Infinity

From Wikipedia, the free encyclopedia

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

Due to the constant light reflection between opposing mirrors, it seems that there is a boundless amount of space and repetition inside of them.

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol ${\displaystyle \mathrm{\infty }}$.

Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.

In physics and cosmology, whether the Universe is spatially infinite is an open question.

History

Further information: Infinity (philosophy)

Ancient cultures had various ideas about the nature of infinity. The ancient Indians and the Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

Early Greek

The earliest recorded idea of infinity in Greece may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".

Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving the infinitude of the prime numbers, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated:

If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles.

Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.

Zeno: Achilles and the tortoise

Main article: Zeno's paradoxes § Achilles and the tortoise

Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".

Achilles races a tortoise, giving the latter a head start.

• Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
• Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
• Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
• Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.

Etc.

Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.

Zeno was not attempting to make a point about infinity. As a member of the Eleatics school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.

Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,

$${\displaystyle a+ax+a{x}^2+a{x}^3+a{x}^4+a{x}^5+\cdots =\frac{a}{1-x}.}$$

Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him

$${\displaystyle 10+0.1+0.001+0.00001+\cdots =\frac{10}{1-0.01}=\frac{10}{0.99}=10.10101\dots \text{ seconds}.}$$

Early Indian

The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

• Enumerable: lowest, intermediate, and highest
• Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
• Infinite: nearly infinite, truly infinite, infinitely infinite

17th century

In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation ${\displaystyle \mathrm{\infty }}$ for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of ${\displaystyle \frac{1}{\mathrm{\infty }}.}$ But in Arithmetica infinitorum (1656), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."

In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.

Mathematics

Hermann Weyl opened a mathematico-philosophic address given in 1930 with:

Mathematics is the science of the infinite.

Symbol

Main article: Infinity symbol

The infinity symbol ${\displaystyle \mathrm{\infty }}$ (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (&infin;) and in LaTeX as \infty.

It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.

Calculus

Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of continuity.

Real analysis

In real analysis, the symbol ${\displaystyle \mathrm{\infty }}$, called "infinity", is used to denote an unbounded limit. The notation ${\displaystyle x\to \mathrm{\infty }}$ means that ${\displaystyle x}$ increases without bound, and ${\displaystyle x\to -\mathrm{\infty }}$ means that ${\displaystyle x}$ decreases without bound. For example, if ${\displaystyle f(t)\ge 0}$ for every ${\displaystyle t}$, then

• ${\displaystyle {\int }_a^{b}f(t)dt=\mathrm{\infty }}$ means that ${\displaystyle f(t)}$ does not bound a finite area from ${\displaystyle a}$ to ${\displaystyle b.}$
• ${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)dt=\mathrm{\infty }}$ means that the area under ${\displaystyle f(t)}$ is infinite.
• ${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)dt=a}$ means that the total area under ${\displaystyle f(t)}$ is finite, and is equal to ${\displaystyle a.}$

Infinity can also be used to describe infinite series, as follows:

• ${\displaystyle \sum _{i=0}^{\mathrm{\infty }}f(i)=a}$ means that the sum of the infinite series converges to some real value ${\displaystyle a.}$
• ${\displaystyle \sum _{i=0}^{\mathrm{\infty }}f(i)=\mathrm{\infty }}$ means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.

In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled ${\displaystyle +\mathrm{\infty }}$ and ${\displaystyle -\mathrm{\infty }}$ can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat ${\displaystyle +\mathrm{\infty }}$ and ${\displaystyle -\mathrm{\infty }}$ as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.

Complex analysis

By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.

In complex analysis the symbol ${\displaystyle \mathrm{\infty }}$, called "infinity", denotes an unsigned infinite limit. The expression ${\displaystyle x\to \mathrm{\infty }}$ means that the magnitude ${\displaystyle |x|}$ of ${\displaystyle x}$ grows beyond any assigned value. A point labeled ${\displaystyle \mathrm{\infty }}$ can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely ${\displaystyle z/0=\mathrm{\infty }}$ for any nonzero complex number ${\displaystyle z}$. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of ${\displaystyle \mathrm{\infty }}$ at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).

