Why do irrational numbers result in uneven divisions?

[cloud_sync]

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. "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?

 

[daveyp225]

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. "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?

The "phenomenon" you speak of is due to our decimal base. It was a choice that was made by man to pick it as a standard. Some people use different bases. Although all bases will have a similar thing going on. 1/3 in base 3 is just 0.1. But 1/10 in base 3 is 0.00220022... repeating.

There is no such thing as an "irrational repeating number." Repeating decimals are rational.

 

[pwsnafu]

For example, if we divide 1/2 we get 0.5, but if we divide 1/3 we get 0.333333...

Well technically "0.5" means 0.50000... which is equal to 0.499999...

We just use the convention that if the decimal expansion terminates, there is an infinite string of zeros. We just don't write them because it gets tedious.

 

[SteveL27]

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?

I assume you know that it's pretty easy to show that in any base, some rationals will have terminating expansions and others won't. And that the ones that terminate are related to factors of the base -- just as in base 10, any rational a/2^n or a/5^n terminates, because 2 and 5 are factors of 10.

So do you mean why? Are you looking for some underlying reason? It's really just a function of the long division algorithm and the factors of the base. It's a homework exercise in undergrad number theory; no great mystery.

 

[Hurkyl]

Anyone ever question why we haven't been able to established a clean-cut, division system that overrides this phenomenon?

We have lots of ways to notate numbers. "2/3", for example, is a perfectly good notation for the number you get when you divide 2 by 3.

Decimal notation for real numbers is taught because:

• It's simple
• It fits well with decimal notation for integers
• It's very easy to trade precision for simplicity. (e.g. just write the first few digits)

Most people don't have any reason to learn notations other than a mix of algebraic expressions with decimals.

 

[cloud_sync]

If you change the base it just shifts where the INP takes place in your written notation. Nothing has been solved, so-to-speak. It is almost as if there is something in nature that won't allow for clean-cut division at particular regions/magnitudes in physics.

Let's say we have a thin strip of wood that is 1 inch and we cut it it up in 3rds. This goes back to my initial fraction. Now this is physically possible. Each of the three pieces of wood will now be 1/3rd (i.e. 0.33) in length except for one. One of the pieces received an additional 0.01 more. Regardless if your at the microscopic level or at the macro level, this phenomenon appears unavoidable, so far.

 

[SteveL27]

Let's say we have a thin strip of wood that is 1 inch and we cut it it up in 3rds. This goes back to my initial fraction. Now this is physically possible.

All physical measurements are inexact. It's not possible to cut a physical object into exactly equal halves or thirds or any other fraction.

I understand the nature of your confusion now. The real numbers do not exist in the physical world. There's a difference between math and physics, and you are confusing the two.

 

[agentredlum]

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. "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?

I think I understand your idea. As far as positive whole numbers are concerned, and the operation of division, you run into a problem very quickly when you try to divide 1 by 3. This appears unsatisfactory to you (and to me)

Methematicians prove that .999999... = 1

Take a number like .5 they say =.500000... but it should also =.49999...

or .42 = .4200000... = .4199999...

or .1439 = .143900000... = .14389999999...

Any decimal number you can think of that can be expressed as a quotient of 2 non zero integers now appears to have at least 3 different representations, although mathematicians prove that all different representations represent the same fraction.

Zero is interesting in this scheme. I guess one can say 0 = .000000... but what is the other representation?

 

[micromass]

If you change the base it just shifts where the INP takes place in your written notation. Nothing has been solved, so-to-speak. It is almost as if there is something in nature that won't allow for clean-cut division at particular regions/magnitudes in physics.

Let's say we have a thin strip of wood that is 1 inch and we cut it it up in 3rds. This goes back to my initial fraction. Now this is physically possible. Each of the three pieces of wood will now be 1/3rd (i.e. 0.33) in length except for one. One of the pieces received an additional 0.01 more. Regardless if your at the microscopic level or at the macro level, this phenomenon appears unavoidable, so far.

