Binary notation had not yet been standardized, and Napier used what he called location numerals to represent binary numbers. Roughly speaking, it used alphabets to stand for various powers of two.
He used a to represent 1, b for 2, c for 4, d for 8, e for 16 and so on. To represent a number as a location numeral, express it as a sum of powers of two and replace the powers by the letters. For example
- 87 = 1 + 2 + 4 + 16 + 64 = abceg
A location numeral can similarly be converted back into standard notation:
- abdgkl = 1 + 2 + 8 + 64 + 512 + 1024 = 1611
He permitted letters to repeat, so the same number could be represented in multiple ways. For example
- abbc = acc = ad = 9
Notice that since each letter is twice the value of the previous one, two occurrences of the same letter can be replaced with one of the next letter without changing the value of the number. Thus you can always remove all repeated letters from a location numeral, and Napier called this the abbreviated form of a number. If on the other hand a location numeral has repeated letters, it is the extended form of the number.
Napier showed ways to convert numbers into and out of abbreviated form which are identical to modern techniques to convert numbers into the binary numeral system and we will not repeat them here.
Location numerals provide a simple way to do addition: just write two numbers in abbreviated form together and abbreviate the result. For example to add 157 (acdeh) to 230 (bcfgh) just write them together
- acdeh + bcfgh = abccdefghh
and abbreviate the result
- abccdefghh → abddefghh → abeefghh → abffghh → abgghh → abhhh → abhi
and abhi = 387 = 157 + 230 as expected.
Subtraction is only a little more complicated. To subtract bcfgh from abhi, first change abhi into its extended equivalent abccdefghh and just remove the letters bcfgh
- abccdefghh - bcfgh = acdeh
to get the result acdeh.
Napier used his non-standard representation of binary numbers to explain his techniques to do arithmetic. However, the rest of this article will rephrase his ideas using the more modern binary notation.
Read more about this topic: Location Arithmetic