The Binary Number System
The key to understanding the Binary Number System is to consider the positioning of the digits.
Similar to the decimal system, each position has a weight. A decimal example would be 142 (One Hundred and Forty Two) or 1 x 100 + 4 x 10 + 2 x 1 = 142. In the Binary System, however, all we have are 0's and 1's.
Therefore, we can only count to 1 in any column. Basically, if a 1 is present or not. A Binary example for decimal 9 would be:
1001 or 1 x 8 + 0 x 4 + 0 x 2 + 1 x 1 = 9.
Notice that each column is double the previous one: 8, 4, 2 and 1. This continues indefinately. A four bit binary number is considered to be a four bit word. In the Binary Number System, each number is considered to be a bit. It takes 8 bits to make a byte. 4 bits (1/2 a byte) is a nibble.
Below is a table converting decimal numbers 0 through 15 to their Binary Number equivalents. We have highlighted each 1 ( in white) to make it easier to see which column is active.
Binary Counting (Base 2)
| Decimal # | 8's | 4's | 2's | 1's | Totals |
|---|
| 0 | 0 | 0 | 0 | 0 | 0+0+0+0=0 |
|---|
| 1 | 0 | 0 | 0 | 1 | 0+0+0+1=1 |
|---|
| 2 | 0 | 0 | 1 | 0 | 0+0+2+0=2 |
|---|
| 3 | 0 | 0 | 1 | 1 | 0+0+2+1=3 |
|---|
| 4 | 0 | 1 | 0 | 0 | 0+4+0+0=4 |
|---|
| 5 | 0 | 1 | 0 | 1 | 0+4+0+1=5 |
|---|
| 6 | 0 | 1 | 1 | 0 | 0+4+2+0=6 |
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| 7 | 0 | 1 | 1 | 1 | 0+4+2+1=7 |
|---|
| 8 | 1 | 0 | 0 | 0 | 8+0+0+0=8 |
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| 9 | 1 | 0 | 0 | 1 | 8+0+0+1=9 |
|---|
| 10 | 1 | 0 | 1 | 0 | 8+0+2+0=10 |
|---|
| 11 | 1 | 0 | 1 | 1 | 8+0+2+1=11 |
|---|
| 12 | 1 | 1 | 0 | 0 | 8+4+0+0=12 |
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| 13 | 1 | 1 | 0 | 1 | 8+4+0+1=13 |
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| 14 | 1 | 1 | 1 | 0 | 8+4+2+0=14 |
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| 15 | 1 | 1 | 1 | 1 | 8+4+2+1=15 |
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