Why do irrational numbers result in uneven divisions?

[agentredlum]

I fear it is you who doesn't understand us... If you ask to divide 1 by 3, then 1/3 is a perfectly valid answer. In fact, I prefer 1/3 to 0.33333... since it is much clearer.

And yes, you do need irrationals to complete the rationals. This is almost by definition so. The rationals are not complete, the reals are.

You need the irrationals to complete the rationals? You're sure about that? I am beginning to think there is now some controversy about the word 'complete'

I understand you perfectly. I never said 1/3 is not valid. But you haven't done any calculations...do you understand that?

 

[micromass]

You need the irrationals to complete the rationals? You're sure about that? I am beginning to think there is now some controversy about the word 'complete'

I understand you perfectly. I never said 1/3 is not valid. But you haven't done any calculations...do you understand that?

http://en.wikipedia.org/wiki/Complete_space

 

[agentredlum]

http://en.wikipedia.org/wiki/Complete_space

I don't understand...the set of rational numbers and the set of irrational numbers are disjoint, so how can you use members of one set to complete the other set?

AFAIK no irrational number belongs in the set of rational numbers and no rational belongs in the set of irrational numbers. Their union completes the set of real numbers in the sense that every real number is either rational or irrational but not both.

Perhaps that link is context sensitive. I was certainly not talking about 'Cauchy Completion' involving complete metric space.

All right, so the next time you ask me to divide 20 by 4 I'm just going to write down 20/4.

 

[micromass]

I don't understand...the set of rational numbers and the set of irrational numbers are disjoint, so how can you use members of one set to complete the other set?

AFAIK no irrational number belongs in the set of rational numbers and no rational belongs in the set of irrational numbers. Their union completes the set of real numbers in the sense that every real number is either rational or irrational but not both.

Perhaps that link is context sensitive. I was certainly not talking about 'Cauchy Completion' involving complete metric space.

Well, what completeness are you talking about then?

All right, so the next time you ask me to divide 20 by 4 I'm just going to write down 20/4.

Well, why not?

 

[agentredlum]

Well, what completeness are you talking about then?
Well, why not?

Well, if I want a decimal approximation to a rational number like 1/3 I don't expect to use irrational numbers anywhere in the calculation.

As an example of the type of completion i was thinking of suppose you have the set of all rational numbers. That set is 'complete' in the sense that no rational number is missing. If you put sqrt(2) in that set then the set still contains all rational numbers but putting sqrt(2) in there somehow makes it seem to me that it implies sqrt(2) is rational, which we clearly know is not the case.

Perhaps one should say 'The set of all rational and only rational' to exclude unwanted extra members?

I understand Cauchy attempt to fill in the holes because a space with holes in it is unsatisfactory and from the link i understand that his method fills in sqrt(2). Does his method also fill in numbers like pi, e, other transcendentals, uncomputable numbers, un-namable numbers?

I guess my 3rd grade teacher would give me an F for 20/4 but my abstract algebra professor would give me an A+ for 20/4.

A long time ago I got a 99 out of 100 in a Calc 2 exam, all series questions and for every problem you had to state the test you were using. I got all the right answers. He made only 2 small marks on my paper in red, -1. One point was taken off because i did not state the test i was using in problem 10, however, i had used the same test in problem 6 and stated it there. No matter how much i argued with him he did NOT give me that point.

The next exam i got 99 out of 100 again because one of my fractions did not match his answer. He did not reduce, i did. When I told him about it he took my paper, looked it over carefully and FOUND ANOTHER ERROR! He gave me a point for the one he took off in error and took off a point for the error he missed when grading. So again the total was 99 out of 100.

I love math but that professor made me hate it for a while. He was a great teacher, a calculating machine, he would assign homework and 3 days later solve every homework problem assigned, he went through more chalk than i have ever seen anyone go through.

 

[Hurkyl]

If you asked me to divide 2 by 3 and i wrote down 2/3 you wouldn't be annoyed?

Only if I implicitly meant I wanted your answer in the form of a decimal numeral.

That is what your problem seems to be -- you think that "and write the answer as a decimal numeral" is part of what "divide" means, rather than an unstated additional requirement.

If i wrote down the problem, the way schoolchildren do, 3 on the outside as divisor, 2 on the inside as dividend and then stopped without doing a single calculation to get the quotient or remainder or long division to get the decimal approximation, would you be happy or would you think i was a smart-aleck?

If I was teaching students long division with remainder, then yes expect them to come up with "0 remainder 2" (or "2/3" depending on how I ask them to write their answer).

And I might even intentionally assign a problem like this, just to disabuse them of the notion that all problems are non-trivial.

 

[agentredlum]

Only if I implicitly meant I wanted your answer in the form of a decimal numeral.

