Numbers: Their Tales, Types, and Treasures.
Chapter 1: Numbers and Counting
1.13.BASE10
Our placevalue system is a written representation of a base10 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 = 10^{2} 

thousand 
= 1,000 = 10 × 10 × 10 = 10^{3} 

ten thousand 
= 10,000 = 10 × 10 × 10 × 10 = 10^{4} 
and so on. With this notation we can write
1776 = 1×10^{3} + 7×10^{2} + 7×10 + 6,
and with the common definition 10 = 10^{1} and 1 = 10^{0} we obtain the unified notation
1776 = 1×10^{3} + 7×10^{2} + 7×10^{1} + 6×10^{0}.
In our placevalue system, any natural number, no matter how large, can be written with the help of a finite number of digits, say,
d_{0}, d_{1}…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}…d_{2} d_{1} d_{0}.
The whole expression is then just a shortcut for digittimesplacevalue addition:
d_{n} ×10^{n} + d_{n}_{–1} ×10^{n}^{–1} +…+ d_{2} ×10^{2} + d_{1} ×10^{1} + d_{0} ×10^{0}.