How Mr. Benjamin squares numbers
This post is a sequel to the post How Mr. Bidder calculated logarithms published a few days ago. As with that post, this post is based on an excerpt from The Great Mental Calculators by Steven B. Smith.
Smith's book says Arthur Benjamin squares large numbers using the formula
n^2 = (n + a)(n - a) + a^2
where a is chosen to make the multiplication easier, i.e. to make n + a or n - a a round number. The method is then applied recursively to compute a^2, and the process terminates when you get to a square you have memorized. There are nuances to using this method in practice, but that's the core idea.
The Great Mental Calculators was written in 1983 when Benjamin was still a student. He is now a mathematics professor, working in combinatorics, and is also well known as a mathemagician.
Major systemSmith quotes Benjamin giving an example of how he would square 4273. Along the way he needs to remember 184 as an intermediate result. He says
The way I remember it is by converting 184 to the word dover' using the phonetic code.
I found this interesting because I had not heard of anyone using the Major system (the phonetic code") in real time. This system is commonly used to commit numbers to long-term memory, but you'd need to be very fluent in the system to encode and decode a number in the middle of a calculation.
Maybe a lot of mental calculators use the Major system, or some variation on it, during calculations. Most calculators are not as candid as Benjamin in explaining how they think.
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