# Math Tricks

## Multiply up to 19x19 in you head.

This is a useful bit of mental arithmatic that is worth remembering, it allows you multiply up to 19 x 19 without the use of a calculator or paper and pencil.

For example consider 19 x 17. Imagine the larger number on the top row and take the last digit of the lower number ie. 7 and add it to 19. 19 + 7 =26.

Multiply the 26 x 10 = 260.

Take the unit digits (9 x 7) = 63 and add them to the 260

260 + 63 = 323.

## Multiplication of Larger Numbers

This trick allows you to multiply large numbers together by using simple mulptiplication and additions. To take a simple example, if we want to multiply 26 by 72. We write the numbers down in the normal way.

26
72

Each of the numbers can be thought of being in one of four cells arranged in a square.

 2 6 7 2

The first step is to multiply the digits in the first column. (6 x 2 = 12)

If there it is greater than 9 remember the value of the tens digits. In this case write down the 2 and remember the 1.

Multiply the two diagonals and add them together. So from the top left to the bottom right and the bottom left and top right. (2 x 2) + (6 x 7) = 4 + 42 = 46. Add the from the last stage = 47. So we write the 7 and remember the 4.

Finally we multiply the right hand column. (2 x 7) = 14 and add the 4 from the last step. = 18.

### Three digits numbers

 4 6 8 5 2 6

We start in the same way, 6 x 8 = 48. Write, 8 and remember, 4.

(6 x 6) + (2 x 8) = 36 + 16 = 52. Don't forget to add the 4 = 56. Write down the 6 and remember the 5.

Now for the third digit we have three small numbers to multiply and sum. Starting from the (4 x 6) + (2 x 6) + (5 x 8) = 24 + 12 + 40 = 76. And 5 from the previous step, 81. Write the 1 and remember the 8.

We go back to using the 2 x 2 cells but this time on the left. So, (4 x 2) + (5 x 6) = 8 + 30 = 38 and 8 from the previous step = 46. So write 5 and remember 4.

Finally, the leftmost column, (4 x 5) = 20. Plus 4 = 24.

You could go on and multiply four numbers in a similar way.

### Four Digit Numbers

Now you should be getting the idea, we use this rather contrived example for 4 digit numbers so it is clear which numbers are being multiplied and just show the working out. We also introduce the notation of w to mean the number which you write down and c to be the number that you carry.

 1 2 3 4 5 6 7 8

(4 x 8) = 32 = w 2, c 3.

(3 x 8) + (7 x 4) = 52. 52 + 3 = 55. w 5, c 5.

(2 x 8) + (6 x 4) + (3 x 7) = 61. 61 + 5 = 66. w 6, c 6

(1 x 8) + (5 x 4) + (2 x 7) + (6 x 3) = 60. 60 + 6 = 66. w 6, c 6

(1 x 7) + (5 x 3) + (2 x 6) = 34. 34 + 6 = 40. w 0, c 4

(1 x 6) + (5 x 2) = 16. 16 + 4 = 20. w 0, c 2

(1 x 5) = 5. 5 + 2 = 7