If I cut something into 3 equal pieces, there are 3 defined pieces. But, 1÷3= .333333~. Why is the math a nonstop repeating decimal when existence allows 3 pieces? Is the assumption that it’s physically impossible to cut something into 3 perfectly even pieces?
In: Mathematics
We start by establishing a value for X.
X = 0.99… (repetend)
We then multiply both sides by 10.
10X = 9.99…
We then subtract 1X from both sides, but on the left side, using X, and on the right, X’s value. This is still equal, and thus mathematically permissible.
9X = 9
We then divide both sides by 9.
X = 1
If X=0.99…, and X=1, 0.99… must necessarily equal 1.
0.99… = 1
By extension, if we divide both sides by 3, we can further extrapolate that…
0.33… = 1/3
Therefore, 0.33… (repetend) is a perfect numerical representation of one third.
In truth, math is a lie essentially. It tries to put a point on a round surface. Math is just approximation of reality. Because as you get smaller and smaller all of the universe is still round (atoms, electrons, etc). So you cannot ever put an exact value on a point in time and space. It is only approximation to help us communicate the natural phenomena.
If you had a piece of wood that is 9cm long, you’d be able to cut it into 3 perfect pieces of 3cm each. Even in your 1 / 3 example, each piece is exactly 0.33333~, so they’re actually still equal.
Here is the math proof for it since no one else seems to have posted it.
Let; X=0.3333…
Therefore; 10X=3.3333…
Thus; (10X-X)=(3.3333…-0.3333…)
Or; 9X=3
Then; X=3/9 (X=1/3)
And as such; 1/3=0.3333…
Because mathematics and every other scientific language and law and principle are only *models* of reality, not exact descriptions of it though sometimes they come very close.
It’s bc it’s athe ratio you’re making by dividing a whole. Just can’t have 3 pieces equal if comparing to the whole you divided them from. That .33333 is that ratio. Jeez I don’t make sense. Sorry. It makes sense to me.
You are touching at the fringes of numbering systems.
When I was taking some networking classes we took 20 minutes to look at Hex and Opt (16 and 8) based numbering systems then we dived into binary.
I went home with this swimming in my head kind of fascinated with the idea.
I spent a few weeks screwing around and arrived at a very fascinating conclusion.
1) There is nothing magical or special about decimal. Nothing. We have 4 fingers and one thumb on each hand – we picked 10 done and done. Not only is it not magical or special it also isn’t really that great.
2) Other number systems are incredibly efficient when utilized properly.
My instinct (cause I am not as gifted in math as I wish I was) is that in a tertiary based system you would always come out with even answers – but when cutting something into half you would run into a problem!
It’s just something that happens when a number has a prime factor that isn’t a prime factor of the base (in this case base 10).
In a base 3 system (where you’d count like this: 0,1,2,10,11,12,20,…) 1/3 would be 0,1. 1/2 however would be 0.1111111111…
You said it yourself in the question, it’s a decimal (base 10) and 10 isn’t divisible by 3. If you used a base 12 system you could do it.
Theoretically a cake weighing 300g (3/3rds) can be equally divided in to 3 perfectly even slices each weighing exactly 100g (1/3rd) with no remainder. As long as you avoid decimal representation.
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