Tool to compute and simplify a square root. The square root for a number N, is the number noted sqrt(N) that, multiplied by itself, equals N.

Square Root - dCode

Tag(s) : Symbolic Computation, Functions

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A square root of $ x $ (or radical of $ x $) is a mathematical concept noted $ \sqrt{x} $ (ou `sqrt(x)`) that refers to the number that, when multiplied by itself, produces the number $ x $.

__Example:__ The square root of $ 9 $ is $ 3 $ that is written $ \sqrt{9} = 3 $, because $ 3 \times 3 = 9 $

Generally, numbers have 2 roots, a positive and a negative, but the negative is usually omitted.

__Example:__ It would be more accurate to write: the square roots of $ 9 $ are $ 3 $ and $ -3 $ which is written $ \sqrt{9} = \pm 3 $, indeed, $ 3 \times 3 = 9 = (-3) \times (-3) = 9 $

There are several methods to calculate a root square.

— By hand framing: the classic method is to estimate the value by calculating which integers squared would give a minimum interval.

__Example:__ Enclosing $ \sqrt{8} $: $ 2^2 = 4 < 8 < 9 = 3^3 $ so $ 2 < \sqrt{8} < 3 $, it is then possible to enclose the first digit after the comma: $ 2.8^2 < 8 < 2.9^2 $ etc.

— By extraction of squares: if the number under the root is factorized with squares, then it is possible to extract them from the root.

__Example:__ Factorization of $ \sqrt{8} = \sqrt{ 4 \times 2 } = \sqrt{ 2^2 \times 2 } = 2 \sqrt{2} $. Since $ \sqrt{2} \approx 1.414 $, then $ \sqrt{8} \approx 2.828 $

— With a square root calculator like this one from dCode:

Enter a positive or negative number (in this case, it will have complex roots).

Choose the format of the result, either an exact value (if it is an integer or variables) or approximate (decimal number with adjustable precision by defining a minimum number of significant digits)

__Example:__ $ \sqrt{12} = 2 \sqrt{3} \approx 3.464 $

__Example:__ $ \sqrt{-1} = i $ (complex root)

For any positive real number $ a \in \mathbb{R}_+^* $

$$ \sqrt{a^2} = a \\ \left( \sqrt{a} \right)^2 = a $$

For any number $ b $

$$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \\ \sqrt{ \frac{a}{b} } = \frac{\sqrt{a}}{\sqrt{b}} \\ \sqrt{a^2 \times b} = a \sqrt{b} $$

The simplification of a square root generally passes by the factorization of the component under the root by one or more squares.

__Example:__ $ \sqrt{20} = \sqrt{ 2^2 \times 5 } = \sqrt{ 2^2 } \times \sqrt{ 5 } = 2 \sqrt{ 5 } $

Use the prime factors decomposition if necessary

If the denominator is a radical, then multiply the numerator and the denominator by it to make it disappear.

$$\frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{\sqrt{b}^2} = \frac{a\sqrt{b}}{b} $$

If the denominator is an addition or subtraction of roots, then apply the remarkable identity: $ (a+b)(a-b) = a^2-b^2 $

$$ \frac{a}{\sqrt{b}+\sqrt{c}} = \frac{a(\sqrt{b}-\sqrt{c})}{(\sqrt{b}+\sqrt{c})(\sqrt{b}-\sqrt{c})} = \frac{a\sqrt{b}-a\sqrt{c}}{b-c} $$

$$ \frac{a}{\sqrt{b}-\sqrt{c}} = \frac{a(\sqrt{b}+\sqrt{c})}{(\sqrt{b}-\sqrt{c})(\sqrt{b}+\sqrt{c})} = \frac{a\sqrt{b}+a\sqrt{c}}{b-c} $$

In Unicode format there is the character `√` (U+221A).

In computer formulas, `sqrt()` function is most often used.

Terms *root*, *radix* ou *radicand* sont équivalents.

Square roots are needed in many areas of mathematics.

__Example:__ In algebra: in algebraic calculations, the roots are used to solve polynomial equations of the type $ x^2 + 2x + 1 = 0 $

__Example:__ In geometry: in length calculations (or vector norms), roots are used to find solutions to the Pythagorean theorem $ a^2 + b^2 = c^2 $

The word `sqrt` is generally used in the formula to indicate a square root, the word comes from the contraction of `square root`.

__Example:__ `sqrt(2)` = $ \sqrt{2} $

A square number is the square of an integer.

__Example:__ $ 3 $ is an integer, $ 3^2 = 3 \times 3 = 9 $ then $ 9 $ is a square number.

If the square root of a number $ x $ is an integer, then $ x $ is a square number.

dCode retains ownership of the "Square Root" source code. Except explicit open source licence (indicated Creative Commons / free), the "Square Root" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Square Root" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Square Root" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

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Cite as source (bibliography):

*Square Root* on dCode.fr [online website], retrieved on 2023-12-05,

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- Why calculate square roots?
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- What is a square number?

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