Nonstandard analysis

Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)

The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).

Set theory

Main articles: Cardinality and Ordinal number

One-to-one correspondence between an infinite set and its proper subset

A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ₀), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.

Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part. (However, see Galileo's paradox where Galileo concludes that positive integers cannot be compared to the subset of positive square integers since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".

Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Cardinality of the continuum

Main article: Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuum ${\displaystyle \mathbf{c}}$ is greater than that of the natural numbers ${\displaystyle {\mathrm{\aleph }}_0}$; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that ${\displaystyle \mathbf{c}=2^{{\mathrm{\aleph }}_0}>{\mathrm{\aleph }}_0}$.

Further information: Cantor's diagonal argument and Cantor's first set theory article

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, ${\displaystyle \mathbf{c}={\mathrm{\aleph }}_1={\beth }_1}$.

Further information: Beth number § Beth one

This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.

The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R.

See also: Hilbert's paradox of the Grand Hotel

The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.

Geometry

Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).

One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.

Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).

In particular, in modern mathematics, lines are infinite sets.

Infinite dimension

The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.

In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.

Fractals

The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.

Mathematics without infinity

Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.

Physics

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.

Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576.^[46] Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."

Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is still an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.

The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.

However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.

The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes. Also, cyclic models posit an infinite amount of Big Bangs, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.

Logic

In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."

Computing

The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.

Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses as sentinel values in algorithms involving sorting, searching, or windowing.

In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible for a programmer to create the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point data type, the infinity values may still be accessible and usable as the result of certain operations.

In programming, an infinite loop is a loop whose exit condition is never satisfied, thus executing indefinitely.

Arts, games, and cognitive sciences

Perspective artwork uses the concept of vanishing points, roughly corresponding to mathematical points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms. Artist M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.

Variations of chess played on an unbounded board are called infinite chess.

Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1,2,3,...>.

El Niño

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/El_Ni%C3%B1o

Normal Pacific pattern: Warm pool in the west drives deep atmospheric convection. In the east local winds cause nutrient-rich cold water to upwell at the Equator and along the South American coast. (NOAA / PMEL / TAO)

 

El Niño conditions: warm water and atmospheric convection move eastwards. In strong El Niños the deeper thermocline off S. America means upwelled water is warm and nutrient poor.

El Niño (/ɛl ˈniːnjoʊ/ el NEEN-yoh, Spanish: [el ˈniɲo]; lit. 'The Boy') is the warm phase of the El Niño–Southern Oscillation (ENSO) and is associated with a band of warm ocean water that develops in the central and east-central equatorial Pacific (approximately between the International Date Line and 120°W), including the area off the Pacific coast of South America. The ENSO is the cycle of warm and cold sea surface temperature (SST) of the tropical central and eastern Pacific Ocean.

El Niño is accompanied by high air pressure in the western Pacific and low air pressure in the eastern Pacific. El Niño phases are known to last close to four years; however, records demonstrate that the cycles have lasted between two and seven years. During the development of El Niño, rainfall develops between September–November. The cool phase of ENSO is La Niña, 'The Girl', with sea surface temperatures (SSTs) in the eastern Pacific below average, and air pressure high in the eastern Pacific and low in the western Pacific. The ENSO cycle, including both El Niño and La Niña, causes global changes in temperature and rainfall.

Developing countries that depend on their own agriculture and fishing, particularly those bordering the Pacific Ocean, are usually most affected. In this phase of the Oscillation, the pool of warm water in the Pacific near South America is often at its warmest about Christmas. The original phrase, El Niño de Navidad, arose centuries ago, when Peruvian fishermen named the weather phenomenon after the newborn Christ.

