I was looking at some fun math tricks for high-speed calculations and I came accross a very nice one involving 11 from Mathemagician Arthur Benjamin
Not very related video
So multiplying 11 by any single digit is pretty easy since you can just write the digit you are mutliplying by twice after the equality sign, for example 11x5=55 and 11x7=77. Easy! So does this easiness carry over when multiplying 11 by something with two digits? 11x11 is certainly not equal to 111 but maybe there is another rule that we can apply.
Multiplying two digit numbers by eleven
Let's try to derive it by example. 11x11=121 and 11x12=132 and 11x13=143. Can you see something special going on here. A secret pattern?
I will give you a hint using the same equations: 11x11=121 and 11x12=132 and 11x13=143. So you might have noticed that the first and last digit of the number multiplied by 11 is the same as the first and last of the solution, respectively.
Rule 1: The first and last digit of the solution are given by the first and last digit, respectively, of the number multiplied by 11.
Well what about the center number in the solution? Can you see a rule?
Again here is a hint: 11x11=121 , 1+1=2 and 11x12=132 , 1+2 = 3 and 11x13=143, 1+3=4. The middle number is just given by adding the digits of the number which is multiplied by 11. That is pretty easy!
Rule 2: The middle number is given adding the first and last digit of the number multiplied by 11.
Having established these two rules you can pretty easily compute 11x34 which is 374 (since 3+4=7), and 11x81 which is 891 (since 8+1=9). How about 11 x 74 = ? Is it equal to 7114? No it is not since it cannot be a four digit number. Does the rule break down? Well maybe we need to extend rule 1 and 2 a bit such that we can consider cases where adding the first and last number is greater or equal to 10.
The general rule
Let's again look at 11x74. With some basic arithmetic you find that 11x74=814. Then observe that 11x74=704 + 110 (7+4=11). Hmm, let's try some other examples: 11x89= 809 + 170 (8+9=17): 11x89= 809 + 170 (8+9=17), 11x48=408 + 120 (4+8=12). So now we have a good candidate for the general rule!
General Rule: The soluton of multiplying a two digit number by 11 is given by putting a zero between the multiplied number and adding both digits multiplied by 10.
We derived this general rule using examples. So we have not really shown that is true for all cases. This requires a short proof. Which you can find it in the technical appendix.
More than two digit numbers?
Let's look at some examples again: 11x123 = 1353 , 11x243=2673, 11x432=4752. Now using the General rule for two digits numbers you might already be able to derive a rule for three digits numbers. Or maybe even for any digit number!
Technical appendix: Proof general rule is true
It is a simple one line proof Denote a two digit number by ab. Then 11 x ab = 10 x ab + ab =a0b + (a+b) x 10 which is exactly the general rule.
Reference
[1] A. Benjamin. Secrets of mental math: The mathemagician's guide to lighting calculations and amazing math tricks.
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