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However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve.

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⴳ ⵢⴰⵜ ⵜⴼⵔⴽⵜ ⴷⴰ ⵜⵜⵓⵙⵏⵜⴰⵢ ⵜⴰⵍⵖⴰ ⵜⴰⵎⴽⴽⵓⵣⵜ ⵎⵉ ⵉⵜⵜⴰⵛⵛⴽⴰ ⵓⵏⵖⴰⴷ; ⵙ ⵓⵣⴳⵍ ⵏ ⵓⵙⵏⵎⵉⵍⴰ, ⵜⵙⵙⵓⴷⵓ ⵜⵖⴰⵔⴰⵙⵜ ⵏ ⵓⵙⵏⵎⵉⵍⴰ ⴳ ⵜⵉⵅⵅⵉⵜⵔⵜ.

Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots.

ⵍⵍⴰⵏⵜ ⵜⴱⵔⵉⴷⵉⵏ ⵏ ⵓⵙⵎⵉⵍⴰ ⴰⵎⵉⴹⴰⵏ ⴷⴰ ⴰⵙ ⵜⵜⵉⵏⵉⵏ ⵜⴰⵖⴰⵔⵜ ⵜⵉⵎⴽⵉⵙⵉⵜ ⵢⴰⴽⴽⴰⵏ ⵉⵙⴰⵏⴼⵏ ⵖⴼ ⵜⵉⴳⴳⵉⵜⵉⵏ ⵉⵜⵜⴰⵎⵥⵏ ⵜⵉⵣⵉ ⴽⵉⴳⴰⵏ, ⵣⵓⵏⴷ ⴰⵙⴼⵓⴽⵜⵉ ⴷ ⵢⵉⵙⵢ ⵏ ⵜⵣⵎⵔⵜ ⴷ ⵓⵣⵖⵕⴰⵏ.

Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics.

ⵉⴼⵙⵙⴰⵢⵏ ⴳⴰⵏ ⴰⴳⵣⵣⵓⵎ ⵉⵙⵜⴰⵡⵀⵎⵎⴰⵏ ⴳ ⵍⵊⵉⴱⵔ ⵉⵣⵔⵉⵔⵉⴳ ⴰⵎⵉⴹⴰⵏ, ⴷ ⵖⵓⵔⵙ ⵜⴰⵡⵉⵍⴰ ⵎⵇⵇⵓⵔⵏ ⴳ ⵜⵏⵣⴳⵉⵜ ⴷ ⴼⵉⵣⵉⴽ ⴷ ⵛⵉⵎⵉ ⴷ ⵜⵓⵙⵏⴰⵎⵙⵙⵓⴷⵙⵜ ⴷ ⵜⴷⴰⵎⵙⴰ.

For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring.

ⵉⵙⵎⴷ ⴰⵎⵎ ⵜⵅⵕⵙⵜ ⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ, ⵏⵖⴷ ⴳ ⵜⴰⵏⵖⵉⵡⵉⵏ ⵏⵏⵉⴹⵏ ⵏ ⵍⵊⵉⴱⵔ, ⵜⵜⵓⵙⴱⵓⵖⵍⵍⴰⵏⵜ ⵜⵎⴰⴳⵓⵏⵉⵏ ⵢⴰⴹⵏ, ⵥⵕ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵣⵔⵉⵔⴳⵜ ⴰⴼⵍⵍⴰ ⵏ ⵜⵅⵕⵙⵜ.

This allows all the language and theory of vector spaces (or more generally, modules) to be brought to bear.

ⵡⴰⴷ ⴰⵔ ⵉⵜⵜⴰⴷⵊⴰ ⴰⴷ ⵉⵜⵜⵓⴳ ⴽⴰ ⵉⴳⴰⵜ ⵜⵓⵜⵍⴰⵢⵜ ⴷ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵉⵙⵜⵓⵎⵏ ⵏ ⵉⵎⵏⵉⴷⵏ ( ⵙ ⵓⵎⴰⵜⴰ ⵉⴼⵔⴷⴰⵙⵏ ⵓⵔⵙⵉⵍⵏ).

Such a system is known as an underdetermined system.

ⵉⵜⵢⴰⵙⵙⵏ ⵓⵏⴳⵔⴰⵡ ⴰⴷ ⵙ ⵢⵉⵙⵎ ⵏ ⴰⵏⴳⵔⴰⵡ ⵓⵔ ⵉⵥⵍⵉⵢⵏ.

The second system has a single unique solution, namely the intersection of the two lines.

ⴰⵏⴳⵔⴰⵡ ⵡⵉⵙⵙ ⵙⵉⵏ ⵖⴰⵔⵙ ⴰⴼⵙⵙⴰⵢ ⵉⵥⵍⵉⵏ ⵉⴳⴰⵏ ⴰⵎⵔⵊⴰⵍ ⵏ ⵉⵣⵔⵉⵔⵉⴳⵏ.

Any two of these equations have a common solution.

ⴽⴰ ⵉⴳⴰⵜ ⵙⵏⴰⵜ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⴳ ⵜⴳⴰⴷⴰⵣⴰⵍⵉⵏ ⴰⴷ ⵖⴰⵔⵙⵏⵜ ⴰⴼⵙⵙⴰⵢ ⵉⵛⵛⴰⵔⵏ.

A system of equations whose left-hand sides are linearly independent is always consistent.

ⴰⵀⴰ ⴷⴰ ⵉⵜⴳⴳⴰ ⵓⵏⴳⵔⴰⵡ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⵏⵏⴰ ⵎⵉ ⵜⴳⴳⴰⵏⵜ ⵜⵙⴳⴳⵉⵏ ⵏⵏⵙ ⵜⵉⵥⵍⵎⴰⴹ ⵜⵉⵎⵥⵍⴰⵢ; ⴰⵣⵔⵉⵔⵉⴳ ⵉⵎⵣⴳⵉ.

This yields a system of equations with one fewer equation and one fewer unknown.

ⴷⴰⴷ ⵢⴰⴽⴽⴰ ⵎⴰⵢⴰ ⴰⵏⴳⵔⴰⵡ ⵏ ⵜⴳⴷⴰⵣⴰⵍⵉⵏ ⴷⵉⴽⵙ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵢⴰⵣⴷⵓⵔⵏ, ⴷ ⵜⴰⵢⴹ ⵓⵔ ⵜⵢⴰⵙⵙⵏ.

Type 3: Add to one row a scalar multiple of another.

ⴰⵏⴰⵡ 3: ⵔⵏⵓ ⴰⵙ ⵉ ⵢⴰⵏ ⵡⴰⴷⵓⵔ ⴰⵙⵍⴰⴳ ⴰⵎⵉⴹⴰⵏ ⵏ ⵡⴰⴷⵓⵔ ⵏⵏⵉⴹⵏ.

For instance, systems with a symmetric positive definite matrix can be solved twice as fast with the Cholesky decomposition.

ⵙ ⵓⵎⴷⵢⴰ, ⵉⵖⵢ ⴰⴷ ⵏⴼⵙⵉ ⵉⴳⵔⵔⴰⵢⵏ ⵉⵍⴰⵏ ⴰⴷⵓⵔⵏ ⵉⵎⵥⵍⴰⵢ ⵓⵎⵏⵉⴳⵏ ⵉⵏⴰⵡⴰⵢⵏ ⵙ ⵣⵣⵔⴰⴱⵉⵜ ⵜⴰⵙⴼⵜⴰⵢⵜ ⵙ ⵜⵙⵍⵟ ⵏⵜⵛⵓⵍⵉⵙⴽⵉ.

A completely different approach is often taken for very large systems, which would otherwise take too much time or memory.

ⴷⴰ ⵡⴰⵍⴰ ⵉⵜⵜⵓⴹⴼⵓⵕ ⵓⵙⴰⵔⴰ ⴰⵎⵣⵉⵔⴰⵢ ⵉ ⵉⴳⵔⵔⴰⵢⵏ ⵎⵇⵇⵓⵔⵏ ⴽⵉⴳⴰⵏ, ⵏⵏⴰ ⵉⵜⵜⴻⵜⵜⵔⵏ ⴽⵉⴳⴰⵏ ⵏ ⵜⵉⵣⵉ ⵏⵖⴷ ⵜⵉⵎⴽⵜⵉⵜ.

This leads to the class of iterative methods.

ⴰⵢⴰ ⴰⵔ ⵉⵜⵜⴰⵡⵢ ⵖⵔ ⵜⴳⵔⵔⵓⵎⴰ ⵏ ⵜⴱⵔⵉⴷⵉⵏ ⵢⵓⵍⵙⵏ.

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

ⴳ ⵜⵓⵙⵏⴰⴽⵜ, ⵜⴳⴰ ⵜⴳⴼⴼⵓⵔⵜ ⵙ ⵜⴰⵍⵖⴰ ⵏ ⵓⵏⵎⵉⵍⵉ; ⴰⵙⵏⵓⵎⵍ ⵉ ⵜⵉⴳⴳⵉⵜ ⵏ ⵜⵓⵔⵏⵓⵜ ⵉⵡⵓⴷⵉⵢⵏ ⵡⴰⵔⵜⵎⵉ ⵉⴳⴳⵓⴷⵉⵏ, ⵢⴰⵜ ⴹⴰⵕⵜ ⵢⴰⵜ ⴰⵔ ⴰⴳⵓⴷⵉ ⵏ ⴽⴰⵏ ⵓⵙⵙⵏⵜⵉ.