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

 

[disregardthat]

The divisions 1/n which will have a repeating decimal expansion are exactly those for which n contain prime factors other than 2 and 5. This is because the prime factors of 10 is 2 and 5. Generally if a/b is a reduced fraction, it will have a repeating decimal expansion if b has any other prime factors than 2 and 5. It is just because we have chosen 10 as our base for representing real numbers.

 

[SubZir0]

You would need a number system with a Field of Elements:
Q[C] = Q + CQ
where C is aleph-one (the infinite cardinal for any point between 0 and 1, for example)If irrational numbers grind your gears then transcendental numbers must twist you up something proper! Also fractal shapes are infinite finite objects.

 

[micromass]

You would need a number system with a Field of Elements:
Q[C] = Q + CQ
where C is aleph-one (the infinite cardinal for any point between 0 and 1, for example)

What is that supposed to mean?? Do you mean the fraction field generated by \aleph_1 elements?

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}.

 

[ramsey2879]

If you change the base it just shifts where the INP takes place in your written notation. Nothing has been solved, so-to-speak. It is almost as if there is something in nature that won't allow for clean-cut division at particular regions/magnitudes in physics.

Let's say we have a thin strip of wood that is 1 inch and we cut it it up in 3rds. This goes back to my initial fraction. Now this is physically possible. Each of the three pieces of wood will now be 1/3rd (i.e. 0.33) in length except for one. One of the pieces received an additional 0.01 more. Regardless if your at the microscopic level or at the macro level, this phenomenon appears unavoidable, so far.

The fact that 1/3 has an infinite decimal expansion in base 10 has nothing to do with whether it is possible to cut a strip of wood into 3 equal lengths in the real world. 1/5 has a finite decimal expansion but it is more difficult to cut a strip of wood into 5 equal lengths than to cut a piece of wood into 3 equal lengths. It is more likely though that a piece of wood will have a length spanned by a multiple of 3 atoms than a multiple of 5 atoms so it is more likely that a piece of wood could be cut into 3 equal lengths than into 5 equal lengths. Just because we may not have the expertise to surely cut a piece of wood into three equal lengths does not mean that it could not be done.

 

[SteveL27]

The fact that 1/3 has an infinite decimal expansion in base 10 has nothing to do with whether it is possible to cut a strip of wood into 3 equal lengths in the real world. 1/5 has a finite decimal expansion but it is more difficult to cut a strip of wood into 5 equal lengths than to cut a piece of wood into 3 equal lengths. It is more likely though that a piece of wood will have a length spanned by a multiple of 3 atoms than a multiple of 5 atoms so it is more likely that a piece of wood could be cut into 3 equal lengths than into 5 equal lengths. Just because we may not have the expertise to surely cut a piece of wood into three equal lengths does not mean that it could not be done.

Even if you thought you'd cut a physical object exactly in three; how would you know? Any measurement could only state a range for each length.

As a thought experiment, imagine the variables you'd have to take into account to divide a object into three parts. If there's slightly more mass on one side of the universe than the other, the object's dimensions would be affected. So first, you'd have to be able to account for the mass, position, and current state of motion of every particle of matter in the universe.

Of course normally we don't need to take that into account ... we know that "on average" the mass in the rest of the universe is about the same in every direction, and anyway the effect would be negligible. So we ignore it, and end up with an approximation.

If we are to be exact, we must take all these things into account. How would you measure the length of an object? The atoms keep bouncing around. How do you define length? Can you do a measurement in one instant of time? Otherwise you'd only be measuring the average length of the bouncing atoms over a period of time. An approximation!

How do you propose to exactly divide a physical object in three?

 

[daveyp225]

Randomly spouting off thoughts here. If whatever method of measurement you are using is divisible by three (say 999999999 identical atoms), given this is just a thought experiment and practical division ignored, manually separate each atom into three separate chambers and reform them into three pieces identical to the original when placed together. If the atoms bouncing around are a cause for concern to obtain adequate length, then if held at the same temperatures/pressures/whataver, they could at least theoretically be assembled in the same conditions (maybe?) so that they have the same movement. So they should be exactly equal in length?

 

[Galron]

There is no innacuracy

At the limit of infinity both are the same thing.

