Measurement

From Wikipedia, the free encyclopedia

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

Four measuring devices having metric calibrations

Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to a basic reference quantity of the same kind. The scope and application of measurement are dependent on the context and discipline. In natural sciences and engineering, measurements do not apply to nominal properties of objects or events, which is consistent with the guidelines of the International vocabulary of metrology published by the International Bureau of Weights and Measures. However, in other fields such as statistics as well as the social and behavioural sciences, measurements can have multiple levels, which would include nominal, ordinal, interval and ratio scales.

Measurement is a cornerstone of trade, science, technology and quantitative research in many disciplines. Historically, many measurement systems existed for the varied fields of human existence to facilitate comparisons in these fields. Often these were achieved by local agreements between trading partners or collaborators. Since the 18th century, developments progressed towards unifying, widely accepted standards that resulted in the modern International System of Units (SI). This system reduces all physical measurements to a mathematical combination of seven base units. The science of measurement is pursued in the field of metrology.

Measurement is defined as the process of comparison of an unknown quantity with a known or standard quantity.

Methodology

The measurement of a property may be categorized by the following criteria: type, magnitude, unit, and uncertainty. They enable unambiguous comparisons between measurements.

• The level of measurement is a taxonomy for the methodological character of a comparison. For example, two states of a property may be compared by ratio, difference, or ordinal preference. The type is commonly not explicitly expressed, but implicit in the definition of a measurement procedure.
• The magnitude is the numerical value of the characterization, usually obtained with a suitably chosen measuring instrument.
• A unit assigns a mathematical weighting factor to the magnitude that is derived as a ratio to the property of an artifact used as standard or a natural physical quantity.
• An uncertainty represents the random and systemic errors of the measurement procedure; it indicates a confidence level in the measurement. Errors are evaluated by methodically repeating measurements and considering the accuracy and precision of the measuring instrument.

Standardization of measurement units

Measurements most commonly use the International System of Units (SI) as a comparison framework. The system defines seven fundamental units: kilogram, metre, candela, second, ampere, kelvin, and mole. Six of these units are defined without reference to a particular physical object which serves as a standard (artifact-free), while the kilogram is still embodied in an artifact which rests at the headquarters of the International Bureau of Weights and Measures in Sèvres near Paris. Artifact-free definitions fix measurements at an exact value related to a physical constant or other invariable phenomena in nature, in contrast to standard artifacts which are subject to deterioration or destruction. Instead, the measurement unit can only ever change through increased accuracy in determining the value of the constant it is tied to.

The seven base units in the SI system. Arrows point from units to those that depend on them.

The first proposal to tie an SI base unit to an experimental standard independent of fiat was by Charles Sanders Peirce (1839–1914), who proposed to define the metre in terms of the wavelength of a spectral line. This directly influenced the Michelson–Morley experiment; Michelson and Morley cite Peirce, and improve on his method.

Standards

With the exception of a few fundamental quantum constants, units of measurement are derived from historical agreements. Nothing inherent in nature dictates that an inch has to be a certain length, nor that a mile is a better measure of distance than a kilometre. Over the course of human history, however, first for convenience and then for necessity, standards of measurement evolved so that communities would have certain common benchmarks. Laws regulating measurement were originally developed to prevent fraud in commerce.

Units of measurement are generally defined on a scientific basis, overseen by governmental or independent agencies, and established in international treaties, pre-eminent of which is the General Conference on Weights and Measures (CGPM), established in 1875 by the Metre Convention, overseeing the International System of Units (SI). For example, the metre was redefined in 1983 by the CGPM in terms of the speed of light, the kilogram was redefined in 2019 in terms of the Planck constant and the international yard was defined in 1960 by the governments of the United States, United Kingdom, Australia and South Africa as being exactly 0.9144 metres.

In the United States, the National Institute of Standards and Technology (NIST), a division of the United States Department of Commerce, regulates commercial measurements. In the United Kingdom, the role is performed by the National Physical Laboratory (NPL), in Australia by the National Measurement Institute, in South Africa by the Council for Scientific and Industrial Research and in India the National Physical Laboratory of India.

