Why Did Thomas Harriot Invent Binary?

Strickland, L.
Why Did Thomas Harriot Invent Binary? Math Intelligencer 46, 57–62 (2024)
https://doi.org/10.1007/s00283-023-10271-9

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Apparently, an English gentleman scientist named Thomas Harriot invented (but did not publish about) binary numerals no later than 1605. This is a little article that gives a plausible reason for why someone living in 1605 might discover the binary numeral system, when there’s no knowledge of electronics or electricity to provide motivation.

As I read the article, Harriot was doing some exacting weighing, trying to determine specific gravity of various common liquids. Unfortunately, the system of weights in use at the time, troy weight, was exceedingly clumsy. For example 1 troy ounce is 8 drachms. Each drachm is 3 scruples. Each scruple is 20 grains. Instead of measuring large numbers of grains, or an unholy mix of ounces, drachms, scruples and grains, Harriot measured in ounces (troy), half, quarter, eighth and sixteenth ounces, plus a few grains, with some kind of balance. Harriot organized his data in columns, and seems to have made the jump to binary notation from the columnar organization.

A few pages in Harriot’s notebooks illustrate some very familiar exercises.

Writing out a few small decimal numbers as binary. It would have made more sense right-justifying the column, and on another page, Harriot does just that. Doing this lets you see how the least significant digits ripple in the same pattern.

Above, the classic positional-numeral-system long multiplication algorithm.

Multiplying 1101₂ (13₁₀) by 1101₂ giving 10101001₂. 1101₂ is a pretty good test case. It has four digits, not all 1s. Four digits gives you several pieces of multiplication to add up, and to see patterns emerge. The least significant digit is a 1, so it’s an odd number. If you work the multiplication incorrectly and get a 0 least significant digit, you know you did something wrong. It has no 2-adjacent-zeros, so there’s not a lot of empty rows to add, but there’s at least one. Harriot got the correct answer, 169₁₀.

An interesting paper for a few reasons.

1. The “Imperial” units that the USA uses are not the most messed up system possible. They’re not even the most irregular and goofy system ever used in an English speaking country.
2. Who knows how many interesting innovations like binary got invented and discarded along the way? The only reason we know about this particular instance of invention and discarding is because Thomas Harriot was patronized by English nobility, thus able to afford notebooks and keep notes.
3. Thomas Harriot himself. Harriot seems to have been an interesting person. He came to North America with Sir Walter Raleigh on the expedition that left behind the Roanoke Colony, among many other things. He died of cancer attributed to excessive tobacco consumption. Ironically, tobacco seems to have come to England when Harriot and Raleigh returned from Roanoke.

2026-05-28 The same author translated Leibniz’s EXPLANATION OF BINARY ARITHMETIC