There is no existing circle where pi = pi at any scale for example, but that doesn't mean that we cannot use infinite limits it just means that a real world circle approximates: A=\pi r^2; in a perfect universe of abstraction it exactly equals A=\pi r^2.

I suppose if you used something like cantors continuum hypothesis you could say that there are infinite infinities all of differing sizes which are the same size. But then you'd disappear up your own axiom.

It might be an idea to google this: http://en.wikipedia.org/wiki/Taylor_series"

And differential geometry rules in general.

[URL]http://upload.wikimedia.org/wikipedia/commons/a/a7/TangentGraphic2.svg[/URL]

 

[SteveL27]

Randomly spouting off thoughts here. If whatever method of measurement you are using is divisible by three (say 999999999 identical atoms), given this is just a thought experiment and practical division ignored, manually separate each atom into three separate chambers and reform them into three pieces identical to the original when placed together. If the atoms bouncing around are a cause for concern to obtain adequate length, then if held at the same temperatures/pressures/whataver, they could at least theoretically be assembled in the same conditions (maybe?) so that they have the same movement. So they should be exactly equal in length?

1. Starting from the problem of creating three identical lengths, now you have to create identical pressure and temperature too? Now you have more problems than before!

2. What is temperature? It's a measure of the average motion of the molecules in a given area. It's a statistical notion. Two objects having the same temperature may have very different configurations of molecules at a given instant.

It's essential to understand that there is no exactness in the physical world. Otherwise you start thinking the real numbers are "real."

 

[Hurkyl]

As far as positive whole numbers are concerned, and the operation of division, you run into a problem very quickly when you try to divide 1 by 3.

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.

 

[cloud_sync]

Regarding different bases, the IDP does not disappear, it just shifts in correspondence to the type of base. Both bases below have the same fraction values:


BASE-3:

1/1 = 1

1/2 = 0.111...

1/10 = 0.1



BASE-10:

1/1 = 1

1/2 = 0.5

1/3 = 0.333...


In regards to physics, we our beginning to embrace nanotechnology which is one billionth of a meter. Unless we are at the quantum level, I believe it is safe to say 0.33" or 0.34" are not measurement issues in regards to my last example. It is understood to be another problem, once we start calculating division of "particular numbers." What we are witnessing is that our number system is relatively flawed or math needs another layer on top of calculus for us to advance forward.

For the moment, it is as if odd and even numbers are at war with each other. Is there a number system that provides clean-cut division for both? Initially, it sounds like an easy task, but no one has been able to establish one. Historically, we decide to add a band-aid solution (rounding) and label them irrational to ease our mind. We need to pay more tribute to what we have put on hold in the past. The patch served our purpose, and we forgot about the INP ever since. Who would be the right person/entity to get in contact with for this problem?

 

[agentredlum]

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.

I think you do. 2/3 doesn't mean anything because you haven't performed the division yet, You don't know the answer to 2/3, no one does until they perform the division. Might as well just call it x and manipulate it using the rules of algebra.

If you are not interested in numerical values of numbers but interested in their abstract representation as quotients of 2 integers then i guess you don't see a problem.

I ask you... what is the difference between .9999.../2 (infinite 9's) and 1/2?

or 1/1.99999...?

Do you understand my point?

1/2 = 1/1.9999... = .9999.../2 = .9999.../1.9999... = 1.000.../2 = 1.000.../2.000... = .9999.../2.000...

= 1/2.000... = 1.000.../1.9999...

Now you have at least 9 different representations of the same UNIQUE value so looking at the fraction as a solution to the problem has not SOLVED the problem but instead has made it more difficult and more aggravating.

See how easy it is to shoot down the fractional abstractions?

Any expression involving any rational number has now become suspect just because you ran into a problem trying to divide 1 by 3

My last comment is related to my previous post. As far as positive integers with the operation of division are concerned, 1 divided by 3 is the first time you run into a problem that forces you to make a correction. The correction is that we must now accept that 1 = .9999... = 1.000...

