y=x², for all integer values of 1≤x≤25. The squares of numbers make a power law. A power law is any Polynomial relationship that exhibits the property of Scale invariance.

In algebra, the square of a number is that number multiplied by itself. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. To square a quantity is to multiply it by itself. Its notation is a superscripted "2"; a number x squared is written as x². This article is about the terms 'subscript' and 'superscript' as used in typography Thus:

If x is a positive real number, the value of x² is equal to the area of a square of edge length x. In Mathematics, the real numbers may be described informally in several different ways Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides

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 integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a square number, sometimes also called a Perfect square, is an Integer that can be written as the square of some other

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 1² + 2² + 3² + 4² + . . . + n² is 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. In Mathematics, a pyramid number, or square pyramidal number, is a Figurate number that represents a Pyramid with a base and four sides . . (sequence A000330 in OEIS)

Uses

Since the product of two real negative numbers is positive, and the product of two real positive numbers is also positive, it follows that no square number is negative. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences A negative number is a Number that is less than zero, such as −2 A negative number is a Number that is less than zero, such as −2 This has important consequences. It follows, in particular, that no square root can be taken of a negative number within the system of real numbers. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, the real numbers may be described informally in several different ways This leaves a gap in the real number system that mathematicians fill by postulating imaginary numbers, beginning with the imaginary unit i, which by convention is one of the square roots of −1. Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

Squaring is also useful for statisticians in determining the standard deviation of a population or sample from its mean. In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values In Statistics, a sample is a Subset of a population. Typically the population is very large making a Census or a complete Enumeration In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean Each datum is subtracted from the mean, and the result is squared. Then an average is taken of the new set of numbers (each of which is positive). This average is the variance, and its square root is the standard deviation -- in finance, the volatility. In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of

See also

• Square root
• Cube (algebra)
• Pythagorean Theorem
• Exponentiation