In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

ⴷ ⵜⵓⵔⵏⵓⵜ ⵅⴼ ⵓⵏⵖⴰⵍ ⵏⵏⵙ ⴳ ⵜⵓⵙⵏⴰⴽⵜ, ⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙⵏⵜ ⵜⵙⵏⵙⵍⵉⵏ ⵡⴰⵔⵜⵎⵉ ⴳ ⵓⴼⴰⵖⵓⵍ ⴰⴱⴰⵔⴰⵡ ⴳ ⵜⵥⵍⴰⵢⵉⵏ ⵏ ⵓⴳⵓⴷⵉⵢ ⵏⵏⵉⴷⵏ ⵣⵓⵏⴷ ⴼⵉⵣⵉⴽ ⴷ ⵜⵓⵙⵙⵏⴰⵎⵙⵙⵓⴷⵙ ⵜⴰⴷⴷⴰⴷⴰⵏⵜ ⴷ ⵓⵙⵙⵥⵕⴼ.

Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on.

ⴷⴰ ⵜⵙⵙⴼⵔⵓ ⵜⵎⵣⴰⵔⴰⵢⵜ ⵏ ⵣⵉⵏⵓ ⴳⵔ ⴰⵅⵉⵍ ⴷ ⴽⴼⵔⵓⵏ ⵜⴰⵥⵍⴰⵢⵜ ⴰⴷ ⵉⵏⵎⵏ ⵖⴼ ⵡⴰⵜⵉⴳⵏ ⵡⴰⵔⵜⵎⵉ, ⴰⵅⵉⵍ ⵉⵜⵜⴰⵣⵍⴰ ⴹⴰⵕⵜ ⴽⴼⵔⵓⵏ, ⵎⴰⴽⴰ ⴰⴷⴷⴰⵢ ⵢⴰⵡⴹ ⴰⵏⵙⴰ ⵏ ⴽⴼⵔⵓⵏ ⴳ ⵜⵉⵣⵡⵉⵔⵉ ⵏ ⵓⵃⵎⵎⵣⵡⵓⵔ, ⴷⴰⵏ ⵉⵜⵜⴰⵡⴹ ⴽⴼⵔⵓⵏ ⴰⵙⵡⵉⵔ ⵡⵉⵙⵙ ⵙⵉⵏ, ⵉⴳⵏ ⵢⵓⵡⴹ ⵡⵉⵙⵙ ⵙⵉⵏ ⴷⴰ ⵜⵜⵉⵍⵉ ⴳ ⵡⵉⵙⵙ ⴽⵕⴰⴹ, ⴰⵔ ⵜⴷⴷⵓ.

This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2.

ⵓⵔ ⴷⴰ ⵉⵣⵣⴳⴰ ⵡⴰⵏⵥⴰ ⵉⵙ ⵜⴰⴳ ⵜⵎⵓⵜⵜⵔⵜ ⵏⵏⵙ 2 (ⵡⴰⵅⵅⴰ ⵉⴳⴰ ⵉⵎⴽⵉⵏⵏⴰⵖ), ⵎⴰⵛⴰ ⴷⴰ ⵜⵣⵣⴳⴰ ⴷⴷⴰⵡ 2.

Tests for uniform convergence include the Weierstrass' M-test, Abel's uniform convergence test, Dini's test, and the Cauchy criterion.

ⵙⵎⵓⵏⵏ ⵢⵉⵔⵉⵎⵏ ⵏ ⵓⵙⵏⵎⴰⵍⴰ ⵏ ⵓⵙⵙⵓⴷⵙ, ⵉⵔⵉⵎ ⵏ “ⵡⴰⵢⵔⵙⵜⵔⵉⵙ ⵎ Weierstrass' M-tes”, ⴷ ⵢⵉⵔⵉⵎ ⵏ ⵓⵙⵏⵎⴰⵍⴰ ⴰⵎⵢⵉⵡⵏ ⵉⵣⵍⵉⵏ ⵙ “ⴰⴱⵉⵍ Abel's” ⴷ “ ⴷⵉⵏⵉ Dini's” ⴷ ⴰⵏⴰⵡⴰⵢ “ⴽⵓⵛⵉ Cauchy ”.

The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets.

ⴰⵙⵏⵎⴰⵍⴰ ⴷⴰ ⵉⵜⵜⵓⵙⵓⴷⵙ ⴳ ⵜⵔⴰⴱⴱⵓⵜⵉⵏ ⵜⴰⵢⵢⴰⵡⵉⵏ ⵉⵇⵏⵏ ⴳⵉⵏⵜ ⵜⵉⵎⵥⴰⵍⵢ ( ⵉⴳ ⴳⴰⵏⵜ ⵜⵓⴷⴷⵉⵣⵉⵏ), ⴳ ⵜⴳⵣⵣⵓⵎⵜ ⵜⴰⴳⵏⵙⵓⵜ ⵏ ⴷⵉⵙⴽ ⵏ ⵓⵙⵏⵎⴰⵍⴰ: ⵉ ⵜⵉⵖⵉⵙⵜ, ⵉⵏⵎⴰⵍⴰ ⵙ ⵜⴰⵍⵖⴰ ⵉⵥⵍⵉⵏ ⴳ ⵜⵔⴰⴱⴱⵓⵜⵉⵏ ⵢⵓⴷⵔⵏ.

The Hilbert–Poincaré series is a formal power series used to study graded algebras.

ⵜⴰⴳⴼⴼⵓⵔⵜ “ⵉⵍⴱⵉⵔ-ⴱⵡⴰⵏⴽⴰⵔⵉ Hilbert–Poincaré” ⵜⴳⴰ ⵜⴰⴳⴼⴼⵓⵔⵜ ⵜⵓⵏⵙⵉⴱⵜ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴳ ⵜⵖⵓⵔⵉ ⵏ ⵍⵊⵉⴱⵔ ⴰⵎⴹⴼⵓⵕ.

In the 17th century, James Gregory worked in the new decimal system on infinite series and published several Maclaurin series.

ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 17 ⵉⵙⵡⵓⵔⵉ “ⴳⵉⵎⵙ ⴳⵔⵉⴳⵓⵔⵉ James Gregory” ⴳ ⵓⵏⴳⵔⴰⵡ ⴰⵎⵔⴰⵡⴰⵏ ⴰⵎⴰⵢⵏⵓ; ⵖⴼ ⵜⴳⴼⴼⵓⵔⵜ ⵜⴰⵔⵜⵎⵉ, ⴷ ⵉⴼⵙⵔ ⴽⵉⴳⴰⵏ ⵏ ⵜⴳⴼⴼⵓⵔⵉⵏ ⵏ “ⵎⴰⴽⵍⵓⵔⵉⵏ Maclaurin”.

Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria.

ⵉⵙⴷⴷⵉⴷ “ⴽⵓⵛⵉ” (1821) ⵖⴼ ⵢⵉⵔⵉⵎⵏ ⵉⵏⵎⴰⵍⴰⵏⴻⵏ ⵓⵇⵊⵉⵔⵏ, ⵉⵙⵙⴰⴽⵣ ⵉⴳ ⵎⵎⵍⵎⴰⵍⴰⵏⵜ ⵙⵏⴰⵜ ⵜⴳⴼⴼⵓⵔⵉⵏ ⴰⵎⵥⴰⵔⴰⵡ ⵏⵏⵙ ⵓⵔ ⵉⴷ ⵛⵛⵉⵍ ⴰⵙ ⴰⴷ ⵎⴰⵢⴰⵏ, ⴰⵔ ⵉⴷⵙ ⵜⵜⵓⵙⵏⵜⴰⵢ ⵜⵓⴼⴰⵢⵜ ⵏ ⵉⵏⴰⵡⴰⵢⵏ ⵉⵎⵕⵡⵉⵜⵏ.

A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence.

ⵜⴰⵖⴰⵔⴰⵙⵜ ⵏ ⵓⵙⵎⵓⵏ ⵜⴳⴰ ⵉⴽⵣ ⴰⵡⵜⵜⵓ ⵉ ⵜⵔⴰⴱⴱⵓⵜ ⵜⴰⵢⵢⴰⵡⵜ ⵏ ⵜⵔⴰⴱⴱⵓⵜ ⵏ ⵜⴳⴼⴼⵓⵔⵉⵏ ⵉⵎⵎⵄⵔⴰⵇⵏ, ⵉⵜⵜⴰⵏⴼⵏ ⵙ ⵜⴰⵍⵖⴰ ⵜⴰⵎⴷⴷⴰⴷⵜ ⴰⵔⵎⵎⵓⵙ ⴰⴽⵍⴰⵙⵉⴽⵉ ⵏ ⵜⵏⵎⵉⵍⴰ.

