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 issue is that the number already exists. 1+1 doesn’t equal 2. 2 has always existed and we are trying to describe it in the way we know how on what that number is. Not to get too philosophical but math doesn’t exist, just the results we end up at – and they’ve always been there we just use what we call math to create understandable predictions. So that weird infinite number is an actual number somewhere and was there before we ever looked at it. We probably made it infinite because we can’t express it correctly
Fun fact, if you cut it in half, it’s 0.500000~.
We can measure lengths that have patterns of repeating decimals. Those are called rational numbers. You know, Ratio-nal. They can be written as fractions.
But not every fraction has a denominator made of twos and fives multiplied together, which is the only way you get a nice stopping point where the rest of your decimal expansion is zeroes.
This is one advantage that fractions have over decimal notation… when handled properly, there is no need to round and there is no loss of precision.
Some numbers just don’t play well together. Our numbering system (Arabic numeral system) is base 10. Three and ten do not play nice, so you get imperfect representations when converted into a floating point format.
I’m addition to the various answers on the math side of this question, I thought I’d chime in on the physical aspect. Simply put, you can’t split an object into 3 equal pieces. There’s always some level of tolerance when cutting pieces, even if that tolerance is really really small. To use the marbles analogy: if you have 10 marbles, you’d end up with 3, 3, and 4. In this case the tolerance is pretty large. Now imagine if you have 10,000 marbles. Same problem, but now that one extra marble makes less of a difference because there are 33,333 others in each pile rather than just 3.
Guy who dropped out of a math major bachelors degree program here lol- no, it’s not because of assumptions that it’s physically anything or whatever.
It’s because our number system is base 10. imagine your have a chocolate bar that is made up of 10 squares of chocolate and you have two others friends that you’re going to share the chocolate fairly with. How many chocolate squares do u give to each person?
Some cultures have used a base 12 or base 60 number system. A base 60 number system is kinda what we use to tell time with clocks. Let me show you what I mean using military time as an example. In military time 12:00 am is 00:00 and 01:00 am is 01:00 and 02:35 pm is 14:35. After 60 minutes go by, the number in the hours place goes up by one. So it’s kinda like the hours place number tells us how many groups of 60 minutes have gone by. Because 60 is a multiple of 3, a base 60 number system wouldn’t have that repeating decimal in the situation you described. Whats 1 hour divided by 3? Why it’s 20 minutes. 1hour/3= 20 minutes because 1 hour = 60 minutes. If I say the time is 13:10 then that means it have been (13*60 + 10) minutes since the day started.
What if we have a number system where none of the digits past 6 existed. So imagine 7,8, and 9 don’t exist and we counted in groups of 6 rather than groups of 10 like in our base 10 number system.
so our ways of counting and representing the numbers we counted is different.
5 (base 10) = 5 (base 6)
6 (base 10) = 10 (base 6)
7 (base 10) = 11 (base 6)
12 (base 10) = 20 (base 6)
35 (base 10) = 55 (base 6)
36 (base 10) = 60 (base 6)
0.5 (base 10) = 0.3 (base 6)
so in base 6 numbers, 10/3 = 2
and 1/3 = 0.2
Not that I know anything about it, but does existence really allows it? Let’s take for example a pice of bread made out of 10 atoms, how can you split that pice of bread in 3 perfectly cut parts?
Want to break your head? 0.999… = 1.
1. 1/3 is 0.333 repeating: 1/3 = 0.333…
2. Multiply both sides by 3 to get rid of the fraction: 1/3 * 3 = 0.333… * 3
3. 3/3 = 0.999…
4. 1 = 0.999…
Want to get weirder? Try multiplying 0.999… by 10, which is just moving the decimal one spot to the right.
1. 10 * 0.999… = 9.999…
2. Now get rid of that annoying decimal by subtracting 0.999… from both sides: 10 * (0.999…) – 1 * (0.999…) = 9.999… – 0.999…
3. The left hand side of the equation is just 9 x (0.999…) because 10 times something minus that something is 9 times the aforementioned thing. And on the right hand side, we’ve canceled out the decimal.
4. 9 * (0.999…) = 9
5. If 9 times something is 9, that thing must be 1.
Lots more fun stuff in the chapter, Straight Logically Curved Globally from the book [How Not to Be Wrong: The Power of Mathematical Thinking](https://smile.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535), by Jordan Ellenberg.
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.
> Is the assumption that it’s physically impossible to cut something into 3 perfectly even pieces?
No, not at all. On the contrary, what that should tell you is that the world doesn’t always conform to base 10.
Math still works mostly the same regardless what base we use for our numbers. In base 12, for example, “10” (12) divided by “3” (3) equals “4” (4), and 1 (1) divided by 3 (3) equals “0.4” (the fraction 4/12, which is equivalent to 1/3).
So you have to consider that regardless of which base you use, the physical qualities of real world things is not affected by that.
All marbles aside.
It’s the total you’d like to separate.
1 is not separable intob3 equal pieces. But 3 is.
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