How do I multiply 3 three digit numbers simultaneously?

Main Question or Discussion Point

Without any rough work

Forget 3-digit numbers. How would you multiply three 1-digit numbers "simultaneously"?

Is there any method like crosswise method of multiplication of two numbers

Three digits at time! In looking at an algorithm, which can be used in order to evaluate

So you are asking for a general algorithm that essentially allows one to compute the product of three numbers in one's head, correct?

Do you understand the crosswise method? I mean understand why it works, not just how to execute the algorithm. Why does the crosswise method for computing the product of two numbers give the correct answer? If you think about that question, it might give you some insight for creating your own method for computing "bigger" products.

Mark44
Mentor
I don't think it's possible to multiply three numbers at the same time. The best you can do is to multiply two of them to get an intermediate product, and then multiply that one times the remaining number.

Okay then!

@gopher

But the thing is that it's too complex

What does "too complex" mean?
- Do you mean it's not clear how one is supposed to proceed? If so, you don't have a well-defined algorithm.
- Do you mean it requires too much working memory to use in practice? If so, change your algorithm to involve writing more things down, so you have less reliance on working memory.
- Do you mean it takes too long? If so, you may just be SOL. If I had to multiply three three-digit numbers without a calculator, it would take me quite a while. Maybe there's a more efficient (i.e. fewer steps, or easier steps) algorithm you can find, maybe not.

I don't think it's possible to multiply three numbers at the same time. The best you can do is to multiply two of them to get an intermediate product, and then multiply that one times the remaining number.
why not?

(100*a+10*b+c)(100*d+10*e+f)(100*g+10*h+i) = 1000000 a d g+100000 a d h+10000 a d i+100000 a e g+10000 a e h+1000 a e i+10000 a f g+1000 a f h+100 a f i+100000 b d g+10000 b d h+1000 b d i+10000 b e g+1000 b e h+100 b e i+1000 b f g+100 b f h+10 b f i+10000 c d g+1000 c d h+100 c d i+1000 c e g+100 c e h+10 c e i+100 c f g+10 c f h+c f i

you can look up the one digit multiplications in a table.

That's how you do it.

Borek
Mentor
1000000 a d g
That's the first multiplication.

100000 a d h
That's the second multiplication.

10000 a d i
That's third, and so on.

Three consecutive multiplications already, and we are still far from the final result.

Basically it depends on how you define "at the same time".

Mark44
Mentor
To add to what Borek said, to do the first multiplication (1000000 a d g) you would probably multiply a and d together, and then multiply that by g, and finally multiply that by 1000000. Multiplication is a binary operation, meaning that it operates on two numbers. That's what I was talking about in post #6.

To multiply three numbers, you can use the fact that multiplication is associative (i.e., (ab)c = a(bc)) to multiply a pair of the numbers and then multiply that product by the third.

To add to what Borek said, to do the first multiplication (1000000 a d g) you would probably multiply a and d together, and then multiply that by g, and finally multiply that by 1000000. Multiplication is a binary operation, meaning that it operates on two numbers. That's what I was talking about in post #6.

To multiply three numbers, you can use the fact that multiplication is associative (i.e., (ab)c = a(bc)) to multiply a pair of the numbers and then multiply that product by the third.
If I multiply two 1 digit numbers, I just use memorized multiplication tables. You could do that with multiplying 3 1 digit numbers also. Mulitiplying with at million is just a left shift.