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
So place value: when we write numbers, we can write the numbers 0 through 9 using one digit. Then we get to ten, and we write it as two digits: 10. This is called base ten, because ten is the smallest 2 digit number.
Why? Tradition. With motivation, but still, it’s just what we’ve decided to do, most of the time (though sometimes we write numbers in other ways).
1 divided by 3 is .33333… for a reason almost like what you said – it’s not that you can’t divided anything into the pieces, it’s that you can’t divide a group of 10 things into 3 equal groups. If you remember your long division, when you try to do one divided by 3, you say “well, 3 doesn’t fit into 1, so let’s put a zero after our and divide 10 by 3 instead (and we’ll put the answer in the tenths place to make up for that)”. So then you can fit the groups of 3 into that 10, but you still have 1 left over. So you do it again, putting the answer in the hundredths place, and still have 1 left over, and so on. It won’t fit evenly because you can’t divide 10 things into 3 even groups.
That’s only a problem because in base ten, everything is thought of in groups of ten. But you don’t have to think of numbers that way. You could think of numbers as groups of three.
And you *can* divide a group of three things into three groups rather easily. One in each group. So you could decide to write numbers in base 3: zero is written 0, one is 1, two is 2, but now you write three as 10, four as 11, and so forth.
If you write numbers this way, one divided by three is written 0.1. But now, 1 divided by 10 is weird. Heck, even 1 divided by 2 is weird, it becomes 0.11111… (One divided by ten is written 1/101 in base 3, which I didn’t feel like working out in my head.)
You can do this with other numbers as well: for any fraction a divided by b (a and b whole numbers, b not 0), there is some base where a/b is goes on forever, and then another where it does not.
And then there are the irrational numbers that are weird in any normal base… And it turns out that there are a bunch more of them than all the nice numbers put together, even though there are an infinite number of both. Math has a lot of cool things.
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