Units and systems

Main articles: Unit of measurement and System of measurement

unit is known or standard quantity in terms of which other physical quantities are measured.

A baby bottle that measures in three measurement systems—metric, imperial (UK), and US customary.

Imperial and US customary systems

Main article: Imperial and US customary measurement systems

Before SI units were widely adopted around the world, the British systems of English units and later imperial units were used in Britain, the Commonwealth and the United States. The system came to be known as U.S. customary units in the United States and is still in use there and in a few Caribbean countries. These various systems of measurement have at times been called foot-pound-second systems after the Imperial units for length, weight and time even though the tons, hundredweights, gallons, and nautical miles, for example, are different for the U.S. units. Many Imperial units remain in use in Britain, which has officially switched to the SI system—with a few exceptions such as road signs, which are still in miles. Draught beer and cider must be sold by the imperial pint, and milk in returnable bottles can be sold by the imperial pint. Many people measure their height in feet and inches and their weight in stone and pounds, to give just a few examples. Imperial units are used in many other places, for example, in many Commonwealth countries that are considered metricated, land area is measured in acres and floor space in square feet, particularly for commercial transactions (rather than government statistics). Similarly, gasoline is sold by the gallon in many countries that are considered metricated.

Metric system

The metric system is a decimal system of measurement based on its units for length, the metre and for mass, the kilogram. It exists in several variations, with different choices of base units, though these do not affect its day-to-day use. Since the 1960s, the International System of Units (SI) is the internationally recognised metric system. Metric units of mass, length, and electricity are widely used around the world for both everyday and scientific purposes.

International System of Units

The International System of Units (abbreviated as SI from the French language name Système International d'Unités) is the modern revision of the metric system. It is the world's most widely used system of units, both in everyday commerce and in science. The SI was developed in 1960 from the metre–kilogram–second (MKS) system, rather than the centimetre–gram–second (CGS) system, which, in turn, had many variants. The SI units for the seven base physical quantities are:

Base quantity	Base unit	Symbol	Defining constant
time	second	s	hyperfine splitting in caesium-133
length	metre	m	speed of light, c
mass	kilogram	kg	Planck constant, h
electric current	ampere	A	elementary charge, e
temperature	kelvin	K	Boltzmann constant, k
amount of substance	mole	mol	Avogadro constant NA
luminous intensity	candela	cd	luminous efficacy of a 540 THz source Kcd

In the SI, base units are the simple measurements for time, length, mass, temperature, amount of substance, electric current and light intensity. Derived units are constructed from the base units, for example, the watt, i.e. the unit for power, is defined from the base units as m²·kg·s⁻³. Other physical properties may be measured in compound units, such as material density, measured in kg/m³.

Converting prefixes

The SI allows easy multiplication when switching among units having the same base but different prefixes. To convert from metres to centimetres it is only necessary to multiply the number of metres by 100, since there are 100 centimetres in a metre. Inversely, to switch from centimetres to metres one multiplies the number of centimetres by 0.01 or divides the number of centimetres by 100.

Length

A 2-metre carpenter's ruler

 

See also: List of length, distance, or range measuring devices

A ruler or rule is a tool used in, for example, geometry, technical drawing, engineering, and carpentry, to measure lengths or distances or to draw straight lines. Strictly speaking, the ruler is the instrument used to rule straight lines and the calibrated instrument used for determining length is called a measure, however common usage calls both instruments rulers and the special name straightedge is used for an unmarked rule. The use of the word measure, in the sense of a measuring instrument, only survives in the phrase tape measure, an instrument that can be used to measure but cannot be used to draw straight lines. As can be seen in the photographs on this page, a two-metre carpenter's rule can be folded down to a length of only 20 centimetres, to easily fit in a pocket, and a five-metre-long tape measure easily retracts to fit within a small housing.

