daniel tammet fascinates me. in his ted talk, different ways of knowing, he explains how he calculates 64 x 75 by using shapes, instead of the traditional way learned in school.
for those of us with a mind operating differently than tammet's, it might be necessary with an extra few steps for a full understanding on how "the tammet way" works. at least i found it useful myself.
representing 64 as a shape is quite straightforward, as that number can be represented as a 8 x 8 chessboard. and as you can see, 8 x 8 is a square, hence 64 is a square number. now you know why we call certain numbers for square numbers.
100 is a square (10 x 10), and 75 is 3/4 of hundred, hence the representation of 75 as shown above.
next up mr tammet states:
"what we do need to do now, is put these two pictures together, in our mind. something like this. 64 becomes 6400. ... 16, 16, 16, 48, 4800, the answer to the sum."
showing the following slide:
this last part was the part that i needed to sit down and think through. below the steps in more details, in fear of oversimplifying though.
squares of squares
how to calculate 8 x 8, using shapes.
we have a 8 x 8 square. this can be divided into four 4 x 4 squares. since 4 x 4 is 16, we get that the sum is 16 + 16 + 16 + 16 equals 64.
what about 64 x 100?
here we have a 8 x 8 square that we are going to "put together" with a 10 x 10 square. the shape is equal in these cases. here it is simple to multiply the sides. so the new shape is 80 x 80. this is the same as four 40 x 40 squares. or 1600 + 1600 + 1600 + 1600 equals 6400.
now, it is easier working with smaller numbers. instead of multiplying the sides of the square you are better off multiplying the result. 8 x 8 is four squares of 16, equals 64. this multiplied with 100 equals 6400.
finally, 64 x 75...
this example is very similar to the one above. however, instead of having four squares of 16 we have three (remember the shape of 75). that is the reason why the end result is 16 + 16 + 16 = 48, and multiplied with 100 this gives 4800.
"beauty is truth" - daniel tammet