Square (algebra)

y = x2, for all integer values of 1 ≤ x ≤ 25. The squares of numbers make a power law.

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 a superscript "2"; a number x squared is written as x2. Thus:

If x is a positive real number, the value of x2 is equal to the area 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 a square number, or more simply a square.

The square of any nonnegative integer n can be represented as the sum

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,

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

This can be written as a series in the following way:

N^2 = \sum_{n=0}^{N-1} (2n+1), or equivalently using recursion, N^2 = (N-1)^2 + 2N-1\, (ends at N=1, does not work for negative numbers).

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 sum of the series

1^2+2^2+3^2+4^2+\cdots+n^2

is

\frac{n(n+1)(2n+1)}{6}.

The first terms of this series (the square pyramidal numbers) 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... (sequence A000330 in OEIS)

Uses

Since the product of real negative numbers is positive, and the product of two real positive numbers is also positive, it follows that no square number is negative. It follows that no square root can be taken of a negative number within the system of real numbers. This leaves a gap in the real number system that mathematicians fill by postulating complex numbers, beginning with the imaginary unit i, which by convention is one of the square roots of −1.

Squaring is used in statistics in determining the standard 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 the variance, and its square root is the standard deviation. In finance, the volatility of a financial instrument is the standard deviation of its values.

Facts

 382 = 1444

 262 = 676

 1112 = 12321
 11112 = 1234321
 111112 = 123454321
 1111112 = 12345654321
 11111112 = 1234567654321
 111111112 = 123456787654321
 1111111112 = 123456787654321
 11111111112 = 12345678987654321

 c2 = 2(a2 + b2) - d2
 112 = 2(72 + 42) - 32

See also