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
The number being infinite doesn’t mean the object is infinite. As we add digits to the number it isn’t getting bigger or smaller, it’s getting more accurate. The object remains finite and fixed.
So…engineer here….who wants to use hex with me?
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.
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.
**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.
* 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
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 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….
Latest Answers