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In [[mathematics]], a '''square root''' of a number ''x'' is another number that, when multiplied by itself ([[squared]]), becomes ''x''. When it is not [[Negative number|negative]], it is represented by the symbol <math>\sqrt{x}</math>, and called the principal square root of ''x''.<ref name=":0">{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/|access-date=2020-08-28|website=Math Vault|language=en-US}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Square Root|url=https://mathworld.wolfram.com/SquareRoot.html|access-date=2020-08-28|website=mathworld.wolfram.com|language=en}}</ref> For example, 3 is the square root of 9, because 3×3=9. Only numbers bigger than or equal to [[zero]] have real square roots. The square root of zero is zero.

An important number is the [[square root of 2]], which is an [[irrational number]]. Its value is around 1.41421.<ref name=":0" /> It is the length of a [[diagonal]] in the square whose side is 1.<ref>{{Cite web|title=Irrationality of the square root of 2.|url=https://www.math.utah.edu/~pa/math/q1.html|access-date=2020-08-28|website=www.math.utah.edu|archive-date=2023-06-05|archive-url=https://web.archive.org/web/20230605084213/http://www.math.utah.edu/~pa/math/q1.html|url-status=dead}}</ref>

Square roots of negative numbers are not [[real number]]s – they are [[imaginary numbers]]. Imaginary numbers are numbers whose square is negative. Every [[complex number]], except 0, has two square roots that are imaginary numbers. For example, −1 has two square roots. We call them <math>i</math> and <math>-i</math>. We also consider the number <math>i</math> as the principal square root of −1, and call it the [[imaginary unit]].

The sign for a square root is made by putting a bent line over a number. For example: <math>\sqrt 4</math>. This is read as "the square root of 4".

A [[whole number]] with a square root that is also a whole number is called a [[square number|perfect square]]. The first few perfect squares are: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225.

Below is a table of square roots (rounded to 3 decimal places).
{| class="wikitable"
|+
!Number
!Square root of number
|-
|1
|1.000
|-
|2
|1.414
|-
|3
|1.732
|-
|4
|2.000
|-
|5
|2.236
|-
|6
|2.449
|-
|7
|2.646
|-
|8
|2.828
|-
|9
|3.000
|-
|10
|3.162
|-
|11
|3.317
|-
|12
|3.464
|-
|13
|3.606
|-
|14
|3.742
|-
|15
|3.873
|-
|16
|4.000
|-
|17
|4.123
|-
|18
|4.243
|-
|19
|4.359
|-
|20
|4.472
|}
== Origin of the symbol ==
It is not really known where the square root symbol <math>\sqrt{\,\,}</math> comes from, but some people believe that it was from the [[letter]] ''r'', which is the first letter of the [[Latin]] word ''radix'' (used in many languages). Radix means the root or base. Thus, radix quadratum from Latin refers most likely to the base of a square. Since the sides of a square are all equal, the word radix may be interpreted as meaning a side of a square.

== Square Root Media ==
<gallery widths='160px' heights='100%' mode='traditional' caption=''>
File:Nuvola apps edu mathematics blue-p.svg|Notation for the (principal) square root of {{mvar|x}}.

File:Five Squared.svg|For example, {{math|{{radic|25}} {{=}} 5}}, since {{math|25 {{=}} 5 &sdot; 5}}, or {{math|52}} (5 squared).

File:Ybc7289-bw.jpg|A black and white rendition of *my own photograph of the Yale Babylonian Collection's Tablet YBC 7289 (c. 1800–1600 BCE),*showing a Babylonian approximation to the square root of 2 (1 24 51 10 w: sexagesimal) in the context of 

File:Square root 0 25.svg|The graph of the function {{math|1=''f''(''x'') = √''x''}}, made up of half a [[parabola]] with a vertical [[Directrix (conic section)#Eccentricity, focus and directrix|directrix]]

File:Imaginary2Root.svg|The square roots of {{mvar|i}}

File:SqrtGeom.gif|[[Straightedge and compass construction|Constructing]] the length x = \sqrt{a}, given the a and the unit length

File:Spiral_of_Theodorus.svg|The [[Spiral of Theodorus]] up to the triangle with a hypotenuse of {{math|{{radic|17}}}}

File:Root_rectangles_Hambidge_1920.png|[[Jay Hambidge]]'s construction of successive square roots using [[root rectangle]]s
</gallery>
== Related pages ==
* [[Methods of computing square roots]]
* [[nth root]]
== References ==
<references />
== Other websites ==
* [http://mathforum.org/library/drmath/sets/select/dm_sqrt.html Table of square and square roots, 1-100] {{Webarchive|url=https://web.archive.org/web/20090208174541/http://mathforum.org/library/drmath/sets/select/dm_sqrt.html |date=2009-02-08 }} -The Math Forum

[[Category:Mathematics]]
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