Difference

Difference, of two numbers, is the result obtained by subtracting the smaller from the greater, and similarly for any two quantities of the same kind. The difference of two physical quantities is often of more importance than the actual value of either; the difference of height of two points, for example, or the difference of electric potential, being frequently required when the absolute height or absolute potential are not wanted. In pure mathematics the law regulating a series of numbers may be derived by obtaining the successive differences between first and second, second and third, and so on. Thus the series representing the squares of the natural numbers is

0, 1, 4, 9, 16, 25

The successive differences are

1, 3, 5, 7, 9,

The differences of these again are

2, 2, 2, 2, 2

and we thus come to a stop, for there is no more complexity in the series. In this way we may gauge the complexity of a series by the number of times that successive differences are to be taken. In the case of the cubes of the natural numbers we have

0 1 8 27 64 125 .. .
First differences 1 7 19 37 61 . . .
Second differences 6 12 18 24 .. .
Third differences 6 6 6 . . .

On these considerations is based it calculus of differences, by means of which the summation of such series may be effected and other dependent problems solved. It is of considerable importance in statistical work, and in the approximate determination of areas, etc.