How can you do e.g. 756 x 675 mentally?

[cyentist]

Hello!

How can you do e.g. 756 x 675 mentally?

Any good trick?

 

[jedishrfu]

Yes, learn the Trachtenberg system and give it a try.



https://en.wikipedia.org/wiki/Trachtenberg_system

 

[hilbert2]

Hello!

How can you do e.g. 756 x 675 mentally?

Any good trick?

To do that, you need a very good short-term memory to not lose track of what you're doing.

Something like $999\cdot 999$ is simpler because then

$(1000-1)^2 = 1000^2 + 2\cdot 1000 \cdot (-1) + (-1)^2 = 998001$

and the numbers here are "special" enough to easily keep track of.

 

[jedishrfu]

Vedic math might have some tricks to allow you to do this too but I think the Trachtenberg method is the best mentally.

https://vedicmathsindia.org/vedic-mathematics/?v=7516fd43adaa

I need to say that Vedic math is not without controversy as the name is a misnomer (ie not from the Vedic period but from a 1965 book on the subject) and its more a collection of arithmetic tricks and not math perse.

As a kid, when I was asked to do this kind of calculation by a friend, I'd ask him why he needed the answer and while he was responding, work it out mentally and then answer. It gave me a few moments but people thought my answer came out instantaneously which was okay by me.

 

[snorkack]

and the numbers here are "special" enough to easily keep track of.

756*675 luckily has special enough numbers. 4*675=2700, and 756 happens to be divisible by 4. So simplifies to 189*27.
And 189 is 200-11. 11*27 is nicely 297. So 5400-297=5103. The answer is 510300.

 

[WWGD]

This may not always work, but does in some cases: first, use the trick for squaring : $a^2=a^2-b^2+b^2$. Seems tautological but then factor as $a^2=(a-b)(a+b)+b^2$. So , e.g., $988^2=(988+12)(988-12)+12^2=(1000)(976)+144=976144$. Now, if you have either two events or two odds, e.g., if you had 756*676,you can find the average and apply the trick: 756*676= (716+40)(716-40)=$716^2-40^2$. And $40^2=1600$ and you find a convenient number to help you square 716. So this works out well with some pairs of numbers. I hope I wrote this clearly.