Indian scholars have been using factorial formulas since at least the 12th century.

ⴷⴰ ⵙⵙⵎⵔⴰⵙⵏ ⵉⵎⴰⵙⵙⵏ ⵉⵀⵉⵏⴷⵉⵢⵏ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵉⵎⴳⴳⵓⵜⵏ ⵙⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 12.

In functional languages, the recursive definition is often implemented directly to illustrate recursive functions.

ⴳ ⵜⵓⵜⵍⴰⵢⵉⵏ ⵜⵉⵙⵖⵏⴰⵏ, ⴷⴰ ⵡⴰⵍⴰ ⵉⵜⵜⵓⴳⴰ ⵙ ⵡⵓⵙⵔⵉⴷ ⵓⵙⵉⵙⵙⵏ ⴰⵎⴰⵖⵓⵍ, ⵎⴰⵔ ⴰⴷ ⵉⵙⵙⴼⵔⵓ ⵜⵉⵙⵖⵏⴰ ⵏ ⵜⵎⴰⵖⵓⵍⵜ.

Other implementations (such as computer software such as spreadsheet programs) can often handle larger values.

ⵖⵉⵏⵜ ⵜⵙⵏⵙⵉⵜⵉⵏ ⵢⴰⴹⵏ (ⵣⵓⵏⴷ ⵉⵖⴰⵡⴰⵙⵏ ⵏ ⵓⵎⵙⵙⵓⴷⵙ ⵣⵓⵏⴷ ⵉⵖⴰⵡⴰⵙⵏ ⵏ ⵉⵙⵎⵢⴰⵍⵍⴰⵢⵏ ⵏ ⵉⵙⵎⵎⴰⵍⵏ) ⴰⵙⵎⴽⴷ ⵏ ⵜⵉⵏⴷⵉⵜⵉⵏ ⵜⵉⵅⴰⵜⴰⵔⵉⵏ ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵡⴰⴷⴷⴰⴷⵏ.

Compared to the Pickover definition of the superfactorial, the hyperfactorial grows relatively slowly.

ⵙ ⵓⵣⵎⵣⴰⵣⴰⵍ ⵏ ⵓⵙⵉⵙⵙⵏ ⴱⵉⴽⵓⴼⵔ ⵏ ⵓⵎⵙⵡⵓⵔ ⴰⵎⴰⴳⵓⵔ, ⴷⴰ ⵉⵜⵜⵓⴳⵎ ⵓⵎⵙⵡⵓⵔ ⵏ ⵓⵙⵉⴳⵣ ⵏ ⵜⵉⵍⴰⵍ ⵙ ⵜⵉⵎⵎⵉⵙⵖⵜ.

There are, relatively speaking, no such simple solutions for factorials; no finite combination of sums, products, powers, exponential functions, or logarithms will suffice to express ; but it is possible to find a general formula for factorials using tools such as integrals and limits from calculus.

ⵙ ⵜⵎⵉⵙⵖⵜ ⵓⵔ ⵍⵍⵉⵏ ⵉⴼⵙⵙⴰⵢⵏ ⵓⵏⵣⵉⵍⵏ ⵉ ⵉⵎⴳⴳⵉⵜⵏ, ⵓⵔ ⵜⴳⵉ ⵜⵔⴰⴱⴱⵓⵜ ⵏ ⵡⴰⵜⵉⴳⵏ ⵜⵓⵥⵍⵉⵢⵜ ⵏⵖⴷ ⵉⵙⵏⴼⵍⵓⵍⵏ ⵏⵖⴷ ⵜⵉⵣⵎⴰⵔ ⵏⵖⴷ ⵜⵉⵎⵔⵙⵉⵏ ⵜⴰⴳⴳⴰⴹⵉⵏ ⵏⵖⴷ ⵍⵓⴳⴰⵔⵉⵜⵎ ⵙ ⵏⵔⴰ ⴰⴷ ⵖⵉⴼⵙ ⵏⵙⵉⵡⵍ, ⵎⴰⴽⴰ ⵏⵖⵢ ⴰⴷ ⵏⴰⴼ ⵜⴰⵍⵖⴰ ⵜⴰⵎⴰⵜⵜⵓⵜ ⵉ ⵓⵎⵙⴼⵓⴽⵜⵉ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵉⵙⴳⴳⵓⵔⵏ ⴰⵎⵎ ⵉⵎⵣⴷⴰⵢⵏ ⴷ ⵉⵡⵜⵜⴰ ⵉ ⵓⵙⵙⵉⵟⵏ ⴰⵎⵢⵓⴼ ⴷ ⴰⵎⵙⵎⴷ.

The integrals we have discussed so far involve transcendental functions, but the gamma function also arises from integrals of purely algebraic functions.

ⵜⵉⵎⵙⵎⴰⴷⵉⵏ ⵅⴼ ⵏⵎⵔⴰⵔⴰ ⴰⵡⴰⵍ ⴰⴷ ⴷⵖⵉ ⵓⵡⵉⵏⴷ ⵜⵉⵎⵔⵙⵉⵜⵉⵏ ⵢⴰⵜⵜⵓⵢⵏ, ⵎⴰⴽⴰ ⵜⴰⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ ⵜⵜⵡⴰⵙⴽⴰⵔ ⴰⵡⴷ ⵏⵜⵜⴰⵜ ⴳ ⵜⵎⵙⵎⴰⴷⵉⵏ ⵏ ⵜⵎⵔⵙⵜ ⵏ ⵍⵊⵉⴱⵔ ⵉⴷⵓⵙⵏ.

By taking limits, certain rational products with infinitely many factors can be evaluated in terms of the gamma function as well.

ⵙⴳ ⵢⵉⵙⵢ ⵏ ⵉⵡⵜⵜⴰ, ⵏⵖⵢ ⴰⴷ ⵏⴳ ⴰⵙⵜⴰⵍ ⵉ ⵉⵜⵙⵏ ⵉⵙⵏⴼⵍⵓⵍⵏ ⵓⵎⴳⵉⵏ, ⴳ ⵉⵍⵍⴰ ⵉⵎⵉⴹ ⵡⴰⵔⵜⵎⵉ ⵏ ⵉⴼⴰⵜⵔⵏ ⵙⴳ ⵜⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ.

Its history, notably documented by Philip J. Davis in an article that won him the 1963 Chauvenet Prize, reflects many of the major developments within mathematics since the 18th century.

ⴷⴰ ⵢⴰⴽⴽⴰ ⵓⵎⵣⵔⵓⵢ ⵏⵏⵙ ⵏⵏⴰ ⵢⴰⵔⵓ ⴼⵉⵍⵉⴱ ⵊ. ⴷⵉⴼⵉⵙ, ⴳ ⵓⵎⴳⵔⴰⴷ ⵢⵓⵡⵉⵏ ⵜⴰⵙⵎⵖⵔⵜ ⵏ ⵛⵓⴼⵉⵏⵉ ⴱⵔⵉⵣ ⴰⵙⴳⴳⵯⴰⵙ ⵏ1963, ⴽⵉⴳⴰⵏ ⵏ ⵉⵙⴱⵓⵖⵍⵓⵜⵏ ⵉⴷⵙⵍⴰⵏ ⴳ ⵜⵓⵙⵏⴰⴽⵜ ⵙⴳ ⵓⵙⴰⵜⵓ 18.

Instead of finding a specialized proof for each formula, it would be desirable to have a general method of identifying the gamma function.

ⵅⴼ ⴰⴷ ⵉⵜⵢⴰⴼ ⵓⵏⵎⵎⴰⵍ ⵉⵥⵍⵉⵏ ⵉ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ, ⵢⵓⴼ ⴰⴷ ⴷⴰⵔⴽ ⵜⵉⵍⵉ ⵜⴱⵔⵉⴷⵜ ⵜⴰⵎⴰⵜⵜⵓⵜ ⵏ ⵢⵉⴼ ⵏ ⵜⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ.

However, the gamma function does not appear to satisfy any simple differential equation.

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ ⵜⴰⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ ⴷⴰ ⵜⵎⵙⴰⵙⴰ ⴷ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴰⵎⵢⴰⴼⵜ ⵜⵓⵏⵣⵉⵍⵜ.

The Bohr–Mollerup theorem is useful because it is relatively easy to prove logarithmic convexity for any of the different formulas used to define the gamma function.

ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏ ⴱⵓⵔⵀ-ⵎⵓⵍⵉⵔⵓ ⵜⵓⴱⵖⵉⵔⵜ, ⴰⵛⴽⵓ ⵉⵡⵀⵏ ⴰⴷ ⵉⵜⵜⵓⵡⵔ ⵡⵓⴹⵓ ⴰⵍⵓⴳⴰⵔⵉⵜⵎⵉ ⵉ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵉⵎⵣⴰⵔⴰⵢⵏ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⵎⴰⵔ ⴰⴷ ⵏⵙⵜⵉ ⵜⴰⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ.

As electronic computers became available for the production of tables in the 1950s, several extensive tables for the complex gamma function were published to meet the demand, including a table accurate to 12 decimal places from the U.S. National Bureau of Standards.

