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 superscripted "2"; a number x squared is written as x². Thus: This article is about the branch of mathematics. ... In mathematics, multiplication is an elementary arithmetic operation. ... This article is about the term superscript as used in typography. ...

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 as numbers that can be given by an infinite decimal representation, such as 2. ... Area is a physical quantity expressing the size of a part of a surface. ... For other uses, see Square. ...

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 are commonly denoted by the above symbol. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ...

It is often also useful to note that the square of any number 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, 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 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... Square pyramidal numbers

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. 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. 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. Look up real in Wiktionary, the free dictionary. ... A negative number is a number that is less than zero, such as −3. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...

Squaring is also useful for statisticians in determining the standard deviation of a population or sample from its 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 and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... A sample is that part of a population which is actually observed. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Volatility most frequently refers to the standard deviation of the change in value of a financial instrument with a specific time horizon. ...

See also

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