Multiplication by 9

I recently saw a video (https://www.youtube.com/watch?v=0x9vx8nfaL4) showing a shortcut for multiplication of two-digit numbers with 9. The typical way of doing this is multiplying the number by 10 and then subtracting it from that product, like so:

36 x 9 = 36 x (10-1) = 360 - 36 = 324

In the video, however, a nice shortcut was presented:

1. Write down the left digit of the number as the left digit of the product: 36 x 9 = 3…
2. Find out how many digits there are between the first and second digit of the number when looking at the series [1 2 3 4 5 6 7 8 9 10]: between 3 and 6 are 4 and 5, so 2 is the next digit of the product: 36 x 9 = 32…
3. Find out, how many digits on that series of numbers are right of the last digit if the number: after 9 there are 7, 8, 9 and 10, so 4 numbers: 36 x 9 = 324

But there is a problem: this shortcut only works if the left digit (tens’ digit) of the number is less than the right digit (ones’ digit), so it does work with 36, 67, 29 but not with 66, 43 or 94. Another problem is, in my opinion, is that you need to somehow visualize and count digits on a series of numbers, making the process slow.

It turns out, the mathematical background of the shortcut is quite easy and without the limitation of the number series from 1-10, it can be used also for numbers like 66, 43 or 94. This is why:

1. Any product of a number with 9 will have a digit sum of 9. In the above example, the product of 324 has a digit sum of 3+2+4=9. So, if you have any two digits of the product, you can easily detect the missing digit by looking for the number that is needed to make the digit sum 9. (By the way, this a phenomenon that is often utilized in mathematical magic tricks).

2. Any number multiplied by 9 always has a ones’ place that is the 10’s complement of the ones’ digit of the multiplier. So, any number xxx8 by 9 ends with a 2 (the 10’s complement of 8). You can easily see this by looking at the multiplies of 9 in the multiplication table: 1x9=9, 2x9=18, 3x9=27, 4x9=36 … At every multiplier the ones’ digit goes one down while the multiplier goes one up.

With these principles, any two-digit number can now easily be multiplied by nine. There is just one thing to remember in addition. If the number’s ones’ digit is higher than the tens’ digit, the first digit (from left) of the product is the same as the tens’ digit, otherwise it is one lower (i.e. you must subtract one).

Examples:

36 x 9 =
3 (first digit of 36),
2 (3+4=7, 2 is missing to make the digit sum equals 9),
4 (10’s complement of 6)
= 324

The difficulty here is only to calculate the ones’ digit as the second step in order to determine the middle digit and thus to speak out the number from left to right. In Germany, we have an advantage since we speak these numbers like „three hundred, four and twenty“ making it easier to calculate the digit sum along the way.

83 x 9 =
7 (first digit of 83 minus 1),
4 (7+7 is digit sum 14, which itself is digit sum 5, so 4 is missing to make it 9)
7 (10’s complement of the 3 in 83)
= 747

44 x 9 =
3 (first digit of 44 minus 1),
9 (3 and 6 have a digit sum of 9, so 0 is missing. However, we need a 9 here)
6 (10’s complement of the 4)

The last example reveals the only weakness of the shortcut. What to do when the digit sum already is 9? Being off my a multiple of 9 in the resulting product is not captured by the digit sum. However, this weakness is in fact a wonderful new shortcut: When the number to be multiplied by 9 has all equal digits, the first digit of the result is again „-1“, the last is again the 10’s complement and all digits in between are always 9. And: there is always one 9 less than the number of digits in the original number.

So, 777 x 9 magically becomes 6993, 9999 x 9 = 89991 and so on.

By the way, in these numbers you will recognize the product of 7x9 (63) and 9x9 (81) written as the first and last digits of the product and the additional 9’s are just put in between the two digits of that product!