Concept

Originally, the term El Niño applied to an annual weak warm ocean current that ran southwards along the coast of Peru and Ecuador at about Christmas time. However, over time the term has evolved and now refers to the warm and negative phase of the El Niño–Southern Oscillation and is the warming of the ocean surface or above-average sea surface temperatures in the central and eastern tropical Pacific Ocean. This warming causes a shift in the atmospheric circulation with rainfall becoming reduced over Indonesia, India and northern Australia, while rainfall and tropical cyclone formation increases over the tropical Pacific Ocean. The low-level surface trade winds, which normally blow from east to west along the equator, either weaken or start blowing from the other direction.

Loop of the 1997–98 El Niño event showing extreme sea surface temperature (SST) anomalies in the east tropical Pacific

It is believed that El Niño has occurred for thousands of years. For example, it is thought that El Niño affected the Moche in modern-day Peru. Scientists have also found chemical signatures of warmer sea surface temperatures and increased rainfall caused by El Niño in coral specimens that are around 13,000 years old. Around 1525, when Francisco Pizarro made landfall in Peru, he noted rainfall in the deserts, the first written record of the impacts of El Niño. Modern day research and reanalysis techniques have managed to find at least 26 El Niño events since 1900, with the 1982–83, 1997–98 and 2014–16 events among the strongest on record.

Currently, each country has a different threshold for what constitutes an El Niño event, which is tailored to their specific interests. For example, the Australian Bureau of Meteorology looks at the trade winds, Southern Oscillation Index, weather models and sea surface temperatures in the Niño 3 and 3.4 regions, before declaring an El Niño. The United States Climate Prediction Center (CPC) and the International Research Institute for Climate and Society (IRI) looks at the sea surface temperatures in the Niño 3.4 region, the tropical Pacific atmosphere and forecasts that NOAA's Oceanic Niño Index will equal or exceed +.5 °C (0.90 °F) for several seasons in a row. However, the Japan Meteorological Agency declares that an El Niño event has started when the average five month sea surface temperature deviation for the Niño 3 region, is over 0.5 °C (0.90 °F) warmer for six consecutive months or longer. The Peruvian government declares that a coastal El Niño is under way if the sea surface temperature deviation in the Niño 1+2 regions equal or exceed 0.4 °C (0.72 °F) for at least three months.

There is no consensus on whether climate change will have any influence on the strength or duration of El Niño events, as research alternately supports El Niño events becoming stronger and weaker, longer and shorter. However, recent scholarship has found that climate change is increasing the frequency of extreme El Niño events.

Occurrences

Timeline of El Niño episodes between 1900 and 2023.

El Niño events are thought to have been occurring for thousands of years. For example, it is thought that El Niño affected the Moche in modern-day Peru, who sacrificed humans in order to try to prevent the rains.

It is thought that there have been at least 30 El Niño events since 1900, with the 1982–83, 1997–98 and 2014–16 events among the strongest on record. Since 2000, El Niño events have been observed in 2002–03, 2004–05, 2006–07, 2009–10, 2014–16, 2018–19, and beginning in 2023.

Major ENSO events were recorded in the years 1790–93, 1828, 1876–78, 1891, 1925–26, 1972–73, 1982–83, 1997–98, and 2014–16.

Typically, this anomaly happens at irregular intervals of two to seven years, and lasts nine months to two years. The average period length is five years. When this warming occurs for seven to nine months, it is classified as El Niño "conditions"; when its duration is longer, it is classified as an El Niño "episode".

During strong El Niño episodes, a secondary peak in sea surface temperature across the far eastern equatorial Pacific Ocean sometimes follows the initial peak.

Cultural history and prehistoric information

Average equatorial Pacific temperatures, published in 2009.

ENSO conditions have occurred at two- to seven-year intervals for at least the past 300 years, but most of them have been weak. Evidence is also strong for El Niño events during the early Holocene epoch 10,000 years ago.

