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Comment by frankus | Hacker News Reader

Our math prof (who moonlighted as a "mathemagician") taught us a trick for two-digit squares:

For e.g. 23 x 23, subtract three from the first number, add three to the second, and then add 3² to the product. So 20 x 26 + 9, the idea being that multiplying by a multiple of 10 is easier to do mentally.

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Neat! This is (a+b)(a-b) + b² = a²-b² + b² = a² and in your eg, a=23 and b=3

In addition to using this trick for getting to multiples of 10, I used it to compute the product of two numbers by leveraging their proximity to a number in between whose square I knew. For eg if I need to multiply 23 by 27, I instead see it as (25-2)(25+2), which is 25²-2² = 621.

(using that same trick from the article to calculate squares of numbers that end in 5)

For ref: https://en.m.wikipedia.org/wiki/Difference_of_two_squares#Me...

That’s a neat one!

(a + b)(a - b) = a^2 - b^2

a^2 = (a + b)(a - b) + b^2

So pick a value of b that makes a - b end in a zero.

FWIW you can also pick a value of b to make a + b end in a zero, which keeps b at 5 or less.

This would have made my college career significantly easier. We were limited on scratch paper (dumb policy) on many tests, and I have to write out arithmetic like this long-hand style, which took up a ton of space.

I think these sort of mental math tricks are mostly useful when you have a case where you’ve got a formula that you often have to run, and it is similar enough every time that you can pre-do the algebra.

The one everyone knows is calculating a tip (bump the decimal place left once and double the result, round depending on how nice you feel). Might be applied multiple times per day, depending on your dining habits.

I bet the formulae you had to use on your tests were worth doing by hand.

They weren't formulae - it was literal arithmetic. Complete waste of time and space.

My formula for calculating a tip is `return 0`, but it's very country-dependant.

(US-defaultism?)

You may tip anywhere you find the service pleasant.

(EU guy who has worked as a waiter)

I recently leafed through "Mathematics for Engineers"[0], originally from 1926, which begins with many such methods for making arithmetic easier. Though maybe not always more space efficient.

Luckily I got to use a calculator for school.

[0]https://archive.org/details/dli.ernet.11725/page/n9/mode/1up

If you look at the way they teach kids math on paper today, they are just teaching them how to do math in your head. The idea being that if your hands were free you’d use the calculator on your phone.

When I first encountered the outrage over New Math my first thought was that this is how I avoid embarrassing myself in checkout lines. Do I have enough cash to pay for this stuff in my hands?

I do the less efficient quadratic form. It works but is slower.

20² + 20 x 3 x 2 + 3²

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