[LouisSayers]

> I can't "deduce" what 7*8 is by any rule

But you can break this into a different problem knowing that 2^3 = 8, and doing 7*2*2*2.

This isn't as fast but is in a way more useful because while 7*8 is fairly easy to remember you're not going to remember 17*8 etc but you can problem solve it fairly quick.

There are other ways of seeing the multiplication table as well. For example 9 times something can be thought of as 9*x = 10*x-x.

I never learnt these, but simply realised over time that there are different approaches to doing calculations.

[Dylan16807]

> But you can break this into a different problem knowing that 2^3 = 8, and doing 7*2*2*2.

Doing that multiplication all the way through is super slow. When they said "can't" they meant in an effective sense, since they did mention repeated addition as an option. And that's not an effective way to get there.

> There are other ways of seeing the multiplication table as well. For example 9 times something can be thought of as 9*x = 10*x-x.

Yes, you can do that one. But that's just about the only fast trick there is.

[lelanthran]

> Yes, you can do that one. But that's just about the only fast trick there is.

I dunno about that. For division, anyway, there's a bunch of fast tricks that give you a decent approximation (i.e. decent precision, maybe to the nearest integer)

Someone recently was surprised that I worked out the VAT (Value Added Tax, 15%) on a very large number in a few seconds. It's because its 10% of the number plus `(10% of the number)/2`.

It's easy to get 10% of any number. It's easy to halve any number. It's a fast trick because there's two easy operations.

There's a bunch of similar "tricks": 1%, 10%, 25% and 50% are fast to calculate in your head (at most 2 easy operations, like `(half of half of N)`). Then you either add or subtract N. Or you multiply N by 2.

At most three easy operations gives you 1%, 2%, 4%, 5%, 10%, 11%, 12%, 14%, 15%, 20%, 21%, 24%, etc

To someone who doesn't know how you are getting the answer it might seem like you are a human calculator because you get so many of those quickly, and they don't see the ones you don't do in 3 easy operations (say, 13%, which is 10% + 1% + (1% * 2)).

IOW, it looks like a very impressive trick, but it isn't.

[Dylan16807]

> For division, anyway

Did you not see the neighbor comment where I made it extra clear I'm talking about times tables in particular?

There are tons of math tricks! But the multiplication table of the numbers between 1 and 10 is mostly rote.

[wizzwizz4]

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

[Dylan16807]

The first one doesn't help with remembering times tables. The second one is outside the 1-10 times table we were talking about.

[wizzwizz4]

6 is 2×3, so 6×5 = 30. Likewise, 17 = 2×8 + 1, so 5×17 = 85.

[tsimionescu]

I completely disagree. First of all, at the time children learn the multiplication table, they definitely don't know the concept of exponentiation. Secondly, 7*2*2*2 is not some immediately obvious shortcut.

Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table. 17*8 is 7*8=56, hold the 5, 1*8 + 5 = 13, so 136.

[LouisSayers]

> Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table

You've actually just proved my point - you used a method of breaking down the problem into a different problem and then solving it rather than simply memorising.

If you give the same question to multiple people there will be numerous ways different people use to go about solving it.

As an example, I might solve this by doing

20*8 = 160 3*8 = 24 160 - 24 = 136

Or 10*8 = 80 7*8 = 56 80+56 = 136

And I might apply different tools like the one I originally mentioned within these calculations. I know that 80+20 is 100 and so "borrow" 20 from 56, so that I can easily add 100 and 36 together.

These ways of calculating happen in your mind very quickly if this is how you get used to calculating.

[tsimionescu]

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.

[lelanthran]

> Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table. 17*8 is 7*8=56, hold the 5, 1*8 + 5 = 13, so 136.

Interestingly, I do those less efficiently:

    17 * 8
     = (10 * 8) + (7 * 8)
     = (80) + (56)
     = (80) + (50 + 6)
     = 130 + 6

I think the reason I do it this way is because I get an approximation sooner when the numbers are very large i.e. I get the most significant digit first, and can stop calculating when I get the precision I require.*