Multiplication by 9

I recently saw a video 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: 1×9=9, 2×9=18, 3×9=27, 4×9=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).


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 7×9 (63) and 9×9 (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!

Engellenner family history

The topic that I am currently most interested in is the fate of my great-grandfather Wilhelm Ernst Eduard Engellenner (later he called himself Ernest Engellenner), born 7 June 1868 in Schleswig, emigrated to New York in 1891, married there and had children, went back to Germany (city of Kiel) in 1902, and died there in 1911. His wife and children then went to England (and probably to Ireland, where his wife Sarah came from), leaving Germany with a ship from Hamburg the 23 December 1911 and during the 1920ies to 1930ies they went back to the US, now with their own young families. A year before Ernest died, in 1910, my grandfather was born, but not by Sarah, but as an illegal child of Ernest and his cousin Friederike Karoline Christine Engellenner (they had the same grandparents). There are many questions connected to this and I am not even sure whether Sarah knew this. So, it would be the enormously interesting to hear what the New York Engellenner people know about him. I do not even have a picture of him and before I started researching him a few years ago, I only knew his name and the name of his cousin, my great-grandmother.

Read more about the Engellenner family in the genealogy section of this site. I will add more and more text in the coming weeks and months.

Multiplication with reference numbers

You may be familiar with the multiplication of two numbers near a threshold, using their reference number. A typical example is:

98 x 96 = (100 – 2) x (100 – 4) =

The threshold number is 100. You would take one of the numbers, subtract from this the difference of the other number to the threshold of 100, and this would be the first part of the product (e.g. 96 – (100 – 98) = 96 – 2 = 94). Then, you would multiply the two numbers’ differences from the threshold with each other (i.e. 2 x 4 = 8) and get the other part of the product. Putting both parts together would give the product of 9408. This method is based on the following relation:

(100 – a) x (100 – b) = 10,000 – 100b -100a +ab = 100 x (100 – b – a) + ab = 100 x ([100 – b] – a) + ab

Here, (100 – b) is the first number and a is the difference of the second number from the threshold 100. At the same time, ab is the product of the two threshold differences.

Now, this method can also be used for other thresholds. A useful application for mental arithmetics is the threshold 50. The only difference to the threshold 100 is that the product of the first number with the threshold difference of the other must be divided by two. Like in this example:

49 x 47 = (50 – 1) x (50 – 3) =

The left part is the difference of 49 and 3 (47 = 50 – 3). 49 – 3 = 46. Divide by two and get 23. The right part is just the product of the differences, i.e. 1 x 3 = 3. The result is 49 x 47 = 2303.

Another example, which needs some more thinking (let me write the left part and right part separated by a ‘|’ character):

49 x 46 = (49 – 4)/2 | 1 x 4 = 45/2 | 04 = 22.5 | 04 = 2254

See the difference? The division resulted in a number ending in ‘.5’. This number needs to be added to the right part of the answer, as 50.

And for the curious, here is the proof:

(50 – a) x (50 – b) = 2500 – 50a – 50b +ab = (5000 – 100a)/2 – 100b/2 +ab = 100 x ( (50 – a)/2 – b/2 ) + ab = 100 x ( (50 – a) – b)/2 ) + ab

… and this last relation is exactly what we do!