El Niño may have led to the demise of the Moche and other pre-Columbian Peruvian cultures. A recent study suggests a strong El Niño effect between 1789 and 1793 caused poor crop yields in Europe, which in turn helped touch off the French Revolution. The extreme weather produced by El Niño in 1876–77 gave rise to the most deadly famines of the 19th century. The 1876 famine alone in northern China killed up to 13 million people.

An early recorded mention of the term "El Niño" to refer to climate occurred in 1892, when Captain Camilo Carrillo told the geographical society congress in Lima that Peruvian sailors named the warm south-flowing current "El Niño" because it was most noticeable around Christmas. Although pre-Columbian societies were certainly aware of the phenomenon, the indigenous names for it have been lost to history. The phenomenon had long been of interest because of its effects on the guano industry and other enterprises that depend on biological productivity of the sea. It is recorded that as early as 1822, cartographer Joseph Lartigue, of the French frigate La Clorinde under Baron Mackau, noted the "counter-current" and its usefulness for traveling southward along the Peruvian coast.

Charles Todd, in 1888, suggested droughts in India and Australia tended to occur at the same time; Norman Lockyer noted the same in 1904. An El Niño connection with flooding was reported in 1894 by Victor Eguiguren (1852–1919) and in 1895 by Federico Alfonso Pezet (1859–1929). In 1924, Gilbert Walker (for whom the Walker circulation is named) coined the term "Southern Oscillation". He and others (including Norwegian-American meteorologist Jacob Bjerknes) are generally credited with identifying the El Niño effect.

The major 1982–83 El Niño led to an upsurge of interest from the scientific community. The period 1990–95 was unusual in that El Niños have rarely occurred in such rapid succession. An especially intense El Niño event in 1998 caused an estimated 16% of the world's reef systems to die. The event temporarily warmed air temperature by 1.5 °C, compared to the usual increase of 0.25 °C associated with El Niño events. Since then, mass coral bleaching has become common worldwide, with all regions having suffered "severe bleaching".

Diversity

Map showing Niño3.4 and other index regions

It is thought that there are several different types of El Niño events, with the canonical eastern Pacific and the Modoki central Pacific types being the two that receive the most attention. These different types of El Niño events are classified by where the tropical Pacific sea surface temperature (SST) anomalies are the largest. For example, the strongest sea surface temperature anomalies associated with the canonical eastern Pacific event are located off the coast of South America. The strongest anomalies associated with the Modoki central Pacific event are located near the International Date Line. However, during the duration of a single event, the area with the greatest sea surface temperature anomalies can change.

The traditional Niño, also called Eastern Pacific (EP) El Niño, involves temperature anomalies in the Eastern Pacific. However, in the last two decades, atypical El Niños were observed, in which the usual place of the temperature anomaly (Niño 1 and 2) is not affected, but an anomaly arises in the central Pacific (Niño 3.4). The phenomenon is called Central Pacific (CP) El Niño, "dateline" El Niño (because the anomaly arises near the International Date Line), or El Niño "Modoki" (Modoki is Japanese for "similar, but different").

The effects of the CP El Niño are different from those of the traditional EP El Niño—e.g., the recently discovered El Niño leads to more frequent Atlantic hurricanes.

There is also a scientific debate on the very existence of this "new" ENSO. Indeed, a number of studies dispute the reality of this statistical distinction or its increasing occurrence, or both, either arguing the reliable record is too short to detect such a distinction, finding no distinction or trend using other statistical approaches, or that other types should be distinguished, such as standard and extreme ENSO.

The first recorded El Niño that originated in the central Pacific and moved toward the east was in 1986. Recent Central Pacific El Niños happened in 1986–87, 1991–92, 1994–95, 2002–03, 2004–05 and 2009–10. Furthermore, there were "Modoki" events in 1957–59, 1963–64, 1965–66, 1968–70, 1977–78 and 1979–80. Some sources say that the El Niños of 2006-07 and 2014-16 were also Central Pacific El Niños.