Some special names

Some non-systematic names are applied for some multiples of some units.

• 100 kilograms = 1 quintal; 1000 kilogram = 1 tonne;
• 10 years = 1 decade; 100 years = 1 century; 1000 years = 1 millennium

Building trades

The Australian building trades adopted the metric system in 1966 and the units used for measurement of length are metres (m) and millimetres (mm). Centimetres (cm) are avoided as they cause confusion when reading plans. For example, the length two and a half metres is usually recorded as 2500 mm or 2.5 m; it would be considered non-standard to record this length as 250 cm.

Surveyor's trade

American surveyors use a decimal-based system of measurement devised by Edmund Gunter in 1620. The base unit is Gunter's chain of 66 feet (20 m) which is subdivided into 4 rods, each of 16.5 ft or 100 links of 0.66 feet. A link is abbreviated "lk", and links "lks", in old deeds and land surveys done for the government.

The Standard Method of Measurement (SMM) published by the Royal Institution of Chartered Surveyors (RICS) consisted of classification tables and rules of measurement, allowing use of a uniform basis for measuring building works. It was first published in 1922, superseding a Scottish Standard Method of Measurement which had been published in 1915. Its seventh edition (SMM7) was first published in 1988 and revised in 1998. SMM7 was replaced by the New Rules of Measurement, volume 2 (NRM2), which were published in April 2012 by the RICS Quantity Surveying and Construction Professional Group and became operational on 1 January 2013. NRM2 has been in general use since July 2013.

SMM7 was accompanied by the Code of Procedure for the Measurement of Building Works (the SMM7 Measurement Code). Whilst SMM7 could have a contractual status within a project, for example in the JCT Standard form of Building Contract), the Measurement Code was not mandatory.

NRM2 Is the second of three component parts within the NRM suite:

• NRM1 - Order of cost estimating and cost planning for capital building works
• NRM2 - Detailed measurement for building works
• NRM3 - Order of cost estimating and cost planning for building maintenance works.

Time

Main article: Time

Time is an abstract measurement of elemental changes over a non spatial continuum. It is denoted by numbers and/or named periods such as hours, days, weeks, months and years. It is an apparently irreversible series of occurrences within this non spatial continuum. It is also used to denote an interval between two relative points on this continuum.

Mass

Main article: Weighing scale

Mass refers to the intrinsic property of all material objects to resist changes in their momentum. Weight, on the other hand, refers to the downward force produced when a mass is in a gravitational field. In free fall, (no net gravitational forces) objects lack weight but retain their mass. The Imperial units of mass include the ounce, pound, and ton. The metric units gram and kilogram are units of mass.

One device for measuring weight or mass is called a weighing scale or, often, simply a scale. A spring scale measures force but not mass, a balance compares weight, both require a gravitational field to operate. Some of the most accurate instruments for measuring weight or mass are based on load cells with a digital read-out, but require a gravitational field to function and would not work in free fall.

Economics

Main article: Measurement in economics

The measures used in economics are physical measures, nominal price value measures and real price measures. These measures differ from one another by the variables they measure and by the variables excluded from measurements.

Survey research

Main article: Survey methodology

 

Measurement station C of EMMA experiment situated at the depth of 75 meters in the Pyhäsalmi Mine.

In the field of survey research, measures are taken from individual attitudes, values, and behavior using questionnaires as a measurement instrument. As all other measurements, measurement in survey research is also vulnerable to measurement error, i.e. the departure from the true value of the measurement and the value provided using the measurement instrument. In substantive survey research, measurement error can lead to biased conclusions and wrongly estimated effects. In order to get accurate results, when measurement errors appear, the results need to be corrected for measurement errors.

Exactness designation

The following rules generally apply for displaying the exactness of measurements:

• All non-0 digits and any 0s appearing between them are significant for the exactness of any number. For example, the number 12000 has two significant digits, and has implied limits of 11500 and 12500.
• Additional 0s may be added after a decimal separator to denote a greater exactness, increasing the number of decimals. For example, 1 has implied limits of 0.5 and 1.5 whereas 1.0 has implied limits 0.95 and 1.05.

