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.
Watch this YouTube video I created to help understand binary counting and why it is connected to computers:
Try using this binary - base 10 conversion app to see how binary numbering works:
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