The hexadecimal numeral system (hexadecimal system), or base-16 number system, represents numeric values using sixteen symbols, usually 0 to 9 and A to F, providing the following symbols in ascending order: 0 1 2 3 4 5 6 7 8 9 A B C D E F
To understand how hexadecimal numbers are constructed, it is useful to consider the decimal numeral system, or base-10 number system, which is the number system most commonly used. Digits in the decimal system are organised into columns multiplied by powers of ten. For example, the columns for 123 are:
| 102 | 101 | 100 | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 |
| 1 | * | 102 | = | 1 | * | ( 1 * 10 * 10 ) | = | 1 | * | 100 | = | 100 |
| 2 | * | 101 | = | 2 | * | ( 1 * 10 ) | = | 2 | * | 10 | = | 20 |
| 3 | * | 100 | = | 3 | * | ( 1 ) | = | 3 | * | 1 | = | 3 |
|
123
|
Decimal number 123 is ( 1 * 102 ) + ( 2 * 101 ) + ( 3 * 100 ).
If more detail is required, refer to the explanation of term decimal.
The hexadecimal system works under the same principles as the decimal system, except that it operates in base-16 rather than base-10. Therefore, the columns for hexadecimal number 7B are:
| 161 | 160 | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7 | B |
| 7 | * | 161 | = | 7 | * | 161 | = | 7 | * | ( 1 * 16 ) | = | 7 | * | 16 | = | 112 |
| B | * | 160 | = | 11 | * | 160 | = | 11 | * | ( 1 ) | = | 11 | * | 1 | = | 11 |
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123
|
Hexadecimal number 7B is ( 7 * 161 ) + ( 11 * 160 ) = 123.
Instead of using digits 0 to 9 as the decimal system does, the hexadecimal system uses digits 0 to 9 and A to F to represent numbers. To put a larger number than F (decimal 15) in column 16n it is neccesary to multiply 16 * 16n, which gives 16n+1 and carry a column to the left. For example, putting 1A in the 160 column is impossible, so a 1 is put in the 161 column and an A in the 160 column, thus using two columns. Twenty six is 26 * 160 = 160( 16 + 10 ) = ( 1 * 161 ) + ( 10 * 160 ).
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