This correction was not needed for 1 divided 1, 1 divided by 2, 2 divided by 1

This is the 'spirit' of my argument. I am not arguing that the results are incorrect. If someone want's to choose 1/2 as the representation of 1 divided by 2, that's fine by me but it doesn't change the fact that other representations are possible and are a consequence of the necessary 'correction'.

 

[micromass]

I think you do. 2/3 doesn't mean anything because you haven't performed the division yet, You don't know the answer to 2/3, no one does until they perform the division. Might as well just call it x and manipulate it using the rules of algebra.

If you are not interested in numerical values of numbers but interested in their abstract representation as quotients of 2 integers then i guess you don't see a problem.

I ask you... what is the difference between .9999.../2 (infinite 9's) and 1/2?

or 1/1.99999...?

Do you understand my point?

1/2 = 1/1.9999... = .9999.../2 = .9999.../1.9999... = 1.000.../2 = 1.000.../2.000... = .9999.../2.000...

= 1/2.000...

Now you have at least 8 different representations of the same UNIQUE value so looking at the fraction as a solution to the problem has not SOLVED the problem but instead has made it more difficult and more aggravating.

See how easy it is to shoot down the fractional abstractions?

Any expression involving any rational number has now become suspect just because you ran into a problem trying to divide 1 by 3

There are a lot more representations than these 8:

\frac{1}{3}=\frac{2}{6}=\frac{3}{9}=\frac{2.9999...}{9}=...

There is an infinite number of such representations.
I don't see why this makes the problem more difficult and why this is now suspect??

Also, note that historically, people only worked with fractional representations. Decimal expansions are far more recent. So I wouldn't call fractional representations to be "more abstract"

 

[agentredlum]

There are a lot more representations than these 8:

\frac{1}{3}=\frac{2}{6}=\frac{3}{9}=\frac{2.9999...}{9}=...

There is an infinite number of such representations.
I don't see why this makes the problem more difficult and why this is now suspect??

Also, note that historically, people only worked with fractional representations. Decimal expansions are far more recent. So I wouldn't call fractional representations to be "more abstract"

Right you are! the representations become infinite, I'm glad i said AT LEAST 9, I didn't even consider fractions with common factors. These other representations certainly make reducing fractions a more difficult nightmare.

Depends on what people you look at and what they were working on. One can say that the most famous problem in math history is getting better decimal approximations to pi so decimal expansions are ancient. The decimal expansion of a rational is easy so once you have that method you can concentrate on other things like getting decimal approximations to irrational numbers like extracting roots, babylonians tried that, archimedes used 2 regular polygons, one inscribed, one circumscribed, both 96 sides and got pi accurate to 3.14, even the chinese approximation using the well known fraction as an approximation to pi of 6 digits. How did they know one fraction is a better approximation than another if they did not get the decimal expansion of both fractions and compare to the KNOWN value of the decimal expansion of pi in their time?

I am only giving a few examples but i am aware of hundreds more cases where decimal expansion and decimal approximations have been very important throughout history so i don't understand your claim that decimal representations are a more recent phenomenon.

Personally i see it as suspect but i don't have a problem if you are not suspicious of an infinite number of representations for the same UNIQUE value. Because i can see it your way... all those representations are proved equal. However...

 

[micromass]

Right you are! the representations become infinite, I'm glad i said AT LEAST 9, I didn't even consider fractions with common factors. These other representations certainly make reducing fractions a more difficult nightmare.

Depends on what people you look at and what they were working on. One can say that the most famous problem in math history is getting better decimal approximations to pi so decimal expansions are ancient. The decimal expansion of a rational is easy so once you have that method you can concentrate on other things like getting decimal approximations to irrational numbers like extracting roots, babylonians tried that, archimedes used 2 regular polygons, one inscribed, one circumscribed, both 96 sides and got pi accurate to 3.14, even the chinese approximation using the well known fraction as an approximation to pi of 6 digits. How did they know one fraction is a better approximation than another if they did not get the decimal expansion of both fractions and compare to the KNOWN value of the decimal expansion of pi in their time?