ⴷⴷⴰ ⴳ ⴳⴰⵏ ⵉⵏⴳⵎⴰⵎⵏ ⵏ ⵓⵎⵙⵙⵓⴷⵙ ⵉⵍⵉⴽⵜⵔⵓⵏⵉ ⵡⵉⵏ ⵓⵙⵏⴼⵍ ⵏ ⵉⵎⵢⴰⵍⵍⴰⵢⵏ ⴳ 1950, ⵜⵜⵢⴰⴼⵙⴰⵔⵏ ⴽⵉⴳⴰⵏ ⵏ ⵉⵙⵏⵎⴰⵍⵍⴰⵢⵏ ⵉⵙⵎⴰⵏ ⵜⴰⵎⵔⵙⵜ ⵏ ⴳⴰⵎⴰ ⵉⵛⵇⵇⴰⵏ ⵎⴰⵔ ⴰⴷ ⵜⵜⵓⴳ ⵜⵓⵜⵜⵔⴰ, ⴳ ⵢⴰⵎⵓ ⵢⴰⵏ ⵓⵙⵏⵎⴰⵍⵍⴰⵢ ⴰⵎⵖⴷⴰⵏ ⵏ 12 ⵏ ⵜⵓⵣⵓⵏⵜ ⵜⴰⵎⵔⴰⵡⵜ, ⵙⴳ ⵖⵓⵔ ⵓⵙⵉⵔⴰ ⴰⵏⴰⵡⴰⵢ ⴰⵏⴰⵎⵓⵔ ⴰⵎⵉⵔⵉⴽⴰⵏⵉ.

In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula.

ⴳ ⵜⵓⵙⵏⴰ, ⵜⴰⴳⴷⴰⵣⴰⵍⵜ ⵜⴳⴰ ⵜⴰⴱⵔⵉⴷⵜ ⵉⵙⴰⵏⴼⵏ ⵏ ⵓⵙⵉⵡⵍ ⵖⴼ ⵉⵏⵖⵎⵉⵙⵏ ⵙ ⵜⴰⵍⵖⴰ ⵜⴰⵎⴰⵜⴰⵔⵜ, ⵉⵎⴽ ⵉⵍⵍⴰⵏ ⴳ ⵜⴰⵍⵖⴰ ⵜⵓⵙⵏⴰⴽⵜ ⴷ ⵜⴰⵍⵖⴰ ⵏ ⵛⵉⵎⵉ.

In mathematics, a formula generally refers to an identity which equates one mathematical expression to another, with the most important ones being mathematical theorems.

ⴳ ⵜⵓⵙⵏⴰⴽⵜ ⴷ ⵜⵎⵎⴰⵍ ⵜⴰⵍⵖⴰ ⵙ ⵓⵎⴰⵜⴰ ⵜⴰⵎⴰⴳⵉⵜ ⵉⴳⴰⵏ ⴰⵙⵉⵡⵍ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵢⴰⴹⵏ, ⵙⵜⴰⵡⵀⵎⵎⴰⵏⵜ ⴷⵉⴽⵙ ⵜⵎⴰⴳⵓⵏⵉⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ.

This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate formulas which are larger and more complex.

ⵜⴰⵎⵢⴰⵇⵇⴰⵏⵜ ⴰⴷ ⵎⵇⵇⴰⵔ ⵓⵔ ⵡⴰⵍⴰ ⵜⵙⵜⴰⵡⵀⵎⵎⴰ ⴳ ⵜⴰⵍⵖⴰ ⵜⴰⵎⵏⵣⴰⵍⵜ, ⴷⴰ ⵜⵜⵉⵏⵉ ⵉⵙ ⵉⵎⵓⵙⵏⴰⵡⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵖⵉⵏ ⴰⴷ ⵜⵎⵛⴰⵛⴽⴰⵏ ⵙ ⴽⵉⴳⴰⵏ ⵏ ⵓⵙⵔⴱⵉ ⴰⴽⴷ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵜⵉⵅⴰⵜⴰⵔⵉⵏ ⴷ ⵜⵉⵏⵏⴰ ⵡⴰⵍⴰ ⵉⵛⵇⵇⴰⵏ.

For example, H2O is the chemical formula for water, specifying that each molecule consists of two hydrogen (H) atoms and one oxygen (O) atom.

ⵙ ⵓⵎⴷⵢⴰ, H2O ⵜⴳⴰ ⵜⴰⵍⵖⴰ ⵜⴰⴽⵉⵎⵉⵢⵜ ⵏ ⵡⴰⵎⴰⵏ ⵏⵏⴰ ⵉⵙⵜⵜⵉⵏ ⵉⴷ ⴽⵓ ⵉⵎⵉⴽ ⴷⵉⴽⵙ ⵙⵏⴰⵜ ⵜⴱⵍⴽⵉⵎⵉⵏ ⵏ ⵀⵉⴷⵔⵓⵊⵉⵏ (H) ⴷ ⵜⴱⵍⴽⵉⵎⵜ ⵏ ⵓⴽⵙⵉⵊⵉⵏ (O).

In empirical formulas, these proportions begin with a key element and then assign numbers of atoms of the other elements in the compound—as ratios to the key element.

ⴳ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵜⵉⵔⵎⵉⵜ, ⴷⴰ ⵜⵜⵓⵙⵏⵜⴰⵢⵏⵜ ⵜⵙⵖⴰⵍⵉⵏ ⴰⴷ ⵙ ⵓⴷⵕⴹⵉⵚ ⴰⴷⵙⵍⴰⵏ ⴷ ⴰⵔ ⵜⵙⵜⵜⵉⵢ ⵎⵏⵏⴰⵡⵜ ⵏ ⵜⴱⵍⴽⵉⵎⵉⵏ ⵏ ⵉⴼⵕⴹⵉⵚⵏ ⵢⴰⴹⵏ ⴳ ⵜⴰⵏⴰⵡⵜ, ⵉⴳⴰⵏ ⵉⵙⵖⴰⵍⵏ ⵏ ⵓⴼⵕⴹⵉⵚ ⴰⴷⵙⵍⴰⵏ.

Some types of ionic compounds, however, cannot be written as empirical formulas which contains only the whole numbers.

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⵓⵔ ⵏⵣⴹⴰⵕ ⴷ ⵏⵓⵔⵓ ⵉⵜⵙⵏ ⵡⴰⵏⴰⵡⵏ ⵏ ⵜⵓⴷⴷⵉⵙⵉⵏ ⵜⴰⵢⵢⵓⵏⵉⵏ ⴰⴷ ⴳⵉⵏⵜ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵜⵉⵔⵎⵉⵜ ⴳ ⵍⵍⴰⵏ ⵉⵎⴹⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ ⴷⴰⵢ.

There are several types of these formulas, including molecular formulas and condensed formulas.

ⵍⵍⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵡⴰⵏⴰⵡⵏ ⴳ ⵜⴰⵍⵖⵉⵡⵉⵏ ⴰⴷ ⴳ ⵜⴰⵎⵓ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵜⵉⴳⵣⵣⵓⵎⵉⵏ ⴷ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵜⴰⵏⵥⵥⵉⵡⵉⵏ.

Functions were originally the idealization of how a varying quantity depends on another quantity.

ⵜⵉⵡⵓⵔⵉⵡⵉⵏ ⴳ ⵓⵥⵓⵕ ⵏⵏⵙⵏⵜ ⴳⴰⵏⵜ ⵜⴰⵙⴼⵉⵍⴰⵡⵜ ⵏ ⵎⴰⵎⵏⴽ ⵙⵏⵣⵖⵣⴰⵏ ⵜⴰⵎⴰⴽⵜⴰ ⵜⴰⵎⵙⵏⴼⵍⵜ ⵖⴼ ⵜⵎⴰⴽⵜⴰ ⵏⵏⵉⴹⵏ.

"This definition of ""graph"" refers to a set of pairs of objects."

ⴷⴰ ⵉⵎⵎⴰⵍ ⵓⵙⵉⵙⵙⵏ ⵏ “ⵓⵏⵓⵖ ⴰⵙⵏⵎⵎⴰⵍ” ⴰⴷ, ⵖⵔ ⵜⵔⴰⴱⴱⵓⵜ ⵏ ⵉⵏⴰⵔⴰⴳⵏ ⵉⵎⵖⵏⴰⵡⵏ.

When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.

ⴰⴷⴷⴰⵢ ⵉⴳ ⵢⵉⴳⵔ ⴷ ⵢⵉⴳⵔ ⵉⵛⵛⴰⵔⵏ ⵜⴰⵔⴱⵉⵄⵜ ⵏ ⵉⵎⴹⴰⵏ ⵏ ⵜⵉⴷⵜ, ⵉⵖⵢ ⴰⴷ ⵉⴳ ⵓⵏⴰⵔⴰⴳ ⴳ ⵉⵏⴰⵔⴰⴳⵏ ⴰⴷ ⵜⵉⵎⵉⵜⴰⵔ ⵜⵉⴷⵉⴽⴰⵔⵜⵉⵢⵢⵉⵏ ⵏ ⵜⵏⵇⵇⵉⴹⵜ ⴳ ⵓⵖⴰⵡⴰⵙ.

