2.2 The power of two
For a system based on 0s and 1s, the number 2 shows up a lot in classical computing. This is not surprising because we use binary arithmetic, which is a set of operations on base 2 numbers.
Most people use base 10 for their numbers. These are also called decimal numbers. We construct such numbers from the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, which we often call digits. Note that the largest digit, 9, is one less than 10, the base.
A number such as 247 is really shorthand for the longer 2 × 102 + 4 × 101 + 7 ×100. For 1,003 we expand to 1 × 103 + 0 × 102 + 0 × 101 + 3×100. In these expansions we write a sum of digits between 0 and 9 multiplied by powers of 10 in decreasing order with no intermediate powers omitted.
We do something similar for binary. A binary number is written as a sum of bits (0 or 1) multiplied by powers of 2 in decreasing order with no intermediate powers omitted. Here are some examples...