Effects on the global climate

Colored bars show how El Niño years (red, regional warming) and La Niña years (blue, regional cooling) relate to overall global warming. The El Niño–Southern Oscillation has been linked to variability in longer-term global average temperature increase, with El Niño years usually corresponding to annual global temperature increases.

El Niño affects the global climate and disrupts normal weather patterns, which as a result can lead to intense storms in some places and droughts in others.

Tropical cyclones

Most tropical cyclones form on the side of the subtropical ridge closer to the equator, then move poleward past the ridge axis before recurving into the main belt of the Westerlies. Areas west of Japan and Korea tend to experience many fewer September–November tropical cyclone impacts during El Niño and neutral years. During El Niño years, the break in the subtropical ridge tends to lie near 130°E, which would favor the Japanese archipelago.

Within the Atlantic Ocean vertical wind shear is increased, which inhibits tropical cyclone genesis and intensification, by causing the westerly winds in the atmosphere to be stronger. The atmosphere over the Atlantic Ocean can also be drier and more stable during El Niño events, which can also inhibit tropical cyclone genesis and intensification. Within the Eastern Pacific basin: El Niño events contribute to decreased easterly vertical wind shear and favours above-normal hurricane activity. However, the impacts of the ENSO state in this region can vary and are strongly influenced by background climate patterns. The Western Pacific basin experiences a change in the location of where tropical cyclones form during El Niño events, with tropical cyclone formation shifting eastward, without a major change in how many develop each year. As a result of this change, Micronesia is more likely to be affected by tropical cyclones, while China has a decreased risk of being affected by tropical cyclones. A change in the location of where tropical cyclones form also occurs within the Southern Pacific Ocean between 135°E and 120°W, with tropical cyclones more likely to occur within the Southern Pacific basin than the Australian region. As a result of this change tropical cyclones are 50% less likely to make landfall on Queensland, while the risk of a tropical cyclone is elevated for island nations like Niue, French Polynesia, Tonga, Tuvalu, and the Cook Islands.

Remote influence on tropical Atlantic Ocean

A study of climate records has shown that El Niño events in the equatorial Pacific are generally associated with a warm tropical North Atlantic in the following spring and summer. About half of El Niño events persist sufficiently into the spring months for the Western Hemisphere Warm Pool to become unusually large in summer. Occasionally, El Niño's effect on the Atlantic Walker circulation over South America strengthens the easterly trade winds in the western equatorial Atlantic region. As a result, an unusual cooling may occur in the eastern equatorial Atlantic in spring and summer following El Niño peaks in winter. Cases of El Niño-type events in both oceans simultaneously have been linked to severe famines related to the extended failure of monsoon rains.

Regional impacts

Observations of El Niño events since 1950 show that impacts associated with El Niño events depend on the time of year. However, while certain events and impacts are expected to occur during events, it is not certain or guaranteed that they will occur. The impacts that generally do occur during most El Niño events include below-average rainfall over Indonesia and northern South America, while above average rainfall occurs in southeastern South America, eastern equatorial Africa, and the southern United States.

Africa

In Africa, East Africa—including Kenya, Tanzania, and the White Nile basin—experiences, in the long rains from March to May, wetter-than-normal conditions. Conditions are also drier than normal from December to February in south-central Africa, mainly in Zambia, Zimbabwe, Mozambique, and Botswana.

Antarctica

Many ENSO linkages exist in the high southern latitudes around Antarctica. Specifically, El Niño conditions result in high-pressure anomalies over the Amundsen and Bellingshausen Seas, causing reduced sea ice and increased poleward heat fluxes in these sectors, as well as the Ross Sea. The Weddell Sea, conversely, tends to become colder with more sea ice during El Niño. The exact opposite heating and atmospheric pressure anomalies occur during La Niña. This pattern of variability is known as the Antarctic dipole mode, although the Antarctic response to ENSO forcing is not ubiquitous.

