Note: This post originally appeared on my discontinued website maartensity.com. The published date and time has been adjusted to mirror the original.
After the piece on double digit mental arithmetic I wrote not long ago, I wondered if the method could be extended to 3 digit multiplication. However, as I found out while trying, it's all too easy to create temporary numbers beyond all practicality, clogging short term memory and pushing the brain onto a rollercoaster of calculation and recalculation to recover numbers from earlier phases!
Nevertheless, it turns out there is indeed a way to build on the 2 digit method that yields results, and requires only a bit more from short term memory than the 2 digit method. (Remember, the goal is to do the calculation without the aid of a pen and paper!).
Without further ado, let's use for our example
569
x 378
The first stage is a repeat of the method we employed in the previous article. We take the two rightmost digits and multiply them (have a look at the previous article to see how this is done).
69
x 78
____
111
72
42
____
5382
This forms the foundation. Our mind has been primed by these numbers, making the subsequent calculations slightly easier to do.
Let's take another look at the original three digit numbers:
569
x 378
In the second stage we multiply the (thus far unused) number in the top left corner with the (already used) bottom two right digits: 5 x 78. That gives 390. Likewise, we then multiply the (unused) number in the bottom left corner with the top right two digits: 3 x 69. That gives 207, and added to 390 gives 597.
569
x 378
The final step in stage two is to multiply the number in the top left with the bottom left: 3 x 5 gives 15.
Now 597 actually represents 59700, and 15 represents 150000, so to calculate the addition we can invoke them visually as follows:
597
15
____
2097
Note: 15+5=20 followed by 97 gives 2097.
The final calculation we do is to add the earlier result (5382) to the latest (2097). Remember that they represent different multiples of 10, so to add them we follow the same kind of visual method as above:
5382
2097
______
215082
And that is the answer!
Seems easy? Actually, it does take more than twice longer than the 2 digit method, but that is largely down to the number of steps involved, and not due to the intrinsic difficulty of any individual step.
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