# Square (algebra)

In

algebra , the**square**of a number is that number multiplied by itself. To**square**a quantity is to multiply it by itself.Its notation is asuperscript ed "2"; a number "x" squared is written as "x"². Thus:If "x" is a positive

real number , the value of "x"² is equal to thearea of a square of edge length "x".A positive

integer that is the square of some other integer, for example 25 which is 5², is known as asquare number , or more simply a square.It is often also useful to note that the square of any number can be represented as the sum (for 0≤n):1 + 1 + 2 + 2 + ... + ("n" − 1) + ("n" − 1) + "n".

For instance, the square of 4 or 4² is equal to

:1 + 1 + 2 + 2 + 3 + 3 + 4 = 16.

This is the result of adding a column and row of thickness 1 to the square graph of three (like a tic tac toe board). You add three to the side and four to the top to get four squared. This can also be useful for finding the square of a large number quickly. For instance, the square of

:52² = 50² + 50 + 51 + 51 + 52 = 2500 + 204 = 2704.

In addition, it can be seen that another equivalent sum may be used to represent the square of a number. The square of a number N is the sum of the first N odd numbers. The square of 1 is 1; the square of 2 is

:1 + 3 = 4;

the square of 7 is

:1 + 3 + 5 + 7 + 9 + 11 + 13 = 49.

and so on. This, of course is the same as the previous sum method but with every two numbers following the initial number added to each other:

:1 + ( 1 + 2 ) + ( 2 + 3 ) + ( 3 + 4 ) + ... = 1 + 3 + 5 + 7 + ...

The general term of the series $1^2+2^2+3^2+4^2+...+n^2$ is $n(n+1)(2n+1)/6$. The first terms of this series (the

Square pyramidal number s) are :0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201... OEIS|id=A000330

**Uses**Since the product of two

real negative number s is positive, and the product of two realpositive number s is also positive, it follows that no square number is negative. This has important consequences. It follows, in particular, that nosquare root can be taken of a negative number within the system ofreal number s. This leaves a gap in the real number system that mathematicians fill by postulatingcomplex number s, beginning with theimaginary unit "i", which by convention is one of the square roots of −1.Squaring is also useful in

statistics in determining thestandard deviation of a set of values. The deviation of each value $x\_i,$ from the mean $overline\{x\}\; ,$ of the set is defined as the difference $x\_i\; -\; overline\{x\}\; ,$. These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is thevariance , and its square root is the standard deviation. Infinance , the volatility of a financial instrument is the standard deviation of its values.**ee also***

Cube (algebra)

*Pythagorean Theorem

*Exponentiation

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