That is what your problem seems to be -- you think that "and write the answer as a decimal numeral" is part of what "divide" means, rather than an unstated additional requirement.If I was teaching students long division with remainder, then yes expect them to come up with "0 remainder 2" (or "2/3" depending on how I ask them to write their answer).

And I might even intentionally assign a problem like this, just to disabuse them of the notion that all problems are non-trivial.

Man A "divide 2 by 3"

Man B "2/3"

Man A "where's the calculation?"

Man B "division does not imply calculation"

If you are all happy about this then who am I to argue? Have a wonderful day, I'm going on youtube to learn Quantum Mechanics from Yale.

 

[ramsey2879]

Well, if I want a decimal approximation to a rational number like 1/3 I don't expect to use irrational numbers anywhere in the calculation.

Why do you think .3333333... is an irrational number. It is perfertly a rational number as are all infinitely repeating decimal strings. I tried to show you that your logic only makes sense when you try to truncate the decimal string instead of putting the 3 dots after the decimal string to show that it is a rational number. You can treat 1/3 just like any other decimal number such as 1/5. You only need to do your calculations until you get either 0 or a repeated decimal string. You know that it is a repeating decimal since you get a remainder having the same string of numbers as before when you are doing long division. Therefore, you put three dots after the decimal string and you are done. All rational numbers can be writtten as a finite non repeating decimal depending upon what base you choose, so all rational fractions are the same in that regard.

 

[Galron]

No you don't. You only run into a problem if you decide you want to write the answer in the form of a terminating decimal numeral.

Can you actually physically terminate it or does that not matter, serious question btw?

I assume you are good at maths.

Know it sounds an odd question, really apologise if it is inapt.

 

[Galron]

Why do you think .3333333... is an irrational number. It is perfertly a rational number as are all infinitely repeating decimal strings. I tried to show you that your logic only makes sense when you try to truncate the decimal string instead of putting the 3 dots after the decimal string to show that it is a rational number. You can treat 1/3 just like any other decimal number such as 1/5. You only need to do your calculations until you get either 0 or a repeated decimal string. You know that it is a repeating decimal since you get a remainder having the same string of numbers as before when you are doing long division. Therefore, you put three dots after the decimal string and you are done. All rational numbers can be writtten as a finite non repeating decimal depending upon what base you choose, so all rational fractions are the same in that regard.

That's true in integral calculus with limits. The away I understand it, even though I personally cannot perceive the limit at it it is purely numbered to be equal to 1/3. An axiom but an obvious one.

 

[SteveL27]

That's true in integral calculus with limits. The away I understand it, even though I personally cannot perceive the limit at it it is purely numbered to be equal to 1/3. An axiom but an obvious one.

It's so much less mysterious than people imagine it to be.

Given some rational number between 0 and 1, the question is whether we can express it as a finite sum of terms of the form a * 10^n where 0 <= a <= 9 and n is a negative integer.

If the rational number happens to be 1/5, then the answer is yes: 1/5 = 2/10.

If the rational number happens to be 1/3, then the answer is no. We can't express 1/3 as a finite sum of negative powers of 10.

The question of whether a particular rational is or isn't expressible as a sum of negative powers of 10 is really not very interesting. It's easy to show exactly which rationals can be so expressed. And the property depends on the base -- you can play the same game for negative powers of 3. In that case, 1/3 has a finite expansion as 1/3.

People are investing this relatively trivial observation with mystical significance, leading to confusion. Better to just realize that some rationals have terminating expansions (to a given base) and some don't. And anything that's base-dependent is not telling us anything about numbers, only about particular representations of numbers. So it's not very important in the scheme of things.

As far as "visualizing" that 1/3 = 3/10 + 3/100 + 3/1000 + ..., once you accept the use of infinite processes, or infinite sets in general, you are pretty much committed to accepting a lot of things that are counterintuitive and difficult to visualize.

As John von Neumann once said:

In mathematics you don't understand things. You just get used to them.

http://en.wikiquote.org/wiki/John_von_Neumann

 

[Galron]

It's so much less mysterious than people imagine it to be.

Given some rational number n/m between 0 and 1, the question is whether we can express it as a finite sum of terms of the form a * 10^n where 0 <= a <= 9 and n is a negative integer.

If the rational number happens to be 1/5, then the answer is yes: 1/5 = 2/10.

If the rational number happens to be 1/3, then the answer is no. We can't express 1/3 as a finite sum of negative powers of 10.

The question of whether a particular rational is or isn't expressible as a sum of negative powers of 10 is really not very interesting. It's easy to show exactly which rationals can be so expressed. And the property depends on the base -- you can play the same game for negative powers of 3. In that case, 1/3 has a finite expansion as 1/3.

People are investing this relatively trivial observation about the expressibility of rational numbers as a big deal. It simply isn't.