Occasionally, it may be identified with the function, but this hides the usual interpretation of a function as a process.

ⴳ ⵜⵉⵣⵉ ⵖⵔ ⵜⴰⵢⴹ, ⵉⵖ ⴰⴷ ⵜⵜⵓⵙⵜⵢ ⴰⴽⴷ ⵜⵡⵓⵔⵉ, ⵎⴰⴽⴰ ⴰⵢⴰ ⴷⴰ ⵉⴷⴷⴰⵍ ⴰⴼⵙⵙⴰⵢ ⵢⴰⴷ ⵉⵍⵍⴰⵏ ⵉ ⵜⵡⵓⵔⵉ ⴰⵎⵎ ⵜⵉⵣⵉⴳⵣⵜ.

A map can have any set as its codomain, while, in some contexts, typically in older books, the codomain of a function is specifically the set of real or complex numbers.

ⵉⵖ ⴰⴷ ⵢⵉⵍⵉ ⴳ ⵜⴽⴰⵕⴹⴰ ⴽⴰ ⵉⴳⴰⵜ ⵜⴰⵔⴰⴱⴱⵓⵜ ⴰⴼⵓⵖⴰⵍ ⴰⵎⴰⵜⴰⵔ ⵏⵏⵙ, ⵎⴰⴽⴰ ⴳ ⵉⵜⵙⵏ ⵉⵙⴰⵜⵉⵍⵏ, ⵡⴰⵍⴰ ⴳ ⵉⴷⵍⵉⵙⵏ ⵉⵇⴱⵓⵔⵏ, ⴷⴰ ⵉⵜⴳⴳⴰ ⵢⵉⴳⵔ ⵜⴰⵎⴰⵜⴰⵔⵜ ⵏ ⵜⵡⵓⵔⵉ, ⵏⵜⵜⴰ ⵏⵏⵉⴽ ⵙ ⵓⵥⵍⴰⵢ; ⵜⴰⵔⴰⴱⴱⵓⵜ ⵏ ⵉⵎⴹⴰⵏ ⵜⵉⴷⵜ ⵏⵖⴷ ⵓⴷⴷⵉⵙⵏ.

Another common example is the error function.

ⴰⵎⴷⵢⴰ ⵢⴰⴷⵏ ⵉⵜⵢⴰⴼⵙⴰⵔⵏ; ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⵉⵣⴳⵍⵏ.

Power series can be used to define functions on the domain in which they converge.

ⵉⵖⵢ ⴰⴷ ⵏⵙⵡⵓⵔⵉ ⵙ ⵜⴳⴼⴼⵓⵔⵜ ⵏ ⵜⵣⴹⴰⵕⵜ ⵎⴰⵔ ⴰⴷ ⵏⵙⵜⵉ ⵜⵉⵡⵓⵔⵉⵡⵉⵏ ⴳ ⵢⵉⴳⵔ ⴳ ⵎⵎⵍⵎⴰⵍⴰⵏⵜ.

Then, the power series can be used to enlarge the domain of the function.

ⴹⴰⵕⵜ ⵓⵢⵏⵏⴰⵖ ⵏⵖⵢ ⴰⴷ ⵏⵙⵡⵓⵔⵉ ⵙ ⵜⴳⴼⴼⵓⵔⵜ ⵏ ⵜⵣⴹⴰⵕⵜ ⵎⴰⵔ ⴰⴷ ⵏⵙⵅⵉⵜⵔ ⵉⴳⵔ ⵏ ⵜⵡⵓⵔⵉ.

Parts of this may create a plot that represents (parts of) the function.

ⵉⵖⵢ ⴰⴷ ⴰⵡⵉⵏ ⵉⴳⵣⵣⵓⵎⵏ ⵙⴳ ⵓⵢⴰ ⵖⵔ ⵓⵙⴽⴰⵔ ⵏ ⵓⵚⵟⵟⴰ ⵉⵙⵎⴷⵢⴰⵏ (ⵜⴰⴳⵣⵣⵓⵎⵜ ⵙⴳ) ⵙ ⵜⵏⴰⵎⴽⴰⵏⵜ.

This is the canonical factorization of .

ⵡⴰⴷ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⴼⴰⵔⵙ ⴰⴷⵙⵍⴰⵏ ⵏ.

At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth.

ⴳ ⵜⵉⵣⵉ ⵏⵏⴰⵖ, ⵜⴰⵏⵏⴰⵢⵜ ⵖⴼ ⵜⵡⵓⵔⵉⵡⵉⵏ ⵉⵍⴰⵏ ⴰⵜⵉⴳ ⵏ ⵜⵉⴷⵜ ⵏ ⵓⵎⵙⵏⴼⵍ ⵏ ⵜⵉⴷⵜ, ⴷ ⵉⵇⵏⴻⵏ ⴰⴷ ⴰⴽⴽⵯ ⴳⵉⵏⵜ ⵜⵏⴰⵎⴽⴰⵜⵉⵏ ⵜⴰⴳⴼⴼⵓⵔⵜ.

Functions are now used throughout all areas of mathematics.

ⴷⴰ ⵜⵜⵓⵙⵎⵔⴰⵙⵏⵜ ⵜⵏⴰⵎⴽⴰⵜⵉⵏ ⴷⵖⵉ ⴳ ⵢⵉⴳⵔⴰⵏ ⵎⴰⵕⵕⴰ ⵏ ⵜⵓⵙⵏⴰⴽⵜ.

This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic.

ⵜⴰⴷⵖ ⵜⴰⴱⵔⵉⴷⵜ ⵏ ⵓⵙⵉⵙⵙⵏ ⵙ ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⵜⴰⵎⴽⵕⴰⴷⵜ ⵜⴰⵎⴰⵖⵓⵍⵜ, ⵙⴳ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵜⴰⵎⴽⵕⴰⴹⵜ, ⴷ ⴷⴰ ⵜⴳⴳⴰ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵜⴰⵎⴽⵕⴰⴹⵜ ⵜⴰⵎⵙⵙⴰⵔⵜⵓⵜ.

Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions.

ⵜⴰⴱⵖⵓⵔⵜ ⵏ ⵓⵙⵉⵙⵙⵏ ⵏ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵎⵉ ⴳⴳⵓⴷⵉⵏⵜ ⵜⵉⵏⴷⵉⵜⵉⵏ, ⴼⴰⵡⵏⵜ ⴽⵉⴳⴰⵏ ⵉⴳ ⴷⴰ ⵏⵙⴽⵙⵉⵡ ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⵜⵓⴷⴷⵉⵙⵜ, ⴷ ⴰⵔ ⵜⴳⴳⴰ ⵓⵍⴰ ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⵜⴰⴼⴰⵔⵙⵜ.

Such a function is called the principal value of the function.

ⵜⵢⴰⵙⵙⵏ ⵜⵏⴰⵎⴽⴰⵏⵜ ⴰⴷ ⵙ ⵡⴰⵜⵉⴳ ⴰⴷⵙⵍⴰⵏ ⵏ ⵜⵡⵓⵔⵉ.

Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions.

ⴰⵙⵖⵉⵡⵙ ⵏ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵉⴳⴰ ⴰⵎⵢⴰ ⵏ ⵓⵙⵖⵉⵡⵙ ⵏⵏⴰ ⴳ ⵜⵍⵍⴰ ⵜⵓⵙⴽⴰ ⵏ ⵉⵖⴰⵡⴰⵙⵏ ⵙ ⵓⵙⵙⵎⵔⵙ ⵏ ⵉⵎⴳⴳⵉⵜⵏ ⴰⵢⵢⴰⵡⵏ ⴷⴰⵢ, ⵏⵏⴰ ⵉⵜⴳⴳⴰⵏ ⵣⵓⵏⴷ ⵜⵉⵡⵓⵔⵉⵡⵉⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ.

"Except for computer-language terminology, ""function"" has the usual mathematical meaning in computer science."

ⵙ ⵓⵙⵍⵉⴷ ⵏ ⵢⵉⵔⵎⵏ ⵏ ⵜⵓⵜⵍⴰⵢⵜ ⵏ ⵓⵎⵙⵙⵓⴷⵙ ⵉⵍⵍⴰ ⵖⵓⵔ ⵜⵏⴰⵎⴽⴰⵏⵜ ⵓⵏⴰⵎⴽ ⵏ ⵜⵓⵙⵏⴰⴽⵜ ⵏⵏⴰ ⵢⴰⴷ ⵉⵍⵍⴰⵏ ⴳ ⵜⵓⵙⵙⵏⴰⵎⵙⵙⵓⴷⵙ.

Terms are manipulated through some rules, (the -equivalence, the -reduction, and the -conversion), which are the axioms of the theory and may be interpreted as rules of computation.