Asia

As warm water spreads from the west Pacific and the Indian Ocean to the east Pacific, it takes the rain with it, causing extensive drought in the western Pacific and rainfall in the normally dry eastern Pacific. Singapore experienced the driest February in 2010 since records began in 1869, with only 6.3 mm of rain falling in the month and temperatures hitting as high as 35 °C on 26 February. The years 1968 and 2005 had the next driest Februaries, when 8.4 mm of rain fell.

Australia and the Southern Pacific

See also: Effects of the El Niño–Southern Oscillation in Australia

During El Niño events, the shift in rainfall away from the Western Pacific may mean that rainfall across Australia is reduced. Over the southern part of the continent, warmer than average temperatures can be recorded as weather systems are more mobile and fewer blocking areas of high pressure occur. The onset of the Indo-Australian Monsoon in tropical Australia is delayed by two to six weeks, which as a consequence means that rainfall is reduced over the northern tropics. The risk of a significant bushfire season in south-eastern Australia is higher following an El Niño event, especially when it is combined with a positive Indian Ocean Dipole event. During an El Niño event, New Zealand tends to experience stronger or more frequent westerly winds during their summer, which leads to an elevated risk of drier than normal conditions along the east coast. There is more rain than usual though on New Zealand's West Coast, because of the barrier effect of the North Island mountain ranges and the Southern Alps.

Fiji generally experiences drier than normal conditions during an El Niño, which can lead to drought becoming established over the Islands. However, the main impacts on the island nation is felt about a year after the event becomes established. Within the Samoan Islands, below average rainfall and higher than normal temperatures are recorded during El Niño events, which can lead to droughts and forest fires on the islands. Other impacts include a decrease in the sea level, possibility of coral bleaching in the marine environment and an increased risk of a tropical cyclone affecting Samoa.

Europe

El Niño's effects on Europe are controversial, complex and difficult to analyse, as it is one of several factors that influence the weather over the continent and other factors can overwhelm the signal.

North America

See also: Effects of the El Niño–Southern Oscillation in the United States

Regional impacts of warm ENSO episodes (El Niño)

Over North America, the main temperature and precipitation impacts of El Niño generally occur in the six months between October and March. In particular, the majority of Canada generally has milder than normal winters and springs, with the exception of eastern Canada where no significant impacts occur. Within the United States, the impacts generally observed during the six-month period include wetter-than-average conditions along the Gulf Coast between Texas and Florida, while drier conditions are observed in Hawaii, the Ohio Valley, Pacific Northwest and the Rocky Mountains.

Historically, El Niño was not understood to affect U.S. weather patterns until Christensen et al. (1981) used entropy minimax pattern discovery based on information theory to advance the science of long range weather prediction. Previous computer models of weather were based on persistence alone and reliable to only 5–7 days into the future. Long range forecasting was essentially random. Christensen et al. demonstrated the ability to predict the probability that precipitation will be below or above average with modest but statistically significant skill one, two and even three years into the future.

Study of more recent weather events over California and the southwestern United States indicate that there is a variable relationship between El Niño and above-average precipitation, as it strongly depends on the strength of the El Niño event and other factors. Though it has been historically associated with high rainfall in California, the effects of El Niño depend more strongly on the "flavor" of El Niño than its presence or absence, as only "persistent El Niño" events lead to consistently high rainfall.

The synoptic condition for the Tehuano wind, or "Tehuantepecer", is associated with a high-pressure area forming in Sierra Madre of Mexico in the wake of an advancing cold front, which causes winds to accelerate through the Isthmus of Tehuantepec. Tehuantepecers primarily occur during the cold season months for the region in the wake of cold fronts, between October and February, with a summer maximum in July caused by the westward extension of the Azores High. Wind magnitude is greater during El Niño years than during La Niña years, due to the more frequent cold frontal incursions during El Niño winters. Its effects can last from a few hours to six days. Some El Niño events were recorded in the isotope signals of plants, and that had helped scientists to study his impact.

