Mathematic and Computer

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