ⴷⴰ ⵉⵜⵜⵓⵙⵓⵔⴰⵔ ⵙ ⵜⴳⵓⵔⵉⵡⵉⵏ ⵙⴳ ⵉⵜⵙⵏ ⵉⵍⴳⴰⵎⵏ (ⵜⴰⴳⴷⴰⵣⴰⵍⵜ, ⴰⵣⵓⴳⵣ, ⴰⵙⵏⴼⵍ), ⴷ ⵏⵉⵜⵏⵉ ⵉⵏⴼⵍⴰⵍⴰⵢⵏ ⵏ ⵜⵎⴰⴳⵓⵏⵜ ⴷ ⵏⵖⵢ ⴰⴷ ⵏⵙⵙⴼⵔⵓ ⵉⵙ ⵜⴳⴰ ⵉⵍⴳⴰⵎⵏ ⵏ ⵓⵙⵙⵉⵟⵏ.

Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel in the 16th century.

ⵉⵙⵙⵎⵔⵙ ⵏⵉⴽⵓⵍⴰⵚ ⵜⵛⵓⴽⵉ ⵜⴰⵍⵖⴰ ⵙⴳ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵏ ⵓⵣⵎⵎⴻⵎ ⵏ ⵜⵙⵉⵍⴰ ⴳ ⵓⵙⴰⵜⵓ ⵡⵉⵙⵙ 15, ⴷ ⵉⵙⵡⵓⵔⵉ ⵉⵙ ⴹⴰⵕⴰⵙ ⵀⵉⵏⵔⵉⴽⵓⵙ ⴳⵔⴰⵎⴰⵜⵓⵙ ⴷ ⵎⵉⵛⵉⵍ ⵙⵜⵉⴼⵍ ⴳ ⵓⵙⴰⵙⵓ ⵡⵉⵙⵙ 16 .

Thus they would write polynomials, for example, as .

ⵉⵎⴽⵉ ⴰⵙ ⵜⵜⵓⵔⵓⵏ ⵜⵉⵎⴳⴳⵓⴷⵢⵉⵏ ⵏ ⵉⵡⵜⵜⴰ, ⵙ ⵓⵎⴷⵢⴰ, ⵣⵓⵏⴷ.

The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents.

ⵜⴰⵢⴰⴼⵓⵜ ⵜⴳⴰ ⴰⵀⴰ ⵓⵟⵟⵓⵏ ⵓⵎⵏⵉⴳ ⵏ ⵜⵉⴷⵜ, ⴳⵉⵎⵉⵏⵜ ⵜⵎⴰⴳⵉⵜⵉⵏ ⴷ ⵉⵏⴰⵎⴰⵥⵏ ⵉⵎⵙⵙⴼⵔⵓⵜⵏ ⴰⴼⵍⵍⴰ, ⵉⴷⵙⵍⴰⵏ ⵉⵎⴷⴷⴰⴷⵏ, ⵉⵎⴷⴷⴰⴷⵏ ⴰⴽⴷ ⵉⵙⵓⵙⵙⵏ ⴰⴷ ⵏ ⵉⴷⵙⵍⴰⵏ ⵏ ⵜⵉⴷⵜ.

This function equals the usual th root for positive real radicands.

ⵜⴰⵏⴰⵎⴽⴰⵏⵜ ⴰⴷ ⴰⵢⴷ ⵉⴳⴰⵏ “ⴰⵥⵓⵕ th”, ⴳ ⴰⵀⴰ ⵍⵍⴰⵏ ⵉⵣⵖⵕⴰⵏ ⵓⵎⵏⵉⴳⵏ ⵏ ⵜⵉⴷⵜ.

This is the starting point of the mathematical theory of semigroups.

ⵜⴰⴷⵖ ⴷ ⵜⴰⵏⵇⵇⵉⴹⵜ ⵜⴰⵎⵣⵡⴰⵔⵓⵜ ⵉ ⵜⵎⴰⴳⵓⵏⵜ ⵜⵓⵙⵏⴰⴽⵜ ⵉ ⵜⵣⵏⴰⵔⴰⴱⴱⵓⵜ.

We can again replace the set N with a cardinal number n to get Vn, although without choosing a specific standard set with cardinality n, this is defined only up to isomorphism.

ⵜⵉⴽⵍⵜ ⵢⴰⴹⵏ ⵏⵖⵢ ⴰⴷ ⵏⵙⵏⴼⵍ ⵜⴰⵔⴱⵉⵄⵜ N ⵙ ⵡⵓⵟⵟⵓⵏ ⴰⴷⵙⵍⴰⵏ n, ⵎⴰⵔ ⴰⴷ ⴰⵖⴷ ⵜⴽ Vn, ⵡⴰⵅⵅⴰ ⵓⵔ ⵏⵙⵜⵉⵢ ⴽⴰⵏ ⵜⵔⴰⴱⴱⵓⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ ⴷ ⵓⵥⵓⵕ n ⴷⴰ ⵉⵜⵜⵓⵢⴰⵙⵙⴰⵏ ⵓⵢⴰ ⴰⵔ ⴰⵎⵜⵔⴰⵏ.

Nicolas Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.

ⵏⵉⴽⵓⵍⴰⵚ ⴱⵓⵔⴱⴰⴽⵉ, ⵉⴼⵕⴹⵉⵚⵏ ⵏ ⵜⵓⵙⵏⴰⴽⵜ, ⵜⴰⵎⴰⴳⵓⵏⵜ ⵏ ⵜⵔⵓⴱⴱⴰ ⵙⴱⵔⵉⵏⵊ-ⴼⵉⵔⵍⴰⴳ, 2004, III.§3.5.

Iterating tetration leads to another operation, and so on, a concept named hyperoperation.

ⴷⴰ ⵉⵜⵜⴰⵡⵢ ⵢⵉⵍⵙ ⵏ ⵜⵙⵏⴰⵡⴰⵢⵜ ⵖⵔ ⵜⵎⵀⵍⵜ ⵏⵏⵉⴹⵏ, ⵉⵎⴽⵉ, ⴰⵔⵎⵎⵓⵙ ⵉⴳⴰⵏ ⵜⴰⵎⵔⵏⵉⵡⵜ ⴳ ⵜⵎⵀⵍⵜ.

In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable.

ⴳ ⵜⵙⵖⵍⵉⵏ ⵏ ⵜⵙⵏⵙⵉ, ⵜⵙⵎⴷⵢⴰ ⵜⵏⴰⵎⴽⴰⵏⵜⵉⵏ ⵏ ⵜⵙⵉⵍⴰ, ⴰⵙⵖⵏ ⵖ ⵢⴰⴽⴽⴰ ⵓⵙⵏⴼⵍ ⴰⵎⵣⴷⴰⵢ ⴳ ⵓⵙⵏⴼⵍ ⴰⵎⵥⵍⴰⵢ; ⴰⵙⵏⴼⵍ ⴰⵎⴰⵙⴰⵖ ( ⵜⵉⴳⵎⵉⴹⵉ ⵏ ⵜⵔⵏⵓⵜ ⵏⵖⴷ ⵓⴽⵓⵙ ), ⴳ ⵓⵎⵙⵏⴼⵍ ⴰⵎⴹⴼⴰⵕ.

This function property leads to exponential growth or exponential decay.

ⴷⴰ ⵉⵜⵜⴰⵡⵢ ⵓⵏⴰⵎⴰⵥ ⵏ ⵜⵎⵔⵙⵜ ⴰⴷ ⵖⵔ ⵜⵉⴳⵎⵉ ⵏ ⵜⵙⵉⵍⴰ ⵏⵖⴷ ⴰⵍⴰⴽⴰⵢ ⵏ ⵜⵙⵉⵍⴰ.

Similarly, the composition of onto (surjective) functions is always onto.

ⵙ ⵓⵔⵡⴰⵙ, ⴰⵙⵎⵓⵜⵜⴳ ⵏ ⵜⵎⵔⵙⵉⵏ ⵜⴰⵢⵢⴰⵡⵉⵏ ( ⵜⵉⵙⵡⵉⵏⴳⵎⴰⵏⵉⵏ) ⴷⴰ ⴰⵀⴰ ⵉⵜⵜⵉⵍⵉ ⴳ ⵜⵡⵓⵔⵉ.

Then one can form chains of transformations composed together, such as .

ⴷ ⵉⵖⵢ ⵓⴼⴳⴰⵏ ⴰⴷ ⵉⵙⵙⴽⵔ ⵜⵉⴳⴼⴼⵓⵔⵉⵏ ⵜⵉⵎⵙⵏⴼⵍⵉⵏ ⵉⵎⴰⵏⴻⵏ, ⵣⵓⵏⴷ.

This alternative notation is called postfix notation.

ⴰⵙⵏⴰⵎⴰⵜⴰⵔ ⵉⵎⴽⴽⵉⵙⵉ ⴰⵢⴷ ⵉⴳⴰⵏ ⴰⵣⵎⵎⴻⵎ ⵏ ⴱⵓⵙⵜⴼⵉⴽⵙ.

The category of sets with functions as morphisms is the prototypical category.