South America

Because El Niño's warm pool feeds thunderstorms above, it creates increased rainfall across the east-central and eastern Pacific Ocean, including several portions of the South American west coast. The effects of El Niño in South America are direct and stronger than in North America. An El Niño is associated with warm and very wet weather months in April–October along the coasts of northern Peru and Ecuador, causing major flooding whenever the event is strong or extreme. The effects during the months of February, March, and April may become critical along the west coast of South America, El Niño reduces the upwelling of cold, nutrient-rich water that sustains large fish populations, which in turn sustain abundant sea birds, whose droppings support the fertilizer industry. The reduction in upwelling leads to fish kills off the shore of Peru.

The local fishing industry along the affected coastline can suffer during long-lasting El Niño events. The world's largest fishery collapsed due to overfishing during the 1972 El Niño Peruvian anchoveta reduction. During the 1982–83 event, jack mackerel and anchoveta populations were reduced, scallops increased in warmer water, but hake followed cooler water down the continental slope, while shrimp and sardines moved southward, so some catches decreased while others increased. Horse mackerel have increased in the region during warm events. Shifting locations and types of fish due to changing conditions create challenges for the fishing industry. Peruvian sardines have moved during El Niño events to Chilean areas. Other conditions provide further complications, such as the government of Chile in 1991 creating restrictions on the fishing areas for self-employed fishermen and industrial fleets.

The ENSO variability may contribute to the great success of small, fast-growing species along the Peruvian coast, as periods of low population removes predators in the area. Similar effects benefit migratory birds that travel each spring from predator-rich tropical areas to distant winter-stressed nesting areas.

Southern Brazil and northern Argentina also experience wetter than normal conditions, but mainly during the spring and early summer. Central Chile receives a mild winter with large rainfall, and the Peruvian-Bolivian Altiplano is sometimes exposed to unusual winter snowfall events. Drier and hotter weather occurs in parts of the Amazon River Basin, Colombia, and Central America.

Galapagos Islands

The Galapagos Islands are a chain of volcanic islands, nearly 600 miles west of Ecuador, South America.  Located in the Eastern Pacific Ocean, these islands are home to a wide diversity of terrestrial and marine species including sharks, birds, iguanas, turtles, penguins, and seals. This robust ecosystem is fueled by the normal trade winds which influence upwelling of cold, nutrient rich waters to the islands. During an El Niño an event where the trade winds weaken and sometimes push from west to east. This causes the Equatorial current to weaken, thus raising surface water temperatures and decreasing nutrients in waters surrounding the Galapagos. El Niño causes a trophic cascade which impacts entire ecosystems starting with primary producers and ending with critical animals such as sharks, penguins, and seals. The effects of El Niño can become detrimental to populations that often starve and die off during these years. Rapid evolutionary adaptations are displayed amongst animal groups during El Niño years to mitigate El Niño conditions.

Socio-ecological effects for humanity and nature

Economic effects

El Niño has the most direct impacts on life in the equatorial Pacific, its effects propagate north and south along the coast of the Americas, affecting marine life all around the Pacific. Changes in chlorophyll-a concentrations are visible in this animation, which compares phytoplankton in January and July 1998. Since then, scientists have improved both the collection and presentation of chlorophyll data.

When El Niño conditions last for many months, extensive ocean warming and the reduction in easterly trade winds limits upwelling of cold nutrient-rich deep water, and its economic effect on local fishing for an international market can be serious.

