How do I do large multiplications quickly

Written Multiplication - Written Multiplication

In contrast to the procedures for written addition and subtraction, only a maximum of two numbers can be multiplied in one step. Of course, you can repeat the process with the resulting product (product is the result of multiplying) as often as you like.

We will see that the procedure is based on the distributive law. It is therefore helpful if you already know this, but not absolutely necessary, as you can learn this procedure very schematically.

One more note: At the beginning it was said that the procedure was applied to multiplications that could not be calculated in your head. We will see, however, that with a little practice and after understanding this procedure, one will be able to multiply large numbers, for example 57 * 83.

But now to the procedure itself.

We want to find the product of 538 and 217.

Step 1: We write the numbers next to one another very neatly, a line is drawn under the product to provide an overview, we will later need as many lines as the number on the right has places and one for carry-over, because later we add. Everything has already been prepared here, but it can also be added gradually as required.

2nd step: We start with the highest digit at the right number (i.e. the hundreds place) and multiply this by the units of the left number. We write the ones of the result under the hundreds of the right number. You remember the tens, here they are shown as subscript numbers, but you usually remember them in your head.

Then multiply the highest digit of the right number by the tens of the left number, after adding it to the carryover, write it to the left of the previous digit, then multiply by the hundreds and, if available, thousands, etc.

So

2 8 = 16 (first digit 6)

2 3 = 6 (+ carry over 1 from the 16, i.e. second digit 7)

2 5 = 10 (no carryover of 7, i.e. third digit 0)

no further product, but the carryover from the 10th, i.e. the fourth digit 1

3rd step: Repeat the 2nd step with the second highest digit of the right number, so:

1 8 = 8 (first digit, comes under the second highest digit, is 8)

1 3 = 3 (second digit 3)

1 5 = 5 (third digit 5)

4th step: Repeat the 2nd step until there are no more places left, so:

7 · 8 = 56

7 3 = 21 (carry 5, i.e. 26)

7 5 = 35 (carry over 2, i.e. 37)

  1. step: Add the lines.

The product 538 · 217 is therefore 116746.

Relationship between written multiplication and distributive law

We'll use the example above and rewrite it a little. We write the right number as a sum: 217 = 200 + 10 + 7 and multiply the following expression in brackets according to the distributive law:

It is noticeable that the products of the divided numbers are equal to the summands from our scheme above. This makes sense when you consider that the distributive law does exactly the same thing here as our method above. Basically, there are two different ways of writing the same thing.

Mental arithmetic: multiplying larger numbers in your head

We now want to calculate the example from above 57 · 83 in our head. We write or think of the numbers 57 and 83 as (50 + 7) and (80 + 3) and multiply the brackets according to the distributive law according to the following scheme:

So you calculate tens times tens plus tens times ones plus the other combination of tens and ones plus one times one.

You could certainly multiply three or four-digit numbers in your head, but you have to memorize a lot of relatively large numbers and then add them up, which is why this procedure will probably be limited to at most two-digit multiplications.