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#EverydayMaths Apparently it's National Numeracy Day. So here are a couple of examples of how an experienced mathematician might go about mental arithmetic. The first one is to multiply 29 by 35. To multiply two close numbers, it is often helpful to use ... 1/
the difference-of-squares trick. Here we see that 29x35=(32-3)(32+3) and that equals 32 squared minus 3 squared. A maths nerd like me will know that 32 is 2 to the five, so its square is 2 to the ten, which is 1024. So we just have to subtract 9 from that, getting 1015. 2/
The point there was that it can be done without one having to hold a lot in one's head at once -- assuming one knows a few useful facts about things like differences of squares and powers of 2.
For a second example, let's try 28x91. Another good thing to do ... 3/
For a second example, let's try 28x91. Another good thing to do ... 3/
for this kind of problem is think about factorizations. Here we notice that 28 and 91 are both multiples of 7. So we need to work out 4x7x13x7. But the 7x7 part is 49, which is just less than 50, which is a "nice" number. And 4x13=52, as any card player knows. 4/
So let's go for 49x52 instead. We can work out 50x52 easily -- we divide 52 by 2 to get 26 and multiply by 100 to get 2600. But that's overshooting by 52. Fortunately, subtracting 52 from 2600 is easy -- it's 2548. So that's the answer. Again, much more comfortable 5/
than trying to do an actual long multiplication in your head. It's surprising how often one can find such tricks to reduce the burden on the short-term memory. Although those two examples were not generated randomly, I think pretty well any two two-digit numbers are ... 6/
easier to multiply in your head if you spend a little time looking for tricks like these than if you try to do it using the conventional long-multiplication algorithm. 7/
Just to clarify -- that last assertion is true only if you have a reasonable repertoire of useful bits of background knowledge. So it's a simple demonstration of the value to mathematicians of playing around with mathematical concepts and noticing little facts about them. 8/
