The standard numbering system used today is based on tens. We have ten digits, 0 - 9. But what if, for example, our digits only went up to four? 0, 1, 2, 3, 4. Then you would start over, using double digits: 10, 11, 12, 13, 14, 20, 21, 22, . . . 43, 44, 100, etc.

If we switched to this numbering system now, it would be quite confusing, since 100 in base-5 would equal our 25, in base-10. When working with numbers with different bases, it is a good idea to label which system a number is based on. Therefore, we use a subscript of the base-number after the value of the actual number.

Binary is just math in base-2. Since there are only 2 digits, 1 and 0, numbers can get quite long. How can you translate a long binary number without sitting there for hours counting your way up? Don't worry, there's a pattern!

Let's look at something we're familiar with, base-10. In base-10, we have place values:

thousands	hundreds	tens	ones	.	tenths	hundredths	=	Value
			0	.	0	1	=	10-2
			0	.	1		=	10-1
			1	.			=	100
		1	0	.			=	101
	1	0	0	.			=	102
1	0	0	0	.			=	103

Look at the values of each number on the chart. Each place value represents a power of ten. The number 6384.4₁₀ (remember we use the subscript to determine the base of the number) can be "tracked", or calculated as:

103	102	101	100	.	10-1
6	3	8	4	.	4

(6 * 10³) + (3 * 10²) + (8 * 10¹) + (4 * 10⁰) + (4 * 10^-1)

6000 + 300 + 80 + 4 + 0.4 = 6384.4₁₀

This pattern found in base-10 is also in any other numerically based system. So now, let's look at the binary number 1100101₂. Since we are in base-2, The place values will be in powers of two:

26	25	24	23	22	21	20
1	1	0	0	1	0	1

We can find it's base-10 value by adding:

2⁶ + 2⁵ + 2² 2⁰ = 64 + 32 + 4 + 1 = 101₁₀.

The most commonly used number systems are 2, 8, 10, and 16. In base-16, also called hexadecimal, we use letters to complete the digits: A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15. A 16 is not needed, as that would be represented by 10₁₆.

Numbers are easy to translate between base-2 and base-8. They are also easy to translate between base-2 and base-16. The number 7, the highest digit in base-8 can be represented by 111₂. Each digit in base-8 can be represented by three digits in base-2. Look at another example:

7	3	6	Base-8
1 1 1 4s 2s 1s	0 1 1 4s 2s 1s	1 1 0 4s 2s 1s	Base-2

Therefore, 736₈ = 111011110₂

It works the same way when changing from hexadecimal to binary:

5	F	A	Hexadecimal
1 0 1 8s 4s 2s 1s	1 1 1 1 8s 4s 2s 1s	1 0 1 0 8s 4s 2s 1s	Binary

Therefore, 5FA₁₆ = 10111111010₂.

Why do we use binary code so often with computers? It's simple: it's simple! The fact that there are only two digits (0 and 1) works out well with the fact that electricity has only two states: on and off. So, if current is flowing, or on, the digit is considered a 1. If there is no current, meaning it's off (that one tiny portion of the memory, not the whole computer!), then the digit is considered a 0. This binary system of ones and zeros, or on and off, is the basis for the entire computer.