Difficulties

Since accurate measurement is essential in many fields, and since all measurements are necessarily approximations, a great deal of effort must be taken to make measurements as accurate as possible. For example, consider the problem of measuring the time it takes an object to fall a distance of one metre (about 39 in). Using physics, it can be shown that, in the gravitational field of the Earth, it should take any object about 0.45 second to fall one metre. However, the following are just some of the sources of error that arise:

• This computation used for the acceleration of gravity 9.8 metres per second squared (32 ft/s²). But this measurement is not exact, but only precise to two significant digits.
• The Earth's gravitational field varies slightly depending on height above sea level and other factors.
• The computation of 0.45 seconds involved extracting a square root, a mathematical operation that required rounding off to some number of significant digits, in this case two significant digits.

Additionally, other sources of experimental error include:

• carelessness,
• determining of the exact time at which the object is released and the exact time it hits the ground,
• measurement of the height and the measurement of the time both involve some error,
• Air resistance.
• posture of human participants

Scientific experiments must be carried out with great care to eliminate as much error as possible, and to keep error estimates realistic.

Definitions and theories

Classical definition

In the classical definition, which is standard throughout the physical sciences, measurement is the determination or estimation of ratios of quantities. Quantity and measurement are mutually defined: quantitative attributes are those possible to measure, at least in principle. The classical concept of quantity can be traced back to John Wallis and Isaac Newton, and was foreshadowed in Euclid's Elements.

Representational theory

In the representational theory, measurement is defined as "the correlation of numbers with entities that are not numbers". The most technically elaborated form of representational theory is also known as additive conjoint measurement. In this form of representational theory, numbers are assigned based on correspondences or similarities between the structure of number systems and the structure of qualitative systems. A property is quantitative if such structural similarities can be established. In weaker forms of representational theory, such as that implicit within the work of Stanley Smith Stevens, numbers need only be assigned according to a rule.

The concept of measurement is often misunderstood as merely the assignment of a value, but it is possible to assign a value in a way that is not a measurement in terms of the requirements of additive conjoint measurement. One may assign a value to a person's height, but unless it can be established that there is a correlation between measurements of height and empirical relations, it is not a measurement according to additive conjoint measurement theory. Likewise, computing and assigning arbitrary values, like the "book value" of an asset in accounting, is not a measurement because it does not satisfy the necessary criteria.

Three type of Representational theory

1) Empirical relation

In science, an empirical relationship is a relationship or correlation based solely on observation rather than theory. An empirical relationship requires only confirmatory data irrespective of theoretical basis

2) The rule of mapping

The real world is the Domain of mapping, and the mathematical world is the range. when we map the attribute to mathematical system, we have many choice for mapping and the range

3) The representation condition of measurement

Information theory

Information theory recognises that all data are inexact and statistical in nature. Thus the definition of measurement is: "A set of observations that reduce uncertainty where the result is expressed as a quantity." This definition is implied in what scientists actually do when they measure something and report both the mean and statistics of the measurements. In practical terms, one begins with an initial guess as to the expected value of a quantity, and then, using various methods and instruments, reduces the uncertainty in the value. Note that in this view, unlike the positivist representational theory, all measurements are uncertain, so instead of assigning one value, a range of values is assigned to a measurement. This also implies that there is not a clear or neat distinction between estimation and measurement.

Quantum mechanics

In quantum mechanics, a measurement is an action that determines a particular property (position, momentum, energy, etc.) of a quantum system. Before a measurement is made, a quantum system is simultaneously described by all values in a range of possible values, where the probability of measuring each value is determined by the wavefunction of the system. When a measurement is performed, the wavefunction of the quantum system "collapses" to a single, definite value. The unambiguous meaning of the measurement problem is an unresolved fundamental problem in quantum mechanics.