ⵜⴰⴳⵔⵔⵓⵎⴰ ⵏ ⵜⵔⵓⴱⴱⴰ ⴳ ⵍⵍⴰⵏⵜ ⵜⵎⵔⵙⵜⵉⵏ ⵣⵓⵏ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵉⴳⴰⵏ ⵜⴰⴳⵔⵔⵓⵎⴰ ⵏ ⵓⵎⴷⵢⴰ ⴰⵎⵏⵣⵓ.

For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example).

ⵙ ⵓⵎⴷⵢⴰ, ⴷⵉⵙⵉⴱⵍ (ⴷⴱ), ⵜⴳⴰ ⵜⴰⴳⵣⵣⵓⵎⵜ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴰⴷ ⵜⵙⵉⵡⵍ ⵖⴼ ⴰⵙⵖⵍ ⴽⵉⵍⵓⴳⴰⵔⵉⵜⵎⴰⵜ, ⴳ ⴽⵉⴳⴰⵏ ⵏ ⵜⵉⴽⴽⴰⵍ ⵉ ⵜⵣⵎⵔⵜ ⵏ ⵜⵎⴰⵜⴰⵔⵜ ⴷ ⵓⴽⵜⵜⵓⵔ ( ⵉⴷⵔ ⵏ ⵉⵎⵙⵍⵉ ⴰⵢⴷ ⵡⴰⵍⴰ ⵉⵍⵍⴰⵏ).

They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.

ⴷⴰ ⵜⵜⴰⵡⵙ ⴳ ⵓⵙⵏⵓⵎⵍ ⵏ ⵓⴳⵍⵓⴳⵍ ⵏ ⵉⵣⵎⴰⵣ ⵏ ⵓⵥⴰⵡⴰⵏ, ⴰⵔⴷ ⵜⴼⴼⴻⵖ ⴳ ⵜⴰⵍⵖⵉⵡⵉⵏ ⵉⵙⵙⵉⵟⵉⵏ ⵉⵎⴹⴰⵏ ⵉⵎⵣⵡⵓⵔⴰ ⵏⵖⴷ ⵜⴰⵏⵎⵉⵍⵉⵜ ⵏ ⵉⴼⵕⴹⵉⵚⵏ, ⴷ ⴰⵔ ⵜⵙⵙⵍⵎⴰⴷ ⵉⵜⵙⵏ ⵉⵎⴷⵢⴰⵜⵏ ⴳ ⵜⵉⴽⵍⵉⵙⵏⵜ ⵜⴰⴼⵉⵣⵉⴽⵜ, ⴷ ⵜⵖⵢ ⴰⴷ ⵜⴰⵡⵙ ⴳ ⵓⵙⵙⵉⵟⵏ ⵏ ⵓⵏⴱⴽⴰⴹ.

The next integer is 4, which is the number of digits of 1430.

ⵉⵎⵉⴹ ⴰⵎⴷⴷⴰⴷ ⴰⴷ ⵉⴳⴰ 4, ⴷ ⵏⵜⵜⴰ ⴰⵢⴳⴰⵏ ⵉⵎⵉⴹ ⵏ ⵜⵓⵣⵓⵏⵉⵏ 1430.

Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600.

ⴷⴰⵜ ⵏ ⴰⴷ ⵉⵜⵜⵓⵀⵢⵢⴰ “ⵏⴰⴱⵉⵔ”, ⵍⵍⴰⵏ ⵜⵉⴽⵏⵉⵜⵉⵏ ⵏⵏⵉⴹⵏ ⵉⵍⴰⵏ ⵉⵖⵓⴼⴰⵍⵏ ⴰⵎ ⵡⵉ, ⵣⵓⵏⴷ “ⵜⵉⵎⵍⵙⵉⵜ”, ⵏⵖⴷ ⴰⵙⵙⵎⵔⵙ ⵏ ⵉⵙⵎⵢⴰⵍⵍⴰⵏⴻⵏ ⵏ ⵜⵉⴹⴼⵓⵔⵜ, ⵏⵏⴰ ⵉⵙⴱⵓⵖⵍⵍⴰ ⵊⵓⵙⵜ ⴱⵓⵔⵊⵉ ⴳ ⵓⵖⵓⴼⴰⵍ ⴰⴱⴰⵔⴰⵡ ⴰⵜⵜⴰⵢⵏ ⵏ ⵓⵙⴳⴳⵯⴰⵙ ⵏ1600.

Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the “order of a number”.

ⵉⴳ ⴷⴰ ⵏⵙⴰⵡⴰⵍ ⵖⴼ ⵡⵓⵟⵟⵓⵏ ⵉⵔⴰⵏ ⴽⵉⴳⴰⵏ ⵏ ⵡⵓⵟⵟⵓⵏ, ⵜⴳⴰ ⵜⴰⵎⴰⵜⴰⵔⵜ ⵏ ⵓⵏⵎⵉⵍⴰ ⵖⵔ ⵍⵓⴳⴰⵔⵉⵜⵎ ⴰⵎⵛⵛⵓⵔ,ⵉⵎⴽ ⵣⴰⵕⵙ ⵉⵏⵄⵜ ⴰⵔⵅⵎⵉⴷ ⵉⴷ ⵏⵜⵜⴰ ⴰⵢⴷ ⵉⴳⴰⵏ “ⴰⵙⵙⵓⴷⵙ ⵏ ⵡⵓⵟⵟⵓⵏ”.

Such methods are called prosthaphaeresis.

ⴷⴰ ⴰⵙⵏ ⵜⵜⵉⵏⵉⵏⵜ ⵉ ⵜⴱⵔⵉⴷⵉⵏ ⴰⴷ; ⵜⵉⵏⴽⵔⴰⵎⵉⵏ.

For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor.

ⵙ ⵓⵎⴷⵢⴰ, ⴽⵓ ⵜⴰⵎⵕⵚⵉⵜ ⴳ ⵉⴼⵔⴽⵉ ⵏ ⵏⵓⵜⵉⵍⵓⵙ, ⵜⴳⴰ ⵜⵓⵏⵖⵉⵍⵜ ⵜⴰⵏⵎⵉⵍⵉⵜ ⵏ ⵜⵎⵕⵚⵉⵜ ⵥⴰⵕⵙ ⵉⵍⵍⴰⵏ ⵜⵜⵓⵙⵖⴰⵍ ⵙ ⴰⵎⵎⴰⴽ ⵉⵣⵣⴳⴰⵏ.

Logarithms are also linked to self-similarity.

ⵍⵓⴳⴰⵔⵉⵜⵎ ⵍⴰⵏⵜ ⴰⵡⴷ ⵜⴰⵙⵖⵏ ⵙ ⴰⵎⵔⵡⵙ ⴰⵏⵉⵎⴰⵏ.

It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in acoustics, and the absorbance of light in the fields of spectrometry and optics.

ⴷⴰ ⵉⵜⵜⵓⵙⵎⵔⴰⵙ ⵎⴰⵔ ⴰⴷ ⵉⵙⵖⵍ ⵉⵙⵡⵉⵔⵏ ⵏ ⵜⵣⵎⵔⵜ ⴳ ⵢⵉⵣⵏ ⵏ ⵜⵎⵓⵍⵉ ⵜⵉⵎⵥⵥⴰⵕⵓⵕⵉⵏ, ⴰⴼⴰⴷ ⵏⵙⵏⵓⵎⵍ ⵉⵙⵡⵉⵔⵏ ⵏ ⵜⵣⵎⵔⵜ ⵏ ⵉⵎⵙⵍⵉ ⴳ ⵜⵎⵙⵍⵉⵜⵉⵏ, ⴷ ⵓⵙⵙⵓⵎ ⵏ ⵜⵔⵉⵙⵉⵏⵜⵉ ⴳ ⵢⵉⴳⵔⴰⵏ ⵏ ⵓⵡⵍⴰⴼ ⴷ ⵉⵎⵡⴰⵍⴰⵏ.

Vinegar typically has a pH of about 3.

ⴷⴰ ⵡⴰⵍⴰ ⵉⵜⵜⵉⵍⵉ ⴳ ⵍⵅⵍⵍ ⴰⵔ ⴰⵜⵜⴰⵢⵏ 3 ⵏ ⵜⵙⴽⴼⴰⵍ ⵏ ⵜⴰⵙⵎⵎⵉ.

"This ""law"", however, is less realistic than more recent models, such as Stevens's power law.)"

ⵡⴰⵅⵅⴰ ⵀⴰⴽⴽⴰⴽ, ⴰⵣⵔⴼ ⴰⴷ ⵓⵔ ⵉⴳⵉ ⴰⵏⵉⵍⴰⵡ ⴰⵎ ⵉⵎⴷⵢⴰⵜⵏ ⵉⵜⵔⴰⵔⵏ, ⵣⵓⵏⴷ ⴰⵣⵔⴼ ⴰⵏⴱⴱⴰⴹ ⵏ ⵙⵜⵉⴼⵏⵙ.

When the logarithm of a random variable has a normal distribution, the variable is said to have a log-normal distribution.