More generally, El Niño can affect commodity prices and the macroeconomy of different countries. It can constrain the supply of rain-driven agricultural commodities; reduce agricultural output, construction, and services activities; create food-price and generalised inflation; and may trigger social unrest in commodity-dependent poor countries that primarily rely on imported food. A University of Cambridge Working Paper shows that while Australia, Chile, Indonesia, India, Japan, New Zealand and South Africa face a short-lived fall in economic activity in response to an El Niño shock, other countries may actually benefit from an El Niño weather shock (either directly or indirectly through positive spillovers from major trading partners), for instance, Argentina, Canada, Mexico and the United States. Furthermore, most countries experience short-run inflationary pressures following an El Niño shock, while global energy and non-fuel commodity prices increase. The IMF estimates a significant El Niño can boost the GDP of the United States by about 0.5% (due largely to lower heating bills) and reduce the GDP of Indonesia by about 1.0%.

Health and social impacts

Extreme weather conditions related to the El Niño cycle correlate with changes in the incidence of epidemic diseases. For example, the El Niño cycle is associated with increased risks of some of the diseases transmitted by mosquitoes, such as malaria, dengue fever, and Rift Valley fever. Cycles of malaria in India, Venezuela, Brazil, and Colombia have now been linked to El Niño. Outbreaks of another mosquito-transmitted disease, Australian encephalitis (Murray Valley encephalitis—MVE), occur in temperate south-east Australia after heavy rainfall and flooding, which are associated with La Niña events. A severe outbreak of Rift Valley fever occurred after extreme rainfall in north-eastern Kenya and southern Somalia during the 1997–98 El Niño.

ENSO conditions have also been related to Kawasaki disease incidence in Japan and the west coast of the United States, via the linkage to tropospheric winds across the north Pacific Ocean.

ENSO may be linked to civil conflicts. Scientists at The Earth Institute of Columbia University, having analyzed data from 1950 to 2004, suggest ENSO may have had a role in 21% of all civil conflicts since 1950, with the risk of annual civil conflict doubling from 3% to 6% in countries affected by ENSO during El Niño years relative to La Niña years.

Ecological consequences

In terrestrial ecosystems, rodent outbreaks were observed in northern Chile and along the Peruvian coastal desert following the 1972-73 El Niño event. While some nocturnal primates (western tarsiers Tarsius bancanus and slow loris Nycticebus coucang) and the Malayan sun bear (Helarctos malayanus) were locally extirpate or suffered drastic reduction in numbers within these burned forests. Lepidoptera outbreaks were documented in Panamá and Costa Rica. During the 1982–83, 1997–98 and 2015–16 ENSO events, large extensions of tropical forests experienced a prolonged dry period that resulted in widespread fires, and drastic changes in forest structure and tree species composition in Amazonian and Bornean forests.

Their impacts do not restrict only vegetation, since declines in insect populations were observed after extreme drought and terrible fires during El Niño 2015–16. Declines in habitat-specialist and disturbance-sensitive bird species and in large-frugivorous mammals were also observed in Amazonian burned forests, while temporary extirpation of more than 100 lowland butterfly species occurred at a burned forest site in Borneo.

Most critically, global mass bleaching events were recorded in 1997-98 and 2015–16, when around 75-99% losses of live coral were registered across the world. Considerable attention was also given to the collapse of Peruvian and Chilean anchovy populations that led to a severe fishery crisis following the ENSO events in 1972–73, 1982–83, 1997–98 and, more recently, in 2015–16. In particular, increased surface seawater temperatures in 1982-83 also lead to the probable extinction of two hydrocoral species in Panamá, and to a massive mortality of kelp beds along 600 km of coastline in Chile, from which kelps and associated biodiversity slowly recovered in the most affected areas even after 20 years. All these findings enlarge the role of ENSO events as a strong climatic force driving ecological changes all around the world – particularly in tropical forests and coral reefs.

In seasonally dry tropical forests, which are more drought tolerant, researchers found that El Niño induced drought increased seedling mortality. In a research published in October 2022, researchers studied seasonally dry tropical forests in a national park in Chiang Mai in Thailand for 7 years and observed that El Niño increased seedling mortality even in seasonally dry tropical forests and may impact entire forests in long run.