As far as "visualizing" that 1/3 = 3/10 + 3/100 + 3/1000 + ..., once you accept the use of infinite processes, or infinite sets in general, you are pretty much committed to accepting a lot of things that are counterintuitive and difficult to visualize.

As John von Neumann once said:

In mathematics you don't understand things. You just get used to them.

http://en.wikiquote.org/wiki/John_von_Neumann

Taylor-Maclaurin series. I am aware of it, yeah intuitively that makes it simple.

I tell you what amazes me, and I kid ye not that the Greek mathematicians worked out the volume of a sphere by using calculus rules, how Eucild et al didn't work out the rules is beyond me: but just wow.

Thanks for that though, nicely explained.

Standing on the Shoulders of Giants.

I guess the proof of the pudding is in the eating, and hence pie.

"In mathematics you don't understand things. You just get used to them."

Eat my goal.

Archimedes.

 

[Mark44]

This has been aggravating me for years. Call it "IDP" as a placeholder name for now, if you will. How come irrational numbers keep propelling forward for particular divisions? My inquiry applies for both repeating and non-repeating irrational numbers.

I looked through all three pages of this thread, and I don't think anyone has commented on this. The title of this thread is misleading, since the discussion is almost entirely about rational numbers. The numbers you are talking about in the following examples are rational numbers, not irrational.

"Just is" or "You're thinking too much into it," are answers I have received in the past. We need to embark a new mindset in math. It is almost as if there is an untold story in physics that ties in with math. Why does uneven division exist for only particular divisions? For example, if we divide 1/2 we get 0.5, but if we divide 1/3 we get 0.333333... I am not asking for the apparent answer to this question. I am asking why our number system creates this inaccuracy for only particular divisions while other divisions come out even. Is it because we use a 10-base number system? Anyone ever question why we haven't been able to established a clean-cut, division system that overrides this phenomenon?

 

[Mark44]

You are correct, it is physically not possible to divide a strip of wood in three parts.

Sure it is. The hard part is dividing the strip into three equal parts.

(I know what you meant to say, though...)

But it is possible in the mathematical world. You can divide 1 by 3 and get 1/3 or 0.3333... This is mathematically correct. But that doesn't mean that you can do it in the real world.
In the same fashion, things like e or infinity do not exist in the real world (as far as I know), but that doesn't prevent us from working with them in mathematics...

 

[Robert1986]

Sure it is. The hard part is dividing the strip into three equal parts.

(I know what you meant to say, though...)

Hahahaha

 

[agentredlum]

I can use e to rob people.

Suppose I'm a bank and you deposit 1 trillion dollars at e% interest compounded anually. If i use 2.718 for e after 1 year i robbed you of 2.8 million dollars.

I wonder if banks tweak the numbers like this so the public is always SCREWED?

It doesn't have to be all in 1 account. Stealing a small fraction of a penny from every dollar of every individual account could add up to a lot of money.

The point is if you have a small account you can't tell the difference, if you have a big account and the difference is a few dollars you don't complain.

 

[SubZir0]

Hey micromass, if you're still in here i wanted to try and clarify what i was babaling on about.

When i was reading up on Phinary (Golden Ratio Base) it said it could finitely represent any elements in Q[√5] = Q + √5Q so I figured to solve Cloud Sync's problem you would need a number that is 'part of' the make-up of every possible number but I also see that the number i gave was nothing to do with that..

..damn.

 

[jambaugh]

Also: $\aleph_1$ is NOT the cardinality of [0,1] (in general). The cardinality of [0,1] is $2^{\aleph_0}$. It is unknown whether $\aleph_1=2^{\aleph_0}$.

I think you have that turned around.
It is a question of axioms (axiom of choice and continuum hypothesis) whether the cardinality of [0,1] or of the Reals is $\aleph_1$.

But by definition: $|\mathbb{Z}| = \aleph_0$ and by definition $2^{\aleph_n}=\aleph_{n+1}$.

 

[SteveL27]

I think you have that turned around.
It is a question of axioms (axiom of choice and continuum hypothesis) whether the cardinality of [0,1] or of the Reals is $\aleph_1$.

But by definition: $|\mathbb{Z}| = \aleph_0$ and by definition $2^{\aleph_n}=\aleph_{n+1}$.

You're thinking of the Beth numbers.

http://en.wikipedia.org/wiki/Beth_number

 

[jambaugh]

You're thinking of the Beth numbers.

http://en.wikipedia.org/wiki/Beth_number

Arrrg! I've had it backwards in my head for 20 years. Thanks for the correction to my correction.

 

[cloud_sync]

I only witnessed two people that understood the logic of INP. I suspect I found a solution (below). This may bring a new layer of math I was speaking of:



Trinity Number System:

0.0.0 = exp . mul . inc

exp = exponential
mul = multiple
inc = incremental = remainder



It is clean and I imagine it is one puzzle piece in developing a new layer of math in relation to calculus. Feel free to find its faults. I have not found any yet.