Biology

In biology, there is generally no well established theory of measurement. However, the importance of the theoretical context is emphasized. Moreover, the theoretical context stemming from the theory of evolution leads to articulate the theory of measurement and historicity as a fundamental notion. Among the most developed fields of measurement in biology are the measurement of genetic diversity and species diversity.

Mathematical fallacy

From Wikipedia, the free encyclopedia

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

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof.

For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation. There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and are not applicable in the cases that are the exceptions to the rules.

The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy. The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the introduction of Pasch's axiom of Euclidean geometry, the five colour theorem of graph theory). Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.

Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a multiple valued function are equated. Well-known fallacies also exist in elementary Euclidean geometry and calculus.

Howlers

${\displaystyle {\begin{array}{l}\;\;\;{\dfrac {d}{dx}}{\dfrac {1}{x}}\\={\dfrac {d}{d}}{\dfrac {1}{x^{2}}}\\={\dfrac {d\!\!\!\backslash }{d\!\!\!\backslash }}{\dfrac {1}{x^{2}}}\\=-{\dfrac {1}{x^{2}}}\end{array}}}$

Anomalous cancellation in calculus

Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion appears to be, is mathematically invalid and is commonly known as a howler. The following is an example of a howler involving anomalous cancellation:

$${\displaystyle {\frac {16}{64}}={\frac {16\!\!\!/}{6\!\!\!/4}}={\frac {1}{4}}.}$$

Here, although the conclusion 16/64 = 1/4 is correct, there is a fallacious, invalid cancellation in the middle step. Another classical example of a howler is proving the Cayley–Hamilton theorem by simply substituting the scalar variables of the characteristic polynomial by the matrix.

Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed "howlers" by Maxwell. Outside the field of mathematics the term howler has various meanings, generally less specific.

Division by zero

The division-by-zero fallacy has many variants. The following example uses a disguised division by zero to "prove" that 2 = 1, but can be modified to prove that any number equals any other number.

1. Let a and b be equal, nonzero quantities

   ${\displaystyle a=b}$

2. Multiply by a

   ${\displaystyle a^{2}=ab}$

3. Subtract b²

   ${\displaystyle a^{2}-b^{2}=ab-b^{2}}$

4. Factor both sides: the left factors as a difference of squares, the right is factored by extracting b from both terms

   ${\displaystyle (a-b)(a+b)=b(a-b)}$

5. Divide out (a − b)

   ${\displaystyle a+b=b}$

6. Observing that a = b

   ${\displaystyle b+b=b}$

7. Combine like terms on the left

   ${\displaystyle 2b=b}$

8. Divide by the non-zero b

   ${\displaystyle 2=1}$

Q.E.D.

The fallacy is in line 5: the progression from line 4 to line 5 involves division by a − b, which is zero since a = b. Since division by zero is undefined, the argument is invalid.

Analysis

Mathematical analysis as the mathematical study of change and limits can lead to mathematical fallacies — if the properties of integrals and differentials are ignored. For instance, a naive use of integration by parts can be used to give a false proof that 0 = 1. Letting u = 1/log x and dv = dx/x, we may write:

${\displaystyle \int {\frac {1}{x\,\log x}}\,dx=1+\int {\frac {1}{x\,\log x}}\,dx}$

after which the antiderivatives may be cancelled yielding 0 = 1. The problem is that antiderivatives are only defined up to a constant and shifting them by 1 or indeed any number is allowed. The error really comes to light when we introduce arbitrary integration limits a and b.

${\displaystyle \int _{a}^{b}{\frac {1}{x\,\log x}}\,dx=1|_{a}^{b}+\int _{a}^{b}{\frac {1}{x\,\log x}}\,dx=0+\int _{a}^{b}{\frac {1}{x\log x}}\,dx=\int _{a}^{b}{\frac {1}{x\log x}}\,dx}$

Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.

