Binary Counting ("Base 2")
Computers do not count with the same counting system we use. Humans have ten fingers, so we use a base-10 counting system--that is, 10 digits. A "digit" is a single-character number; the ten digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Computers use switches to keep information. A switch can be off or on. On is 1; off is 0. SO computers only use those two digits. One number--a 1 or a 0--is called a binary digit, or a "bit" for short.
When we count, we start with a single digit, going from 0 to 9. After 9, there are no more digits. So we have to use two digits. Each digit represents a "column," as shown in the chart below. In the number "10," the "1" is in the "tens" column, and equals "1 x 10"; the "0" is in the "ones" column, and equals "0 x 1."
As an example, take the number 256:
|
Start at the right, and move to the left |
||||||
| Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones |
| 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 |
| 106 | 105 | 104 | 103 | 102 | 101 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| 5 | 5 | 5 | 5 | 5 | 5 | 5 |
| 6 | 6 | 6 | 6 | 6 | 6 | 6 |
| 7 | 7 | 7 | 7 | 7 | 7 | 7 |
| 8 | 8 | 8 | 8 | 8 | 8 | 8 |
| 9 | 9 | 9 | 9 | 9 | 9 | 9 |
Therefore, the number "256" means that you have 2 hundreds, 5 tens, and 6 ones. Each digit has ten possibilities, so with each new digit, you multiply by ten. A 2-digit number has 100 possibile combinations. A 3-digit number has 1000 combinations, and so on.
However, if you only have two numbers instead of ten, your counting has to look like this:
|
Start at the right, and move to the left |
|||||||||
| five hundred and twelves | two hundred and fifty- sixes | one hundred and twenty- eights | sixty- fours | thirty- twos | sixteens | eights | fours | twos | ones |
| 512 | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 29 | 28 | 27 | 26 | 25 | 24 | 23 | 22 |
21 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Notice the numbers: 8, 16, 32, 64, 128, 256, 512. These are numbers seen very often with computer memory.
To write the number 7, you would write "111"--that is, 1 four, 1 two, and 1 one. To write the number 8, you would write "1000"--1 eight, and 0 fours, twos and ones. The number 256, therefore, is "10000000."
Notice that the second column is the "base," and all other columns are powers of the base--base2, base3, base4, etc.
Here is an example of counting from one to ten in binary:
binary base 10 0 0 1 1 10 2 11 3 100 4 101 5 110 6 111 7 1000 8 1001 9 1010 10
Each digit has two possibilities, so with each new digit, you multiply by two. For example, a 2-digit binary number has four possible combinations (00,01,10, and 11). A 3-digit number has eight combinations; a 4-digit number has 16 combinations, and so on.
