Page:IntroductionToMathematicsWhitehead.pdf/68

value of a digit depends on the position in which it occurs. Consider, for example, the digit ${\displaystyle 5}$, as occurring in the numbers ${\displaystyle 25}$, ${\displaystyle 51}$, ${\displaystyle 3512}$, ${\displaystyle 5213}$. In the first number ${\displaystyle 5}$ stands for five, in the second number ${\displaystyle 5}$ stands for fifty, in the third number for five hundred, and in the fourth number for five thousand. Now, when we write the number fifty-one in the symbolic form ${\displaystyle 51}$, the digit ${\displaystyle 1}$ pushes the digit ${\displaystyle 5}$ along to the second place (reckoning from right to left) and thus gives it the value fifty. But when we want to symbolize fifty by itself, we can have no digit 1 to perform this service; we want a digit in the units place to add nothing to the total and yet to push the ${\displaystyle 5}$ along to the second place. This service is performed by ${\displaystyle 0}$, the symbol for zero. It is extremely probable that the men who introduced ${\displaystyle 0}$ for this purpose had no definite conception in their minds of the number zero. They simply wanted a mark to symbolize the fact that nothing was contributed by the digit's place in which it occurs. The idea of zero probably took shape gradually from a desire to assimilate the meaning of this mark to that of the marks, ${\displaystyle 1}$, ${\displaystyle 2}$,. . . ${\displaystyle 9}$, which do represent cardinal numbers. This would not represent the only case in which a subtle idea has been introduced into mathematics by a symbolism which in its origin was dictated by practical convenience.