When working with computers, we typically use three [and to some degree, four] numbering systems. The most common to the reader will be Decimal, a numbering system with ten digits. Digital computers work with a second numbering system—Binary—with only two digits. The third most common numbering system is Hexadecimal, a sixteen digit system. Hexadecimal is often abbreviated as “hex,” which should not be confused with the six digit system, Heximal or Hexal. The last and least common numbering system is Octal, an eight digit numbering system.
All the numbering systems (not just computer-related) have two features in common. First, all start at ZERO, rather than one; for that reason, the highest value for a digit is ONE LESS than the total number of digits allowed.
| Base or “Radix” | Allowed Digits |
|---|---|
| Binary | 0,1 |
| Octal | 0,1,2,3,4,5,6,7 |
| Decimal | 0,1,2,3,4,5,6,7,8,9 |
| Hexadecimal | 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F |
Note that hexadecimal uses letters for the last six digits beyond decimal “9”. This is merely a convention that eliminates the need for “inventing” a new character to represent one of those digits.
The second common feature is the how individual digits are “lined up” to form a number in a given base. In a series of digits, each has a “place” which indicates relative magnitude in relation to the base number. In decimal, we’re taught that we have the “one’s place”, the “ten’s place”, the “hundred’s place”, etc., for as many places as we want to add (we won’t worry about fractional representations in this discussion) . In reality, however, we’re really indicating the magnitude of that place based on the powers of the base.
That is to say, the “one’s place” is based on “ten to the zero power”, or 10º = 1. The “ten’s place” is based on “ten to the first power”, or 10¹ = 10. The “hundred’s place” is based on “ten to the second power”, or 10² = 100.
We do likewise in all the numbering systems: If the base or radix is N, then the places (in increasing magnitude) are Nº, N¹, N², etc. The sum of the digits multiplied by the magnitude gives the “true value” of a number. If the bushel-basket contains seventeen apples, then that number of apples can be expressed in the following manner:
| Base or “Radix” | “seventeen” |
|---|---|
| Binary | 10001b |
| Octal | 21o |
| Decimal | 17d |
| Hexadecimal | 11h |
Note the use of a lower-case letter to indicate which numbering system is in use for all the representations of “seventeen”. Where it may be ambiguous or my intention is specific, I’ll be using that convention to indicate the different numbering systems.
Because we “live & breathe” the decimal system, our best method of converting an arbitrary number from one base to another requires two steps: convert the number to decimal, then convert decimal to the new base.
Converting to decimal should be simple: The digit in each place is converted to decimal (which is trivial in all but Hex) and multiplied by the “place” in decimal. Again, the place is numbered from right-to-left, starting at zero. Then the base is taken to the place number and multiplied by the digit.:
| Step | FE04h | |||
|---|---|---|---|---|
| 1) separate all the places | F | E | 0 | 4 |
| 2) convert each place to decimal | 15 | 14 | 0 | 4 |
| 3) Assign a magnitude multiplier | 16³ = 4096 | 16² = 256 | 16¹ = 16 | 16º = 1 |
| 4) multiply the magnitude times the place value | 15 × 4096 = 61440 | 14 × 256 = 3584 | 0 × 16 = 0 | 4 × 1 = 4 |
| 5) add all the results together | 61440 + 4584 + 0 + 4 = 65028 | |||
2705o = (2 × 8³) + (7 × 8²) + (0 × 8¹) + (5 × 8º) = 1477d
1101b = (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2º) = 13d
Conversion from decimal is simply a sequence of divisions, and rules for using the results of the division. In any conversion, the decimal number is divided by the decimal value for the base: 2, 8, 16 (or any other base). The remainder of the division is then converted to the base, and starting from the Nº place, the new number is “built” to the Nª place after “a” steps. If the quotient of the division is greater than the base, then another division is performed, and the remainder is converted and added to the next place of the new number. This proceeds until the quotient is less than the base. At this point, the quotient is converted from decimal to the base, and is added to the next place in the new number.:
1447d in binary:
| Step | Number |
|---|---|
| 1) 1477 ÷ 2 = 738 remainder 1 | 1 |
| 2) 738 ÷ 2 = 369 remainder 0 | 01 |
| 3) 369 ÷ 2 = 148 remainder 1 | 101 |
| 4) 184 ÷ 2 = 92 remainder 0 | 0101 |
| 5) 92 ÷ 2 = 46 remainder 0 | 00101 |
| 6) 46 ÷ 2 = 23 remainder 0 | 000101 |
| 7) 23 ÷ 2 = 11 remainder 1 | 1000101 |
| 8) 11 ÷ 2 = 5 remainder 1 | 11000101 |
| 9) 5 ÷ 2 = 2 remainder 1 | 111000101 |
| 10) 2 ÷ 2 = 1 remainder 0 | 0111000101 |
| 11) 1 < 2, so convert and place | 10111000101 |
1447d in hexadecimal:
| Step | Number |
|---|---|
| 1) 1477 ÷ 16 = 92 remainder 5 | 5 |
| 2) 92 ÷ 16 = 5 remainder 12 | C5 |
| 3) 5 < 16, so convert and place | 5C5 |
The conversion between hex or octal and binary is actually far easier than the steps I’ve shown above, which is actually why the two numbering systems became widely used in computing: octal and hex become “shorthand” for binary numbers.
As it turns out, the first three binary places (2², 2¹, 2º) can convert to exactly all the digits in the octal system, which represents the first place in that system. For each three additional places in the binary system, another one place in the octal system is generated. This gives programmers a three-to-one “compression” of the number of characters needed to represent a given binary number.
Hexadecimal takes this shortcut one place farther, by encompassing four binary places per hexadecimal place. In practice, that gives you a direct conversion between the two systems and binary.
From the above example, 2705o is converted to binary...
| Step | 2705o | |||
|---|---|---|---|---|
| 1) separate all the places | 2 | 7 | 0 | 5 |
| 2) convert each place to a three-place binary, adding leading zeros if needed | 010 | 111 | 000 | 101 |
| 3) string the 3-place binaries together | 010 111 000 101 | |||
| 4) remove or add leading zeroes (as applicable) | 10111000101b | |||
Another example: 10111000101b converted to hex...
| Step | 10111000101b | ||
|---|---|---|---|
| 1) separate into blocks of four places, adding leading zeros if needed | 0101 | 1100 | 0101 |
| 2) convert each block to a hex digit | 5 | C | 5 |
| 3) string the hex digits together | 5 C 5 | ||
| 4) remove or add leading zeroes (as applicable) | 5C5h | ||
The last “trick” is to use binary as the “pivot” between octal and hex (instead of decimal), by combining the above two examples:
| Step | 2705o | |||||
|---|---|---|---|---|---|---|
| 1) separate all the places | 2 | 7 | 0 | 5 | ||
| 2) convert each place to a three-place binary, adding leading zeros if needed | 010 | 111 | 000 | 101 | ||
| 3) string the 3-place binaries together | 010111000101 | |||||
| 4) separate into blocks of four places, adding leading zeros if needed | 0101 | 1100 | 0101 | |||
| 5) convert each block to a hex digit | 5 | C | 5 | |||
| 6) string the hex digits together | 5 C 5 | |||||
| 7) remove or add leading zeroes (as applicable) | 5C5h | |||||
| Comments? Email jim3@millard.org | Last updated October 20, 2001 |