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
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.
It’s just something that happens when a number has a prime factor that isn’t a prime factor of the base (in this case base 10).
In a base 3 system (where you’d count like this: 0,1,2,10,11,12,20,…) 1/3 would be 0,1. 1/2 however would be 0.1111111111…
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.
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.
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
You are touching at the fringes of numbering systems.
When I was taking some networking classes we took 20 minutes to look at Hex and Opt (16 and 8) based numbering systems then we dived into binary.
I went home with this swimming in my head kind of fascinated with the idea.
I spent a few weeks screwing around and arrived at a very fascinating conclusion.
1) There is nothing magical or special about decimal. Nothing. We have 4 fingers and one thumb on each hand – we picked 10 done and done. Not only is it not magical or special it also isn’t really that great.
2) Other number systems are incredibly efficient when utilized properly.
My instinct (cause I am not as gifted in math as I wish I was) is that in a tertiary based system you would always come out with even answers – but when cutting something into half you would run into a problem!
It’s bc it’s athe ratio you’re making by dividing a whole. Just can’t have 3 pieces equal if comparing to the whole you divided them from. That .33333 is that ratio. Jeez I don’t make sense. Sorry. It makes sense to me.
Take math as a language, a representation of how we describe the universe. If you cut 1 thing into 3 equal pieces, you’ll have 1/3 a piece. Like language, you can also say that it’s 0.3333~, simply 0.33. With the number system we’re using, there are times where we come up with answers that barely represent the real thing, but that doesn’t mean it isn’t correct and we can’t use it.
Take 0.3333~ for example.
> **Is the assumption that it’s physically impossible to cut something into 3 perfectly even pieces?**
It is possible in this case. They’re all equal and they’re all 0.3333~. Most of the people are just not used to arriving at repeating decimals as answers. There’s nothing wrong about it. At one point in our lives, we were always expecting for whole numbers as the correct answers
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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