The simplest trick for converting from one base to another is grouping. To convert between base b and base b^k, group numbers in sets of k and convert one group at a time. To convert from binary to octal, for instance, group bits in sets of three, starting from the right end, and convert each group to octal.

11110010_two → (11)(110)(010) → 362_eight

For an example going the other direction, let’s convert 476 in base nine to base three.

476_nine → (11)(21)(20) → 112120_three

In general, conversion between bases is too tedious to do by hand, but one important case where it’s a little easier than it could be is converting between decimal and octal. In combination with the grouping trick above, this means you could, for example, convert between decimal and binary by first converting decimal to octal. Then the conversion from octal to binary is trivial.

The key to converting between decimal and octal is to exploit the fact that 10 = 8 + 2, so powers of 10 become powers of (8 + 2), or powers of 8 become powers of (10 − 2). These tricks are easier to carry out than to explain. You can find descriptions and examples in Knuth’s TAOCP, volume 2, section 4.4.

Knuth cites a note by Walter Soden from 1953 as the first description of the trick for converting octal to decimal.

The trick for moving between base 9 and base 10 (or by grouping, between base 3 and base 10) is simpler and left as an exercise by Knuth. (Problem 12 in section 4.4, with a solution given at the back of the volume.)

Related posts

• Radix conversion calculator
• When is round-trip floating point radix conversion exact?
• Hexadecimal floating point
• How to memorize a Bitcoin address
• Testing divisibility by 7, 11, and 13

The post Tricks for radix conversion by hand first appeared on John D. Cook.