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
Just because it’s a non-stop repeating decimal, it’s still a rational number because it can still be expressed as a fraction.
0.3 repeating can be written as 1/3.
This also forms the basis for one of the easier proofs that .9 repeating = 1
1/3 + 1/3 + 1/3 = 3/3 = 1
.333333333333 + .333333333333 +.333333333333 = .9999999999999 = 1
**Math is not reality, it’s just a *description* of reality.** You can cut your Thing into three perfectly equal pieces, and then describe each piece as:
* 1/3
* .33…
* 1 ÷ 3
…and no matter which description you pick, it doesn’t change the Thing. If you choose .33… then you’re picking a description which is an infinite repeating series. If you pick 1/3 then you’re picking a fraction which is a perfectly even piece of a whole. Either way, your Thing was still cut into three equal pieces, and no description will change that.
You could even describe each piece as 1, and all three together as 3 – that wouldn’t mean that your Thing has tripled from its original size! It just means you’ve changed the way you’re describing reality.
Just to add a little to other answer … we think of 1/3 is a nonstop repeating sequence due to our predominant use of base-10 to represent numbers. In base-3 1/3 is not a repeating sequence, it’s 0.1.
It’s a quirk of the way we represent numbers. In numerical bases other than decimal, you can express one third without infinitely repeating digits.
* The problem isn’t math.
* The problem isn’t the laws of nature either.
* It’s just a quirk of the number system we invented.
* Imagine this:
* You have 10 marbles.
* Using all 10 marbles, make three equal groups.
* You can’t since if you did three groups of three you still have a marble left over.
* Now imagine each marble is made up of 10 smaller marbles stuck together.
* Now try it again. You still can’t do it because you’d still have one of those smaller marbles left over.
* Now image you had 9 marbles instead.
* You can easily split those up in to three equal groups.
* But what if you had to split them into two groups?
* You can’t because you’d still have a marble left over.
* What if each marble was really a group of 9 marbles stuck together that could be broken apart?
* You still have the same issue of a marble left over.
* With any number system (Base 10, Base 9, Base Whatever) you’re going to run into numbers that are hard to represent cleanly
The fact that 1/3 is a repeating decimal is an artifact of the completely arbitrary base 10 system we use to represent numbers, and has nothing to do with physical reality. If we used base 9 instead of base 10, 1/3 could just be written as 0.3, while 1/2 would be written as the infinitely repeating decimal 0.444….
So…engineer here….who wants to use hex with me?
Dividing the number ‘1’ into thirds isn’t the same as dividing a cake into thirds. Simply put, numbers aren’t cakes and cakes aren’t numbers. The number ‘1’ has the repeating decimal, a cake does not. Different things divide different ways.
No. You’re confusing the fundamental nature of reality with the peculiar details of our human-created numbering system. One-third is only represented by repeating decimals because we’re using base-10 numbers. There’s no good way to represent a third with base-10. However, if you use a better base numbering system like base-12, a third can be represented quite easily indeed (although, suddenly, it’s quite hard to represent one-fifth.)
The real lesson to learn is that it’s hard to represent all numbers using any system that humans have been able to create.
I think even the best explanations so far aren’t quite getting to the level of a 5 year old. So here goes:
Think of the thing you are cutting as being 3 feet long. You then cut it into 3 equal pieces, each are 1 foot long. If you then cut one of the 1 foot bits into 3 equal pieces it would still be equal, we just don’t have a number to represent that length.
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