People say that out of the four basic mathamatical operations, the one which they like the most is addition. And they have an obvious explanation behind their choice, it is the easiest. But if you ask me, my favourite is Division. It is not because I find it easy. In fact, I find it to be the toughest to perform with a pen and a paper. The reason this one attracts me is how mysteriously harmonic all the results are. Here's an interesting observation I made the other day while performing a calculation:
Let me first begin by explaining how division actually works, for those of you who don't know. The basic definition of division, as we all learnt in the third grade, is 'Repeated Subtraction'. The number of times you have to subtract to get a zero, becomes the quotient. Let us say we have to divide 15 by 5. We subtract 5 from 15 once, it becomes 10. Subtract 5 once more, it becomes 5. Subtract 5 the third time, it becomes 0(zero). Hence, 15 divided by 5 equals 3. This was a case of perfect division(I don't really know the technical term to be used over here, but I'm sure you understand what I mean!). Let us now consider a different case. Let's say we have to divide 20 by 8. We first subtract 8 from 20 to get a 12, subtract once more, we get 4. No more subtraction is possible without going into negative numbers. Hence, the quotient in this case is 2, and the remainder is 4. Now, to get the exact value of 20 divided by 8, we divide the remainder by 8 as well. 4 divided by 8 can be reduced to 1 divided by 2, which we all know equals 0.5. Hence the exact value of 20 divided by 8 is 2 + 0.5 = 2.5.This can also be understood by splitting 20 into two parts, 16 and 4. 16 divided by 8 is 2, and 4 divided by 8 is 0.5. Hence 20 divided by 8 is 2.5.
We're now ready to observe a really interesting fact about numbers.
Note: Division by any of the numbers used here produces a recurring and infinite series of digits after the decimal. The given values repeat themselves after the last digit.
If you divide any number by 7, there can only be six possible remainders, excluding zero. Have a look at what their values in decimals are:
1/7 = 0.142857142857
2/7 = 0.285714285714
3/7 = 0.428571428571
4/7 = 0.571428571428
5/7 = 0.714285714285
6/7 = 0.857142857142
Did you notice any pattern? Yes, the digits '142857' keep repeating themselves in each of the divisions. Just the starting point is different for each one of them. Strange?
Let us try it again with 13!
1/13 = 0.076923076923
2/13 = 0.153846153846
3/13 = 0.230769230769
4/13 = 0.307692307692
5/13 = 0.384615384615
6/13 = 0.461538461538
7/13 = 0.538461538461
8/13 = 0.615384615384
9/13 = 0.692307692307
10/13 = 0.769230769230
11/13 = 0.846153846153
12/13 = 0.923076923076
If you're not a keen observer, you would say that the numbers don't follow any particular pattern over here. But look closely. For the numerators 1,3,4,9,10 and 12, the digits obey the following sequence of numbers '076923'. For the numerators 2,5,6,7,8 and 11, they obey '153846'. Why?
If you do it with 17, ALL the numbers obey the following 16-digit pattern '0588235294117647'.
What makes these numbers so disciplined?
How probable is it that you multiply a number (1/17) with another number, the digits simply re-arrange themselves? Multiply the first number with another number, the digits rearrange themselves once again. This happens not once, not twice, but SIXTEEN times in the case of the number 17.
Why do 7 and 17 have 6 and 16-digit patterns respectively, while 13 has two 6-digit patterns?
(7 - 1 = 6, 17 - 1 = 16, 13 - 1 = 12 = 6 * 2)
Why do the numbers 9 and 11 have only 1 and 2-digit, fairly simple patterns? (Check them out on a calculator)
What differentiates numbers 1,3,4,9,10 and 12 from the numbers 2,5,6,7,8 and 11, in the pattern for the number 13?
If you have the answer to any of the above questions, do leave a comment on this blog post with your name and e-mail address.
Thank you for sparing your time.
Note: If there is number incorrectly typed in, do let me know! And rest assured, its just a typing error. I've double checked all the numbers on my calculator. Some of you may not like the fact that I spent so much time on this during my board exams, but really, I think its worth it. Plus, I really couldn't wait another month to write this. =)