 

[pwsnafu]

Feel free to find its faults.

Fault: you haven't explained how the notation is supposed to work. All I see are three zeros.

Trinity Number System:

0.0.0 = exp . mul . inc

exp = exponential
mul = multiple
inc = incremental = remainder

So how do you write

1. one third,
2. square root 2,
3. pi,

in this system.

Secondly, is this just notation, or are you trying to define a new number system?

 

[Deveno]

what a quagmire...

the question: "what is 1 divided by 3?", and the question: "what is the size of 1/3?" are two different questions.

1/3 (as defined to be a certain equivalence class in the field of fractions of the integral domain of the grothendieck groupification of the free monoid formed by creating a minimal inductive set from the one set postulated existentially by the zermelo fraenkel axioms...isn't THAT a mouthful? and did i miss anything there?) is a perfectly good answer to the first.

the answer to the second is a bit more complicated. 1/3 is a rational expression (literally, a ratio of 1 to 3), and the greeks measured such expressions through comparison. so, it's easy to say, given a/b and c/d, which is "bigger", but relating these quantities to some common scale depends on finding lcm(b,d), and then using that as a unit of measurement. and perhaps here you can see some practical dificulty, because thre is no "universal" scale that will work for every rational number (or for every finite set of measurements).

the decimal system is a compromise of sorts, in that it allows us to establish a "scalable" scale of measurement, good to whatever precision we're satisfied with. it is rather troubling that we cannot represent what we consider an "exact" quantity (such as 1/3) exactly in this system (designed mainly for ease of computation).

but changing our arbitrarily chosen base of 10 will not help matters, because NO natural number is divisible by every smaller natural number, except two. the natural expression of fractions in base 2 is depicted very succintly in the subdivisons of an inch-ruler, and carpenters (for example) have been know to decide on a "scale of resolution" and call out their measurements (say 1/8 of an inch) as 4-8-4 (meaning 4 feet 8 inches and 1/2 inch). in the truss industry i used to work in, feet-inches-sixteenths was the standard (a peculiar system to do arithmetic in, i assure you).

but while base 2 may represent some kind of "ideal" system, for fractions with odd denominators, it does spectacularly poorly. for small numbers, one might be satisfied with something like the base 60 the babylonians used (and they were pretty handy with fractions), but the number of prime integers is infinite, so no "greatest common denominator" for all integers can be found, to use as a common base.

and these are just inherent difficulties with doing arithmetic operations with rational expressions, the situation gets very out of hand quite quickly in dealing with solutions of even fairly simple polynomial equations (such as x^2 - 2 = 0).

even considering the sides of triangles based on "even" divisions of a 360 degree circle, lead us in short order to consideration of various irrational quantities, and when we extend the ratio of the sides of such triangles to a continuous function, we encounter numbers that aren't even solutions of polynomial equations. and yet a circle is such a clean, "whole" thing, so it is very counter-intuitive that it should imply we need numbers that are "unmeasurable".

to go even further, i don't think we have ever really agreed amongst ourselves, as to what should properly qualify as a "number". are matrices numbers? (if you think not, then what about this one:

[a -b]
[b a] ?)

are polynomials numbers (don't be too hasty answering this one, either)? what about power series? how about functions themselves, surely they have "sizes" we can measure at least some of them do)? where do we draw the line, and say: "these are proper numbers, these are not?" my point is, this is actually a hard question to get a handle on, especially when we take for granted a number system the vast majority of whose members are totally unknowable to us (so how can we say we know what real numbers are? I've only personally made the acquaintance of a handful).

in the end, the more sophisticated among us exclaim: the things we shall consider as numbers, are those things which have the properties we want! and such things are unique up to isomorphism! which kind of works, conceptually, but is somewhat at odds with how we actually use numbers to calculate stuff. so we wind up with a situation where we say: "in a perfect world, i would have this, but i am finite, so here is my best approximation". there is something deeply unsatisfactory about that, but smarter people than i have attempted various reconciliations between our abstractions and our abilities, and failed.

@OP: it doesn't sound like you have something truly novel or useful, but that's no reason not to give us the details, so at the very least we can mock you :)

 

[SteveL27]

what a quagmire...

the question: "what is 1 divided by 3?", and the question: "what is the size of 1/3?" are two different questions.

1/3 (as defined to be a certain equivalence class in the field of fractions of the integral domain of the grothendieck groupification of the free monoid formed by creating a minimal inductive set from the one set postulated existentially by the zermelo fraenkel axioms...isn't THAT a mouthful? and did i miss anything there?) is a perfectly good answer to the first.

You must be a fan of Linderholm's wonderful Mathematics Made Difficult.

https://www.amazon.com/dp/0529045524/?tag=pfamazon01-20

 

[Borek]

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