This article introduces a mental division method that gives very precise results at ligntning speed. The divisor is limited to numbers finishing with a '9', such as 69, 299, etc.
He was born as the eldest of seven children on April 1st 1895 in Otago, New Zealand. In 1920, he graduated in Otago in French, Latin, and mathematics. The same year he married Mary, a botany teacher in Otago. Eventually they migrated to Edinburgh, Scotland where he became a professor at the University of Edinburgh in 1925.
He was very fond of mathematics, and specifically mental calculations. He is one of the few performers that shared his speed math secrets with others. What you read in the following paragraphs is one of them.
Division is a rather tedious operation, and mental calculation techniques are sparse. The technique Aitken used applies to two- or three digit divisors ending with a 9, and assuming the 'performer' can divide mentally with a one digit divisor (1-9) or a two-digit divisor (12, 30, etc).
The idea behind his method is to round up the divisor to the next number ending with a zero, and do the simpler division mentally. In order to compensate for the rounding up of the divisor, the quotient must be appended to the original number and the division continues indefinitely, but quickly and accurately.
Let us try the division 45/29.
Step 1: round up the divisor to 30, and perform the division to 2 digits (same number as the divisor) to give 1.5.
Step 2: Divide 1.5 by 30, or .15 by 3 to give .05, append to the quotient to give 1.55
Continue adding the quotient and dividing the added digit by 30.
45/30=1.5
46.5/30=1.55 (15/3=5, remainder 0)
46.55/30=1.551 (5/3=1, remainder 2, to be carried over to make 21)
46.551/30=1.5517 (21/3=7, remainder 0)
46.5517/30=1.55172 (7/3=2, remainder 1)
One can continue the division just by looking at the quotient and continue the division by 30 (or mentally, 3).
The answer is 45/29=1.55172413793103448275 to 20 decimal places. You could verify your own calculations using the arbitrary precision arithmetic page at this site.