When we divide by 11

Hello guys, it has been a long time... well, to be exact it has been 17 days or 408 hours or 24,480 minutes or 1,468,800 seconds.. some calculation, huh? I used a calculator! So I love to experiment with numbers and arithematics and all that thing. I mean I love math. I love playing with numbers and all that stuff. Due to this reason only I have been able to remember the Multiplication Table upto 25! So... in one of my previous blogs titled 'When we divide by 9' we observed a pattern in the series of division of natural numbers by 9. It was a very fascinating... Well this one's going to be more fascinating than that! So let's roll.

Dividing by 11[]

Now, if you are interested in the process of this problem then please keep a rough notebook and a pen/pencil with you to get an idea of the unimaginable result. SO here goes the series:-

$\frac{1}{11} = 0.\overline{09}$

$\frac{2}{11} = 0.\overline{18}$

$\frac{3}{11} = 0.\overline{27}$

$\frac{4}{11} = 0.\overline{36}$

$\frac{5}{11} = 0.\overline{45}$

$\frac{6}{11} = 0.\overline{54}$

$\frac{7}{11} = 0.\overline{63}$

$\frac{8}{11} = 0.\overline{72}$

$\frac{9}{11} = 0.\overline{81}$

$\frac{10}{11} = 0.\overline{90}$

$\frac{11}{11} = 1.0$

$\frac{12}{11} = 1.\overline{09}$

and so on!

This pattern is continuous unlike the one when we divide by 9. You might have observed that this pattern is following the multiplication table of nine after the decimal point and the pattern recurs after a complete division happens (11 divided by 11).

Another way to learn the pattern[]

But there is another way to learn the pattern. All the digits follow the pattern of multiplication table of 9 (including the digits before decimal point). Let me elaborate- After a complete division step occurs, the product that we get in the future steps(till another complete division step occurs) is sum of the corresponding multiple of nine and the product of the previous complete-divison step. For example,

$\frac{11}{11} = 1.00$ This is a complete-division step. Now, when we go further let's say I want to find the product of 13/11. What I will do is this:- The 13^th multiple of 9 is 117. And the product of 11/11(most recent complete-division step) is 1. I will add them both and get 118. Now I will put a decimal point between the two 1s and get something like this : $1.\overline{18}$ . Note that the overline above 18 represents the recurring decimal value. And this is the answer for 13/11. Easy huh!

Next in the Series[]

So I will be posting some more blogs in this series of 'When we divide by a #'. The next in the series are:-