Square (algebra)
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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:
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.
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 + ...
[edit] 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 uniti, which by convention is one of the square roots of −1.
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.