Multivalued functions

Main article: Multivalued function

Many functions do not have a unique inverse. For instance, while squaring a number gives a unique value, there are two possible square roots of a positive number. The square root is multivalued. One value can be chosen by convention as the principal value; in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root given as the principal value of the square of a number will be equal to the original number (e.g. the principal square root of the square of −2 is 2). This remains true for nth roots.

Positive and negative roots

Care must be taken when taking the square root of both sides of an equality. Failing to do so results in a "proof" of 5 = 4.

Proof:

Start from

${\displaystyle -20=-20}$

Write this as

${\displaystyle 25-45=16-36}$

Rewrite as

${\displaystyle 5^{2}-5\times 9=4^{2}-4\times 9}$

Add 81/4 on both sides:

${\displaystyle 5^{2}-5\times 9+{\frac {81}{4}}=4^{2}-4\times 9+{\frac {81}{4}}}$

These are perfect squares:

${\displaystyle \left(5-{\frac {9}{2}}\right)^{2}=\left(4-{\frac {9}{2}}\right)^{2}}$

Take the square root of both sides:

${\displaystyle 5-{\frac {9}{2}}=4-{\frac {9}{2}}}$

Add 9/2 on both sides:

${\displaystyle 5=4}$

Q.E.D.

The fallacy is in the second to last line, where the square root of both sides is taken: a² = b² only implies a = b if a and b have the same sign, which is not the case here. In this case, it implies that a = –b, so the equation should read

${\displaystyle 5-{\frac {9}{2}}=-\left(4-{\frac {9}{2}}\right)}$

which, by adding 9/2 on both sides, correctly reduces to 5 = 5.

Another example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity

${\displaystyle \cos ^{2}x=1-\sin ^{2}x}$

which holds as a consequence of the Pythagorean theorem. Then, by taking a square root,

${\displaystyle \cos x={\sqrt {1-\sin ^{2}x}}}$

Evaluating this when x = π , we get that

${\displaystyle -1={\sqrt {1-0}}}$

or

${\displaystyle -1=1}$

which is incorrect.

The error in each of these examples fundamentally lies in the fact that any equation of the form

${\displaystyle x^{2}=a^{2}}$

where ${\displaystyle a\neq 0}$, has two solutions:

${\displaystyle x=\pm a}$

and it is essential to check which of these solutions is relevant to the problem at hand. In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos x is positive. In particular, when x is set to π, the second equation is rendered invalid.

Square roots of negative numbers

Invalid proofs utilizing powers and roots are often of the following kind:

${\displaystyle 1={\sqrt {1}}={\sqrt {(-1)(-1)}}={\sqrt {-1}}{\sqrt {-1}}=i\cdot i=-1.}$

The fallacy is that the rule ${\displaystyle {\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}}$ is generally valid only if at least one of ${\displaystyle x}$ and ${\displaystyle y}$ is non-negative (when dealing with real numbers), which is not the case here.

Alternatively, imaginary roots are obfuscated in the following:

${\displaystyle i={\sqrt {-1}}=\left(-1\right)^{\frac {2}{4}}=\left(\left(-1\right)^{2}\right)^{\frac {1}{4}}=1^{\frac {1}{4}}=1}$

The error here lies in the third equality, as the rule ${\displaystyle a^{bc}=(a^{b})^{c}}$ only holds for positive real a and real b, c.

Complex exponents

When a number is raised to a complex power, the result is not uniquely defined (see Failure of power and logarithm identities). If this property is not recognized, then errors such as the following can result:

${\displaystyle {\begin{aligned}e^{2\pi i}&=1\\\left(e^{2\pi i}\right)^{i}&=1^{i}\\e^{-2\pi }&=1\\\end{aligned}}}$

The error here is that the rule of multiplying exponents as when going to the third line does not apply unmodified with complex exponents, even if when putting both sides to the power i only the principal value is chosen. When treated as multivalued functions, both sides produce the same set of values, being {e^2πn | n ∈ ℤ}.

