Basic Method — for Small Numbers
When calculating a multiplication where one of the numbers is small, such as 68435 × 18, it may be fastest to simply add together multiples of the smaller number:
- 5 × 18 = 90 ⇒ …………0
- 9 + 3 × 18 = 63 ⇒ ……….30
- 6 + 4 × 18 = 78 ⇒ ……..830
- 7 + 8 × 18 = 151 ⇒ ……1830
- 15 + 6 × 18 = 123 ⇒ 1231830
In fact when I coach beginners in Mental Math, one of the first things I work on is expanding the knowledge of times tables to other useful numbers — such as 18 — to make this easier to perform.
But when using this basic method for larger multiplications, such as 29136 × 5847, we don’t have enough working memory to calculate each multiple of e.g. 5847 without forgetting the numbers we’ve already calculated! So we need another method — one that’s more efficient in terms of memory.
Below I’ll show you the cross-multiplication method that most advanced mental calculators use for multiplications.
Cross-Multiplication Method
Some nice things about this method are:
- You only need to know your times tables up to 9 × 9
- However large the multiplications, you never have to remember many numbers at once
- It’s very straightforward once you know the simple pattern
To see how this works, we’ll take our example of 29136 × 5847:
1st digit — units place:
To begin, we simply multiply 6 × 7 = 42. Then the rightmost digit of the answer is 2 and we can “carry” the 4 for the next step:
29136 × 5847 2 (carrying 4 from the 42)
2nd digit — factors with 10:
Next we consider 40 × 6 and 7 × 30, as these are the digit products (4 × 6 and 7 × 3) that come with a factor of 10, just like the 40 we remembered from the previous step.
The quickest way is to start with the 4 from the 40 that we carried, then add on the 4 × 6 and 7 × 3:
- 4 + 4 × 6 = 28
- 28 + 7 × 3 = 49
These addition-multiplication pairs are quick to do with practice.
Again we can write down the “9” in the tens place of the final answer, and keep the 4 for the following step.
29136 × 5847 92 (carrying 4 from the 49)
3rd digit — factors with 100:
We continue with 800 × 6, 40 × 30 and 7 × 100, as these are the digit products that come with a factor of 100.
Again — start with the 4 from the 400 that we carried, then add on the other products:
- 4 + 8 × 6 = 52
- 52 + 4 × 3 = 64
- 64 + 7 × 1 = 71
Write down the “1” in the 100s place of the final answer, and keep the 7 for the following step:
29136 × 5847 192 (carrying 7 from the 71)
Notice that in each step, the order of the colors on the top is the mirror image of the colors on the bottom, as each matching pair of digits must represent the same power of 10.
You can add the digits in any order you like, but I find it helpful to always start with the bottom-left and top-right product (here the 8 × 6) and systematically move simultaneously rightwards on the bottom number and leftwards on the top number.
4th digit — factors with 1000:
By now the pattern should be fairly clear, so I’ll continue with minimal commentary:
- 7 + 5 × 6 = 37
- 37 + 8 × 3 = 61
- 61 + 4 × 1 = 65
- 65 + 7 × 9 = 128
29136 × 5847 8192 (carrying 12 from the 128)
5th digit — factors with 10,000:
This time, notice that the “6” has already been multiplied by every digit of the bottom number, so will not be active for the rest of the calculation:
- 12 + 5 × 3 = 27
- 27 + 8 × 1 = 35
- 35 + 4 × 9 = 71
- 71 + 7 × 2 = 85
29136 × 5847 58192 (carrying 8 from the 85)
6th digit — factors with 100,000:
From now on the calculation gets simpler as there are fewer and fewer products of the same magnitude:
- 8 + 5 × 1 = 13
- 13 + 8 × 9 = 85
- 85 + 4 × 2 = 93
29136 × 5847 358192 (carrying 9 from the 93)
7th digit — factors with 1,000,000:
- 9 + 5 × 9 = 54
- 54 + 8 × 2 = 70
29136 × 5847 0358192 (carrying 7 from the 70)
8th digit — factors with 10,000,000:
- 7 + 5 × 2 = 17
As this is the final stage, we don’t have to carry anything, and simply write down the remaining digits:
29136 × 5847 170358192
Summary
So that is the standard cross multiplication method used by amateur human calculators, as well as current and past multiplication world-record holders such as Freddis Reyes and Marc Jornet Sanz! (Jeonghee Lee prefers a left-to-right method instead).
To conclude — the method in general is:
- Start with the rightmost digit of each number:
- calculate their product
- write down the units digit
- carry the tens digit for the next stage
- For every subsequent digit of the answer:
- take the carried number from before
- add all products of the same magnitude by working systematically
- write down the units digit in the final answer
- carry the rest for the next stage
In the Memoriad competition, and in the Mental Calculation World Cup, competitors must multiply 8-digit numbers, such as 12345678 × 98702468, and the fastest competitors can do these in 15-30 seconds!
To practice this you can use the Memoriad software — although I recommend to start with smaller products of 3- and 4-digit numbers before working your way up.
To get in touch with me (Daniel Timms) about mental calculation training or anything on this site, you can contact me here.