Chapter Five: Digital Audio

2. Binary Numbers

A rudimentary knowledge of how binary numbers work is required in order to understand the mechanism of digital audio. A good way to start is with decimal numbers, which are much more familiar to most of us. Each "place" of a decimal number is filled by a digit. Our decimal system is called base-10, meaning that each digit can express 10 values, ranging from zero to nine. To express a quantity greater than 9, we need an additional digit or digits (we are ignoring decimals for the time being). Each place of a base-10 number represents a power of 10 (with 100 = 0–9), so 1's, 10's, 100's, etc.

Binary numbers developed as a symbolic representation of computer circuits, which can be thought of as a series of switches or logic gates that are either on or off. It seemed logical to use our first two familiar symbols, 0 and 1 to represent these two states . A single-place binary number is called a bit, which is short for "Binary digIT." Binary numbers are base-2, with each place representing the powers of two (as opposed to ten in our decimal system). The places for a binary number from right to left are 1's, 2's, 4's, 8's, 16's, 32's, etc. or 20, 21, 22, 23, 24, 25, 26, etc. which add up cumulatively if there is a '1' in that particular place.

powers of 2 23 22 21 20
equivalent decimal values 8's 4's 2's 1's
sample 4-bit binary number 1 0 1 1
how to solve 8 + 0 + 2 + 1 = 11 (decimal)

Below is a chart of some equivalent decimal to 4-bit binary values
0 = 0000
4 = 0100
8 = 1000
12 = 1100
1 = 0001 5 = 0101 9 = 1001 13 = 1101
2 = 0010 6 = 0110 10 = 1010 14 = 1110
3 = 0011 7 = 0111 11 = 1011 15 = 1111

For a more extensive printable chart, click here.

human ear
Fun Factoid: Origin of the Term Computer Bug

Early computers had physical, electromechanical switches prior to the modern silicon-based electronic ones for storing and calculating bits. Grace Hopper, legendary computer programming pioneer, documented the first use of the term computer bug in programming or execution by taping a moth, which had been found by others in one of the Harvard Mark II electromechanical switches creating a glitch or program error, to the computer's log book with the inscription "First actual case of bug being found."