Geometry

Many mathematical fallacies in geometry arise from using an additive equality involving oriented quantities (such as adding vectors along a given line or adding oriented angles in the plane) to a valid identity, but which fixes only the absolute value of (one of) these quantities. This quantity is then incorporated into the equation with the wrong orientation, so as to produce an absurd conclusion. This wrong orientation is usually suggested implicitly by supplying an imprecise diagram of the situation, where relative positions of points or lines are chosen in a way that is actually impossible under the hypotheses of the argument, but non-obviously so.

In general, such a fallacy is easy to expose by drawing a precise picture of the situation, in which some relative positions will be different from those in the provided diagram. In order to avoid such fallacies, a correct geometric argument using addition or subtraction of distances or angles should always prove that quantities are being incorporated with their correct orientation.

Fallacy of the isosceles triangle

The fallacy of the isosceles triangle, from (Maxwell 1959, Chapter II, § 1), purports to show that every triangle is isosceles, meaning that two sides of the triangle are congruent. This fallacy was known to Lewis Carroll and may have been discovered by him. It was published in 1899.

Given a triangle △ABC, prove that AB = AC:

1. Draw a line bisecting ∠A.
2. Draw the perpendicular bisector of segment BC, which bisects BC at a point D.
3. Let these two lines meet at a point O.
4. Draw line OR perpendicular to AB, line OQ perpendicular to AC.
5. Draw lines OB and OC.
6. By AAS, △RAO ≅ △QAO (∠ORA = ∠OQA = 90°; ∠RAO = ∠QAO; AO = AO (common side)).
7. By RHS,^{[note 2]} △ROB ≅ △QOC (∠BRO = ∠CQO = 90°; BO = OC (hypotenuse); RO = OQ (leg)).
8. Thus, AR = AQ, RB = QC, and AB = AR + RB = AQ + QC = AC.

Q.E.D.

As a corollary, one can show that all triangles are equilateral, by showing that AB = BC and AC = BC in the same way.

The error in the proof is the assumption in the diagram that the point O is inside the triangle. In fact, O always lies on the circumcircle of the △ABC (except for isosceles and equilateral triangles where AO and OD coincide). Furthermore, it can be shown that, if AB is longer than AC, then R will lie within AB, while Q will lie outside of AC, and vice versa (in fact, any diagram drawn with sufficiently accurate instruments will verify the above two facts). Because of this, AB is still AR + RB, but AC is actually AQ − QC; and thus the lengths are not necessarily the same.

Proof by induction

There exist several fallacious proofs by induction in which one of the components, basis case or inductive step, is incorrect. Intuitively, proofs by induction work by arguing that if a statement is true in one case, it is true in the next case, and hence by repeatedly applying this, it can be shown to be true for all cases. The following "proof" shows that all horses are the same colour.

1. Let us say that any group of N horses is all of the same colour.
2. If we remove a horse from the group, we have a group of N − 1 horses of the same colour. If we add another horse, we have another group of N horses. By our previous assumption, all the horses are of the same colour in this new group, since it is a group of N horses.
3. Thus we have constructed two groups of N horses all of the same colour, with N − 1 horses in common. Since these two groups have some horses in common, the two groups must be of the same colour as each other.
4. Therefore, combining all the horses used, we have a group of N + 1 horses of the same colour.
5. Thus if any N horses are all the same colour, any N + 1 horses are the same colour.
6. This is clearly true for N = 1 (i.e. one horse is a group where all the horses are the same colour). Thus, by induction, N horses are the same colour for any positive integer N. i.e. all horses are the same colour.

The fallacy in this proof arises in line 3. For N = 1, the two groups of horses have N − 1 = 0 horses in common, and thus are not necessarily the same colour as each other, so the group of N + 1 = 2 horses is not necessarily all of the same colour. The implication "every N horses are of the same colour, then N + 1 horses are of the same colour" works for any N > 1, but fails to be true when N = 1. The basis case is correct, but the induction step has a fundamental flaw.