Friday, February 27, 2015

What's 14 subtracted from 432?

Sorry, I just had to get this out of my head. I learnt about binary system in JC but I was very bad at it. However, that new number system is so vastly different from the current decimal one that I'm obsessed and fascinated by it at the same time. It had always been on my mind and recently more so. Did I tell you that being moody makes me ultra creative in unexpected ways? It can be in the form of art, poem, music or whatever other forms it takes. It just happens to take the form of a elaborate sibei eng new number system. 


We’re all fixated with the decimal system. We have 10 digits on our hands, and that makes it easier to count, so I guess that’s the reason why our number system is based on 10 digits from 0 to 9. But let’s say we are all aliens with 4 digits, how would our ‘normal’ viewpoint of numbers change? Let’s start from first principles.

If we have 4 digits in our hands, the only numbers that we can work on is 0,1,2,3 and 4 – essentially a 5 digit system.  There’s no digits after 4, so there's no 5, 6, 7, 8 or 9.

Looking at our current decimal system, how do we get the next number after 9? We add one more placing to the left of our single digit, and reset the number 9 back to 0. So after 9, we can 10. Back to our 5 digit system, what’s the number after 4? We add one more digit to the left of our number and reset our number back to 0. So, after 4, we get 10. Let’s run through the whole number sequence:

For decimal:

0,1,2,3,4,5,6,7,8,9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21…

For our 5 digit system:

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30…

So what happens after we reach 40+? 40, 41, 42, 43, 44. What’s the next number?

By right, we should add one to the left of the digit we’re currently working on, and reset the digit back to 0. But since both digits are 4, we have to reset the numbers and add another digit to the left again. We end up with 43, 44, 100, 101, 102, 103, 104, 110, 111 and so on and on.

That’s for addition.

In principal, subtraction is the same as addition, except you’re going backwards. How do we do 432 – 14 in our new 5 digit number system? The best way to do this is to convert back to decimal system. Every 10 in our new system is equivalent to a 5 in decimal system, so 20 is 10 in decimal, 30 is 20 in decimal, and 40 is 30 in decimal and so on and on

Every 100 in our new system is actually 25 in decimal, so 200 is 50 in decimal and 300 is 75 and so on.

In other words, the hundreds placing of our new system is really how many multiples of 25; the tens placing is how many multiples of 5 and the ones placing is just ones. In our new system, if we have a number xyz, then the conversion of this number to decimal is just 25x + 5y + z

Therefore 432 is equivalent to 4x25 + 3*5 + 2 = 117 and 14 is 1*5 + 4 = 9. Since 117 – 9 = 108 in decimal system, we need to convert back to our new 5 digit system.

108 is not entirely divisible by 25, so the best we can do is 4 x 25, and we’ll have a remainder of 8. Next, we try dividing 8 by 5, the best we can do is 1 x 5 and we’ll have a remainder of 3. So, 108 = 4x25 + 1x5 + 3. In other words, 108 in decimal is 413 in our new system.

432 - 14 = 413

Can we derive a new method of doing subtraction by hand? Sure.

This is essentially the problem.  Looking at the ‘ones’ placing, 4 cannot be subtracted from 2, so we need to take away the ‘tens’ placing (i.e. 3), knock off 1 and add 5 (not 10 because we’re doing a 5 digit number system) to the ‘ones’ placing. Like this:

Looking back at the ‘ones’ placing, 5+2-4 = 3. And 2-1 = 1, so we have:

Essentially the same answer if we were to convert the number 432 and 14 back to decimal, carry out the subtraction, then convert back to 5 digit system.

This means that our operation of numbers is very robust in the sense that besides being able to work on the current decimal system, even by restricting the digits available at hand (from 10 to 5), it can still work. I haven't tried multiplication, which is really another form of addition, and also division, which is really another form of subtraction. From there on, it can develop in powers (another kind of multiplication) and what other operation out there.

Is this important? Nope. But it's certainly fun for me. I'm trying to answer this age old question. Why is 1 + 1 = 2? The answer is that it need not be 2. It really depends on which number system you're using. It can really be 10 if you're using binary for example.


Hey, while showering, I just came up with a formula to convert any base system to our standard decimal one! If the base is y and the numbers are abcd, then to convert it, you just have to do this:


For example, if we want to convert the binary number (base 2) 10110 to decimal, we just do this:

1(2^4)+0(2^3)+1(2^2)+1(2^1)+0(2^0) = 22.

Or if you want to convert 432 (base 5) to decimal, you just do this: 4(5^2) + 3(5^1) + 2(5^0) = 117!



Sillyinvestor said...

How about doing that in roman numeric system?


How?? Can try to do??


la papillion said...

Hi SI,

HAHA, roman numerals are super super inefficient! Not going to do lol!

la papillion said...

In every post of mine, there's a literal content and a metaphorical content. The literal content is usually not as important as the metaphorical one, because it's hidden, abstract and usually transferable to other fields.

What's important in this post, really, is how you can deconstruct underlying assumptions so that you can analyse what's really important, construct it back and see if it works.

Ferns said...

Hi LP,

First time I'm posting on your blog, but I've been following it for a while.

I'm a big fan of mathematics too. Judging from your love of reading and possibly mathematics, there are a few books I'd recommend, had a great time reading them.

Search for authors "Alex Bellos", "marcus du sautoy" and "Ian stewart", I found alex bellos and marcus more interesting though.

la papillion said...

Hi ferns!

I'm so glad that you gave me some book recommendations! I Googled Alex bellos and I'm quite interested in some of the books :) I'll definitely read up on the authors you've mentioned here!

Thanks so so much for the book recommendation - it's the best kind of gift for me!