Binary Math Home

Whether it be from math class or computer science class, eventually we will need to learn about number systems, and the mathematics that is involved.

Binary numbers

, as with decimal, octal, and hexadecimal numbers, are organized into columns. To learn binary math, we first need to understand how number systems operate. Let's take a look at the decimal system first, since it is simple and easier to think about. We can consider the number "1234" as,

Thousands	Hundreds	Tens	Ones
1	2	3	4

Which means,
1234 = 1x1000 + 2x100 + 3x10 + 4x1

Given,

1000	= 10^3 = 10x10x10
100	= 10^2 = 10x10
10	= 10^1 = 10
1	= 10^0 (any number to the exponent zero is one, except for zero)

The table above can be represented as,

Thousands	Hundreds	Tens	Ones
10^3	10^2	10^1	10^0
1	2	3	4

such that,
1234 = 1x1000 + 2x100 + 3x10 + 4x1
= 1x10^3 + 2x10^2 + 3x10^1 + 4x10^0

The decimal system, as with decimal math, operates in "base 10" (dec being the Latin prefix for ten) using the digits 0-9 to represent numbers, whereas the binary system, as well as its math, operates in "base 2" (bi being the Latin prefix for two) using the digits 0-1 to represent numbers. The base is also known as the radix. In other words, the table above can be represented as,

	Thousands	Hundreds	Tens	Ones
Decimal	10^3	10^2	10^1	10^0
Binary	2^3	2^2	2^1	2^0

In base 10, we put the digits 0-9 in columns 10^0, 10^1, 10^2, and so on. To put a number that is greater than 9 into 10^n, we must add to 10^(n+1). For example, adding 10 to column 10^0 requires us to add 1 to the column 10^1.

In base 2, we put the digits 0-1 in columns 2^0, 2^1, 2^3, and so on. To put a number that is greater than 1 into 2^n, we must add to 2^(n+1). For example, adding 3 to column 2^0 requires us to add 1 to the column 2^1.

Here are some decimal numbers represented in binary.

Decimal	Binary
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
10	1010