Counting in Binary
| Decimal pattern | Binary numbers |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 8 | 1000 |
| 9 | 1001 |
| 10 | 1010 |
| 11 | 1011 |
| 12 | 1100 |
| 13 | 1101 |
| 14 | 1110 |
| 15 | 1111 |
| 16 | 10000 |
Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar decimal counting system as a frame of reference.
- Decimal counting
Decimal counting uses the ten symbols 0 through 9. Counting primarily involves incremental manipulation of the "low-order" digit, or the rightmost digit, often called the "first digit". When the available symbols for the low-order digit are exhausted, the next-higher-order digit (located one position to the left) is incremented, and counting in the low-order digit starts over at 0. In decimal, counting proceeds like so:
- 000, 001, 002, ... 007, 008, 009, (rightmost digit starts over, and next digit is incremented)
- 010, 011, 012, ...
- ...
- 090, 091, 092, ... 097, 098, 099, (rightmost two digits start over, and next digit is incremented)
- 100, 101, 102, ...
After a digit reaches 9, an increment resets it to 0 but also causes an increment of the next digit to the left.
- Binary counting
In binary, counting follows similar procedure, except that only the two symbols 0 and 1 are used. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:
- 0000,
- 0001, (rightmost digit starts over, and next digit is incremented)
- 0010, 0011, (rightmost two digits start over, and next digit is incremented)
- 0100, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented)
- 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...
Since binary is a base-2 system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. To determine the decimal representation of a binary number simply take the sum of the products of the binary digits and the powers of 2 which they represent. For example, the binary number 100101 is converted to decimal form as follows:
- 1001012 = + + + + +
- 1001012 = + + + + +
- 1001012 = 3710
To create higher numbers, additional digits are simply added to the left side of the binary representation.
Read more about this topic: Binario
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