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
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
You said it yourself in the question, it’s a decimal (base 10) and 10 isn’t divisible by 3. If you used a base 12 system you could do it.
Theoretically a cake weighing 300g (3/3rds) can be equally divided in to 3 perfectly even slices each weighing exactly 100g (1/3rd) with no remainder. As long as you avoid decimal representation.
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.
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?
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.
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.
We start by establishing a value for X.
X = 0.99… (repetend)
We then multiply both sides by 10.
10X = 9.99…
We then subtract 1X from both sides, but on the left side, using X, and on the right, X’s value. This is still equal, and thus mathematically permissible.
9X = 9
We then divide both sides by 9.
X = 1
If X=0.99…, and X=1, 0.99… must necessarily equal 1.
0.99… = 1
By extension, if we divide both sides by 3, we can further extrapolate that…
0.33… = 1/3
Therefore, 0.33… (repetend) is a perfect numerical representation of one third.
.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.
In truth, math is a lie essentially. It tries to put a point on a round surface. Math is just approximation of reality. Because as you get smaller and smaller all of the universe is still round (atoms, electrons, etc). So you cannot ever put an exact value on a point in time and space. It is only approximation to help us communicate the natural phenomena.
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