ⵉⴳ ⵉⴳⴰ ⵍⵓⴳⴰⵔⵉⵜⵎ ⴰⵎⵙⵏⴼⵍ ⴰⵎⵅⴱⴱⴹ ⵏ ⵉⴱⵟⵟⵓ ⴰⵖⴰⵔⴰⵏ, ⴷⴰ ⵉⵜⵢⴰⵏⵏⴰ ⵉⴷ ⴰⵎⵙⵏⴼⵍ ⵖⵓⵔⵙ ⴰⴱⵟⵟⵓ ⴰⵍⵓⴳⴰⵔⵉⵜⵎ ⵓⵏⵣⵉⵍ.

For such a model, the likelihood function depends on at least one parameter that must be estimated.

ⴳ ⵣⵓⵏⴷ ⴰⵎⴷⵢⴰ ⴰⴷ, ⴷⴰ ⵜⴱⴷⴷⴰ ⵜⵎⵔⵙⵜ ⵏ ⵉⵖⵢ ⵖⴼ ⵢⵓⵡⵜ ⵜⵙⵖⵍⵜ, ⵙ ⵉⵍⴰⵇ ⴰⴷ ⵜⵜⵓⵡⵇⵇⵔ.

Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results.

ⵣⵓⵏⴷ, ⴷⴰ ⵜⴳⴳⴰ ⴰⵍⴳⵓⵔⵉⵜⵎ ⴰⵙⵓⴼⵖ ⵏ ⵓⵎⵎⴽⵛⵎ, ⵙ ⵡⵓⴼⵓⵖ ⵏ ⵜⵍⴳⴰⵎⵜ ⴷ ⵓⵔ ⵉⴼⴼⵉⵖⵏ ⵙ ⵜⴱⵔⵉⴷⵜ ⵏ ⵜⵍⴳⴰⵎⵜ ⵖⴼ ⵙⵉⵏ, ⴷ ⵡⵓⴼⵓⵖ ⵏⵏⵙ ⴷⴰⵜ ⵓⵙⵎⵓⵏ ⵏ ⵜⵢⴰⴼⵓⵜⵉⵏ.

Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system.

ⴷⴰ ⵙⵙⵎⵔⴰⵙⵏ ⵡⵉⵏⵏⴰ ⵉⵇⵔⴰⵏ ⵙ ⵍⵢⴰⴱⵓⵏⵓⴼ ⵍⵓⴳⴰⵔⵉⵜⵎ ⵎⴰⵔ ⴰⴷ ⵙⵙⵖⴰⵍⵏ ⵜⴰⵙⴽⴼⵍⵜ ⵏ ⵓⵔⵡⴰⵢ ⴳ ⵓⵏⴳⵔⴰⵡ ⴰⴷⵉⵏⴰⵎⵉⴽ.

The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length.

ⵏⵖ ⴰⴷ ⵏⴷⵍ ⴰⵎⴽⵕⴰⴹ ⵙⵢⵉⵔⴱⵉⵏⵙⴽⵉ ( ⴳ ⵜⵡⵍⴰⴼⵜ), ⵙ ⴽⵕⴰⴹⵜ ⵜⵓⵏⵖⵉⵍⵉⵏ, ⴽⵓ ⵢⵓⵡⵜ ⴷⵉⴽⵙⵏⵜ ⵙ ⵓⵣⴳⵏ ⵏⵜⴰⵖⵣⵉ ⴰⵥⵓⵕⴰⵏ.

Another example is the p-adic logarithm, the inverse function of the p-adic exponential.

ⴰⵎⴷⵢⴰ ⵏⵏⵉⴹⵏ ⵉⴳⴰⵜ p-adic ⵍⵓⴳⴰⵔⵉⵜⵎ ⵏ ⵜⵡⵓⵔⵉ ⵜⴰⵎⴰⵖⵓⵍⵜ ⵜⴰⵙⵉⵍⴰ ⵏ p-adic.

Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups.

ⵏⵖⵢ ⴰⴷ ⵏⴳ ⵜⴰⵎⵀⵍⵜ ⵏ ⵜⵙⵉⵍⴰ ⵙ ⵜⵣⵎⵔⵜ, ⵎⴰⴽⴰ ⴷⴰ ⵉⵜⵢⴰⵖⵉⵍ ⵉⵙ ⵉⵛⵇⵇⴰ ⵓⵙⵙⵉⵟⵏ ⵏ ⵍⵓⴳⴰⵔⵉⵜⵎ ⴰⵎⴱⴹⵉ ⴳ ⵉⵜⵙⵏⵜ ⵜⵔⵓⴱⴱⴰ.

Square roots of negative numbers can be discussed within the framework of complex numbers.

ⵉⵖⵢ ⵓⵎⵔⴰⵔⴰ ⵏ ⵡⴰⵡⴰⵍ ⵖⴼ ⵉⵥⵓⵕⴰⵏ ⵉⵎⴽⴽⵓⵥⵏ ⵏ ⵉⵎⴹⵏ ⵓⵣⴷⵉⵔⵏ ⴰⴳⵏⵙⵓ ⵏ ⵉⵎⴹⴰⵏ ⵓⴷⴷⵉⵙⵏ.

In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier).

ⴳ ⵍⵀⵉⵏⴷ ⵜⴰⵇⴱⵓⵔⵜ ⵜⴽⴽⴰⵜ ⵜⵓⵙⵙⵏⴰ ⵙ ⵜⵙⴳⴳⵉⵏ ⵜⵉⵎⴰⴳⵓⵏⵉⵏ ⴷ ⵉⵙⴽⴽⵉⵔⵏ ⵉⵥⵓⵕⴰⵏ ⵉⵎⴽⴽⵓⵥⵏ ⴷ ⵓⵥⵓⵕ ⴰⵎⴽⴽⵓⵥ ⵉⴳⴰ ⴰⵇⴱⵓⵔ ⵣⵓⵏⴷ ⵙⵓⵍⴱⴰ ⵙⵓⵜⵔⴰⵙ ⵏⵏⴰ ⵉⴽⴽⴰⵏ ⴰⵜⵜⴰⵢⵏ ⵏ 800 ⴰⵔ 500 ⵏ ⵓⵙⴳⴳⵯⴰⵙ ⴷⴰⵜ ⵜⵍⴰⵍⵉⵜ ⵏ ⵍⵎⴰⵙⵉⵃ (ⵉⵖⵢ ⵉⴷ ⵓⴳⴳⴰⵔ).

The letter jīm resembles the present square root shape.

ⴰⵙⴽⴽⵉⵍ jīm ⵢⴰⵖ ⴳ ⵜⴰⵍⵖⴰ ⵏ ⵓⵥⵓⵕ ⴰⵎⴽⴽⵓⵥ ⵏ ⴷⵖⵉ.

It defines an important concept of standard deviation used in probability theory and statistics.

ⴷⴰ ⵉⵙⵜⵜⵉ ⴰⵔⵎⵎⵓⵙ ⵉⵙⵜⴰⵡⵀⵎⵎⴰⵏ ⵏ ⵡⵓⵏⵓⴼ ⴰⵏⴰⵡⴰⵢ ⵉⵜⵜⵓⵙⵎⵔⴰⵙⵏ ⴳ ⵜⵎⴰⴳⵓⵏⵜ ⵏ ⵢⵉⵖⵉⵢ ⴷ ⵜⴰⴷⴷⴰⴷⴰⵏⵜ.

Most pocket calculators have a square root key.

ⵜⵍⵍⴰ ⴳ ⴽⵉⴳⴰⵏ ⵜⵎⵙⵙⵉⵟⵏⵉⵏ ⵜⵙⴰⵔⵓⵜ ⵏ ⵓⵥⵓⵕ ⴰⵎⴽⴽⵓⵥ.

The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.

ⴰⵙⵔⵔⵓⵏⴽⵙ ⴰⵣⵎⵣⴰⵏ ⵎⴰⵔ ⴰⴷ ⵏⵙⵙⵉⵟⵏ ⵓⵥⵓⵕ ⴰⵎⴽⴽⵓⵥ ⵙ ⵡⵓⵟⵟⵓⵏ n ⵙ ⵓⵏⵖⴰⴷ, ⵢⴰⴽⵙⵓⵍ ⴷ ⵓⵙⵔⵔⵓⴽⵙ ⵏ ⵓⵙⴼⵓⴽⵜⵉ ⵏ ⵙⵉⵏ ⵉⵎⴹⴰⵏ ⵙⴳ ⵙⵉⵏ ⵡⵓⵟⵟⵓⵏ n.

Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.

ⵉⵎⵓⴽⵔⵉⵙⵏ ⵏ ⵀⵉⴱⵔⵜ ⴳⴰⵏ ⴰⴳⵏⴰⵔ ⴷ ⴽⵕⴰⴹ ⵉⵎⵓⴽⵔⵉⵙⵏ ⴳ ⵜⵓⵙⵏⴰⴽⵜ, ⵉⴼⵙⵔⵜⵏⵜ ⵓⵎⵓⵙⵏⴰⵡ ⴰⵍⵎⴰⵏⵉ ⴷⵉⴼⵉⴷ ⵀⴰⵍⴱⵔ ⴰⵙⴳⴳⵯⴰⵙ ⵏ 1900.