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For example, since .

This example suggests how square roots can arise when solving quadratic equations such as or, more generally, .

Extending the square root concept to negative real numbers gives rise to imaginary and complex numbers.

Square roots of integers are often irrational numbers, i.e., numbers not expressible as one integer over another. (It is a misconception that mathematicians define irrational number to be one whose decimal expansion is infinite and non-repeating. That is equivalent, but nothing is sacred about base-10 numerals as opposed to other bases.) For example, cannot be written exactly as m/n, where n and m are integers (and so cannot be written in finite or repeating decimal form, although that is a fact of less interest to mathematicians.) Nonetheless, it is exactly the length of the diagonal of a square with side length 1.

The discovery that is irrational is attributed to Hippasus, a disciple of Pythagoras. After the number was revealed to be irrational, the Pythagoreans killed Hippasus, not wishing to believe this fundamental number could be infinitely long and nonrepeating. Other Greek philosopherA philosopher is a person devoted to studying and producing results in philosophy. The word, "philosopher," literally means "lover of wisdom. Popular Western philosophers in (approximate) historical order Not listed above: (some of) The Presocratics Epicus celebrated the discovery with a sacrifice of 100Integers Composite numbers 100 (the Roman numeral is C for centum is the natural number following 99 and preceding 101. Prefixes for 100 include hecta- ( Greek) and cent- ( Latin). Cardinal one hundred Ordinal100th (one hundredth) Factorization Divisors 2 oxen (a hecatombIn Ancient Greece, a Hecatomb was the sacrifice to the gods of 100 cattle hecaton one hundred). In the Iliad hecatombs are described in a formulaic way. Here is one instance. T]hey ranged the holy hecatomb all orderly round the altar of the god. They wash).

The square root symbolIn mathematics, a set of symbols is frequently used in mathematical expressions. As mathematicians are familiar with these symbols, they are not explained each time they are used. So, for mathematical novices, the following table lists many common symbols (√) was first used during the 16th century15th century 16th century 17th century more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. Events Beginning of the " Little Ice Age" a cooling period that resulted in lower crop yi. It has been suggested that it originated as an altered form of lowercase rR is the eighteenth letter of the Latin alphabet. Semitic ReS (the head) developed into Greek (Ro). The sound value /r/ however was maintained in Greek as well as Etruscan and Latin. The finishing stroke was added to the Greek Rho to distinguish it from a, representing the Latin radix (meaning " root").

1 Properties

The following important properties of the square root functions are valid for all positive real numbers and :

for every real number (see absolute value)

The square root function generally maps rational numbers to algebraic numbers; is rational if and only if is a rational number which, after cancelling, is a quotient of two perfect squares. In particular, is irrational.

In geometrical terms, the square root function maps the area of a square to its side length.

Suppose that and are reals, and that , and we want to find . A common mistake is to "take the square root" and deduce that . This is incorrect, because the square root of is not , but the absolute value , one of our above rules. Thus, all we can conclude is that , or equivalently .

In calculus, for instance when proving that the square root function is continuous or differentiable or when computing certain limits, the following identity often comes handy:

It is valid for all non-negative numbers and which are not both zero.

The function has the following graph, made up of half a parabola lying on its side:

The function is continuous for all non-negative , and differentiable for all positive (it is not differentiable for since the slope of the tangent there is ∞). Its derivative is given by

Its Taylor series about can be found using the binomial theorem:

for .

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