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 big number quickly. For instance, the square of
52 = 502 + 50 + 51 + 51 + 52 = 2500 + 204 = 2704.
Since the product of two negative numbers is positive, and the product of two 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 i, which by convention is the square root 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.