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:

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,

d₀, d₁…d_n-1, d_n,

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

d_n d_n–1…d₂ d₁ d₀.

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

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