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 reason is because we use base 10 as our counting system. It’s totally arbitrary. If we used base 12, 1/3 would be written exactly as 0.4. Not 0.4444…just 0.4.
Repeating decimals aren’t some kind of philosophical conundrum. They’re just the result of what number system we choose to use.
You’re trying to shoe horn something that’s really simple into a decimal numbering system.
They’re just thirds, .33333… is just 1/3. 1/3 x 3 = 3/3 = 1.
Nothing is missing.
The answer is simply that 10 isn’t evenly divisible by three. That’s it.
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.
> Why is the math a nonstop repeating decimal when existence allows 3 pieces?
It’s a representation problem because you are using [base 10](https://en.wikipedia.org/wiki/Decimal). Like the other poster said, imagine every marble can be split into 10 sub-marbles, and those submarbles can be split up into 10 sub-sub marbles, and so-on.
So if you split up into **any multiple of 2 or 5**, it will be exact. (For example, if I have one marble, I can split into 10 sub-marbles. Now I can divide them by 5 or 2, or by 10.) That means splitting up into 2, 4, 5, or 8 is easy.
On the other hand, splitting thing up into **3, 6, 7 or 9** pieces requires dividing by 3 or 7, which is not possible when all you have are fractions of 10. Ditto for any larger numbers not evenly divisible by 2 and 5. For example, if you look at all the numbers between 20 and 30, only 20, 25 and 30 are OK. The rest will repeat. (NOTE: Numbers like 1/22 or 1/3 are still [rational](https://en.wikipedia.org/wiki/Rational_number) numbers: They can repeat, but only in a repeated pattern, unlike irrational numbers which have no pattern.)
Computers have the same problem, but worse: They use binary, which is base 2. Anything that is not divisible by 2 is a problem. That means “1/10” [can not be represented in floating point](https://www.exploringbinary.com/why-0-point-1-does-not-exist-in-floating-point/) — it requires an infinite decimal.
> Is the assumption that it’s physically impossible to cut something into 3 perfectly even pieces?
Well, it depends. Let’s say you have a pie, and you are trying to divide it up into 3 equal pieces. How would you know if they are equal? The truth is, the only way you get it 100% evenly divided is if the number of atoms/molecules in the pie was **exactly** a multiple of 3. If there were an even number of Atoms, it is impossible to slice the pie into 3 equal parts.
For the same reason you just eyeball the pie and say “those 3 parts look are equal”, you shouldn’t worry about the accuracy of truncating a repeating decimal. For example, NASA only needs [15 decimal places of PI](https://kottke.org/16/03/how-many-digits-of-pi-does-nasa-use) to send out probes to the entire solar system.
.3333 repeating is just a numerical representation of 1/3. It doesn’t mean the sum can’t add back up to the whole. What it does imply, is that there is NO such thing as perfect. No matter the precision, there is no such thing as perfection.
Define the item as having length 3. Then you can partition it into three pieces of length 1.
Define the item as having length 1. Then you partition it into three pieces of length 1/3=0.333…
They’re all real numbers. The last one is a real number too. Don’t get hung that it’s a repeating decimal. That’s a consequence of base 10 here.
Similarly, if decimal places are infinite, then moving two fingers closer together should mean they can never touch. There is infinite space between them. But there’s not.
Math assumes you can divides something at an infinitely small precision.
The real world does not have infinite precision.
why limit your answer to decimal form? As noted by others, fractions work great. 1/3 or one third solves your immediate problem elegantly.
Are they perfectly symmetrical pieces? There lies your problem.
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