Numbers: Their Tales, Types, and Treasures.

Chapter 1: Numbers and Counting

 

1.13.BASE-10

Our place-value system is a written representation of a base-10 numeral system. The first number after nine plays a particular role because it is the first number that requires a symbolic representation consisting of more than one digit—namely 10, which is one times ten plus zero times one. Moreover, the other place values can be obtained as powers of 10:

 

hundred

= 100 = 10 × 10 = 102

 

thousand

= 1,000 = 10 × 10 × 10 = 103

 

ten thousand

= 10,000 = 10 × 10 × 10 × 10 = 104

and so on. With this notation we can write

1776 = 1×103 + 7×102 + 7×10 + 6,

and with the common definition 10 = 101 and 1 = 100 we obtain the unified notation

1776 = 1×103 + 7×102 + 7×101 + 6×100.

In our place-value system, any natural number, no matter how large, can be written with the help of a finite number of digits, say,

d0d1dn-1dn,

and each of these digits is taken from the set of ten symbols {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, except dn, which is usually assumed to be not zero. The numeral representing the given number is then formed by writing the digits in a row:

dn dn–1d2 d1 d0.

The whole expression is then just a shortcut for digit-times-place-value addition:

dn ×10n + dn–1 ×10n–1 +…+ d2 ×102 + d1 ×101 + d0 ×100.