^2\frac{b}{2R}-2{\sin }^2\frac{a}{2R}{\sin }^2\frac{b}{2R}.}$

Hyperbolic geometry

Hyperbolic triangle

In a hyperbolic space with uniform Gaussian curvature −1/R², for a right triangle with legs a, b, and hypotenuse c, the relation between the sides takes the form:^[64]

${\displaystyle \cosh \frac{c}{R}=\cosh \frac{a}{R}\cosh \frac{b}{R}}$

where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles:

${\displaystyle \cosh \frac{c}{R}=\cosh \frac{a}{R}\cosh \frac{b}{R}-\sinh \frac{a}{R}\sinh \frac{b}{R}\cos \gamma ,}$

with γ the angle at the vertex opposite the side c.

By using the Maclaurin series for the hyperbolic cosine, cosh x ≈ 1 + x²/2, it can be shown that as a hyperbolic triangle becomes very small (that is, as a, b, and c all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras' theorem.

For small right triangles (a, b << R), the hyperbolic cosines can be eliminated to avoid loss of significance, giving

${\displaystyle {\sinh }^2\frac{c}{2R}={\sinh }^2\frac{a}{2R}+{\sinh }^2\frac{b}{2R}+2{\sinh }^2\frac{a}{2R}{\sinh }^2\frac{b}{2R}.}$

Very small triangles

For any uniform curvature K (positive, zero, or negative), in very small right triangles (|K|a², |K|b² << 1) with hypotenuse c, it can be shown that

${\displaystyle c^2=a^2+b^2-\frac{K}{3}{a}^2{b}^2-\frac{K^2}{45}{a}^2{b}^2(a^2+b^2)-\frac{2K^3}{945}{a}^2{b}^2(a^2-b^2{)}^2+O(K^4{c}^{10}).}$

Differential geometry

Distance between infinitesimally separated points in Cartesian coordinates (top) and polar coordinates (bottom), as given by Pythagoras' theorem

The Pythagorean theorem applies to infinitesimal triangles seen in differential geometry. In three dimensional space, the distance between two infinitesimally separated points satisfies

${\displaystyle d{s}^2=d{x}^2+d{y}^2+d{z}^2,}$

with ds the element of distance and (dx, dy, dz) the components of the vector separating the two points. Such a space is called a Euclidean space. However, in Riemannian geometry, a generalization of this expression useful for general coordinates (not just Cartesian) and general spaces (not just Euclidean) takes the form:

$d{s}^2=\sum _{i,j}^n{g}_{ij}d{x}_{i}d{x}_j$

which is called the metric tensor. (Sometimes, by abuse of language, the same term is applied to the set of coefficients g_ij.) It may be a function of position, and often describes curved space. A simple example is Euclidean (flat) space expressed in curvilinear coordinates. For example, in polar coordinates:

$d{s}^2=d{r}^2+r^{2}d{\theta }^2.$

History

The Plimpton 322 tablet records Pythagorean triples from Babylonian times.

There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof. Historians of Mesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during the Old Babylonian period (20th to 16th centuries BC), over a thousand years before Pythagoras was born. The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system.

Written c. 1800 BC, the Egyptian Middle Kingdom Berlin Papyrus 6619 includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. The Mesopotamian tablet Plimpton 322, also written c. 1800 BC near Larsa, contains many entries closely related to Pythagorean triples.

In India, the Baudhayana Shulba Sutra, the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of the isosceles right triangle and in the general case, as does the Apastamba Shulba Sutra (c. 600 BC).

Byzantine Neoplatonic philosopher and mathematician Proclus, writing in the fifth century AD, states two arithmetic rules, "one of them attributed to Plato, the other to Pythagoras", for generating special Pythagorean triples. The rule attributed to Pythagoras (c. 570 – c. 495 BC) starts from an odd number and produces a triple with leg and hypotenuse differing by one unit; the rule attributed to Plato (428/427 or 424/423 – 348/347 BC) starts from an even number and produces a triple with leg and hypotenuse differing by two units. According to Thomas L. Heath (1861–1940), no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted. Classicist Kurt von Fritz wrote, "Whether this formula is rightly attributed to Pythagoras personally, but one can safely assume that it belongs to the very oldest period of Pythagorean mathematics." Around 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented.

Geometric proof of the Pythagorean theorem from the Zhoubi Suanjing

With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the Chinese text Zhoubi Suanjing (周髀算经), (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle — in China it is called the "Gougu theorem" (勾股定理). During the Han Dynasty (202 BC to 220 AD), Pythagorean triples appear in The Nine Chapters on the Mathematical Art, together with a mention of right triangles. Some believe the theorem arose first in China, where it is alternatively known as the "Shang Gao theorem" (商高定理), named after the Duke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in the Zhoubi Suanjing.