Mental Math Tricks

Divide by six, multiply by ten. And vice versa. Quicker in the head when an approximation is required.

ChatGPT's original answer was surprisingly close.Both humans and LLMs fail to multiply those numbers in their heads.Some people can multiply them using a special algorithm. Either for mind or for pen and paper. Your example shows that LLM knows some of those tricks and can carry out the task using "pen and paper" (context in this case).Both humans and LLMs are able to do that multiplication correctly using a calculator.

I like the shove it to the nearest 10 approach. It makes a lot of calculations much simpler b/c they can be transformed to a simple multiplication by 10 and a addition or subtraction or two.1. 6⋅7 = (6⋅10) - (6⋅3) = 60 - 18 = 422. 7⋅7 = (7⋅10) - (7⋅3) = 70 - 21 = 493. 13⋅19 = (13⋅20) - (13⋅1) = 260 - 13 = 2474. 58⋅61 = (58⋅60) + (58⋅1) = 3480 + 58 = 3538If we go up another order of magnitude, then the system starts really grinding to a halt though tbh :)

I can't reliably multiply four digit numbers in my head either, what's your point?

Hmm that's interesting - I'd tend to round one of the numbers, whichever is closest to a '5' or '10', to make an 'easier' multiplication, then add / subtract the extra value lost / gained by rounding.Example 9 x 14 = ( * 14) - ([1] * 14) = 140 - 14 = 126key: easier rounded number [] deal with rounding 'error'

Multiplication and division by 2 and 5 correspond to each other. Multiplication by 11 is shift-and-add. There are many such tricks.

Then you haven't realized how well what I wrote actually scales. In base 10, 2 operations gets you the whole of single digit multiplication (versus memorizing the whole table just to get it down to 1). It also gets you to 11, 12, 20, 21, 19, 30, 40 (by various use of the append 0 trick to multiply by 10 for free anywhere you want). Now try to work out which multiplications are possible within 3 additions/subtractions? How about 4? You should find the accessible fraction of the number l

Is this the best way to do quick arithmetic, or are there other ways?

Unrelated note, you can do simple multiplication by 9 with your fingers: place your palms on a table and imagine your fingers are numbered 1-10 from left to right.4 * 9? Lift up your fourth finger - you have 3 to the left and 6 to the right - answer is 36.8 * 9? Lift up your eight finger - 7 to the left and 2 to the right - answer is 72.I hope it's not totally common knowledge thing.

Sure, but all of those work for numbers higher than 10, and all assume you know the multiplication table by heart. The multiplication table (the result of multiplying every number between 1 and 10 with each other) is something you have to memorize. You can get away with memorizing only some of these results and computing the others based on them, but it's basically impossible to do any more complex arithmetic if you don't know most of it by rote memorization.