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There's a very simple way of doing foolproof long multiplication, provided you can do simple multiplication up to 9×9. This one is so far as I know Mediaeval in origin, but I learned it about thirty years ago and don't remember where I found it. It's complicated to explain, and needs diagrams, but once you know how to do it it's really fast and can be used for figures with decimal points as well.
In fact, you don't even need to know the 9×9 table. For those readers who don't know how to do mental arithmetic (and there are bound to be some), know that all you need to know is that ten times any number is the original number with a zero on the end; how to halve a number; how to do simple addition and subtraction and the two-times table.
So, for example:
7×7 is the same as 5×7 plus 2×7. 5×7 is half of 10×7, that is, half of 70, which is 35. 2×7 is 14. So 7×7 is 35 plus 14, which is 49.
3×9 can be either:
2×9 plus 1×9. 2×9 is 18. So 3×9 is 18 plus 9, which is 27.
or:
5×9 minus 2×9. 5×9 is half of 10×9, that is, half of 90, which is 45. 2×9 is 18. 45 minus 18 is 27.
Suppose you want to multiply 4271 by 317. Draw up a box divided into four cells lengthways and three in height, and write 4271 along the top and 317 down the right-hand side, with 3 at the top end and 7 at the bottom, viz.:
[If you preferred, you could have three columns and four rows, with 317 along the top and 4271 down the side. The important thing is that the number which is written vertically should be written from top to bottom, not bottom to top.]
Divide the individual cells with diagonal lines from top right to bottom left, creating seven diagonally-sloping columns running across the table and down onto a line below it, viz.:
For each square cell in the table, multiply the number of its column by the number of its row - so for example for the third cell from the left in the top row, multiply 7 x 3, because it's in the column underneath the 7 of 4271, and in the row next to the 3 of 317. If the result is a single figure, write it in the triangle at bottom right of the cell. If the result is in double figures, write the first figure in the triangle at top left of the cell and the second figure at bottom right, viz.:
Now add up the diagonally-sloping columns from right to left, writing the total of each one on the bottom line and carrying any tens into the next column. So the first column is just 7, then the next is 9 + 1 so you write 0 and carry the 1, then the next column is 4 + 4 + 7 + 3 plus the 1 you carried, making 19, write down the 9 and carry the 1 to the next column etc., viz.:
And there's your answer along the bottom - 1353907. If your leftmost diagonal column came to 10 or more you would just write the 1 to the left of the diagonal line.
To work out figures with decimal points in, just treat them as long whole numbers until you get an answer, as above, and then do a rough calculation to tell you where the point should go. If the figures being multiplied here were actually 42.71 and 31.7, you would just do a rough mental calculation - 30x42 is 3x40 plus 3x2 with an extra zero on the end, so it's 1260 - only we're really multiplying 42-and-a-bit by 31-and-a-bit so the answer will be in the same general region as 1260 but a bit more, so it's going to be 1353.907.