Well, Archimedes did not yet have acces to decimal approximations, so he must have did this by another method. I'm not sure when decimal representation was invented, but I guess somewhat around the time of Fibonacci. Or perhaps by the Indians. It's worth looking up.

 

[agentredlum]

Well, Archimedes did not yet have acces to decimal approximations, so he must have did this by another method. I'm not sure when decimal representation was invented, but I guess somewhat around the time of Fibonacci. Or perhaps by the Indians. It's worth looking up.

Yeah you are right about Archimedes. I wonder if the ancient greeks stopped when they got a remainder in the Euclidean Algorithm? They didn't even have numbers, they used letters to represent quantities, Hindu-Arabic number system and invention (discovery?) of zero came much later. I guess if recent means 13th or 14th century, then decimal representation is a more recent phenomenon. It must be very hard to compute 1 divided into 7 equal parts if you don't have Hindu -Arabic number system and trying to do it by using letters of your alphabet, or symbols like Egyptians, Mayans, Roman Numerals, notches representing positional base 60 by babylonians.

This is an example where notation becomes king!

 

[agentredlum]

The decimal point was invented by this man.

http://inventors.about.com/od/nstartinventors/a/John_Napier.htm

So apparently, the ancients didn't have decimal representations and must have represented numbers like 3.37 as addition of a whole number and fractions. Micromass is right.

 

[Robert1986]

Speaking of bases, why do mathematicians get Halloween and Christmas confused?

 

[micromass]

speaking of bases, why do mathematicians get halloween and christmas confused?

31oct=25dec?

 

[mishrashubham]

The decimal point was invented by this man.

http://inventors.about.com/od/nstartinventors/a/John_Napier.htm

So apparently, the ancients didn't have decimal representations and must have represented numbers like 3.37 as addition of a whole number and fractions. Micromass is right.

en.wikipedia.org/wiki/Hindu-Arabic_numeral_system. Check the paragraph about positional notation.
en.wikipedia.org/wiki/Counting_rods

 

[Hurkyl]

Now you have at least 9 different representations of the same UNIQUE value so looking at the fraction as a solution to the problem has not SOLVED the problem but instead has made it more difficult and more aggravating.

So what? This criticism only has relevance if, for some strange reason, I need to use a notation where everything is notated in a unique way.

While such a property is nice and occasionally useful, it is nowhere near as important as you are making it out to be.

As an aside, it is a trivial exercise to tweak decimal notation for real numbers so that every real number really does have a unique numeral form. (the two most common ways are to forbid decimals ending in repeated 0's, or to forbid decimals ending in repeated 9's).

You don't know the answer to 2/3, no one does until they perform the division. Might as well just call it x and manipulate it using the rules of algebra.

I do know the answer; I did the division and I got the result "2/3". This isn't decimal notation, but you didn't ask for that.

Notating things as arithmetic expressions has the advantage that arithmetic is very, very easy. One practical application is that this notation is of absolutely crucial importance in efficient C++ linear algebra packages -- when you add two vectors v and w, it effectively stores the result as a triple "(plus, v, w)". It doesn't convert the result into an actual vector unless you (or some library routine) ask it to store the result in a vector.

(why is it crucial? Because if you wrote a C++ program to do x = u + v + w in a naive way, you would waste a lot of time and memory creating unnecessary intermediate value vectors)

 

[K^2]

In regards to physics, we our beginning to embrace nanotechnology which is one billionth of a meter. Unless we are at the quantum level, I believe it is safe to say 0.33" or 0.34" are not measurement issues in regards to my last example. It is understood to be another problem, once we start calculating division of "particular numbers." What we are witnessing is that our number system is relatively flawed or math needs another layer on top of calculus for us to advance forward.

So why are your 3 pieces at 0.33", 0.33", and 0.34"? Why not make each piece 0.333" and 0.334"? Or go to 0.3333" and 0.3334"? There is no problem with numbers. It's the problem with what you measure.

Furthermore, why should the piece of wood pick base 10? Maybe it likes base 12. In base 12, 1/3 is exactly 0.4, so you'd have 3 x 0.4" base-12 pieces. No problem with terminating decimals.

If you need to divide a length into N equal segments, just go with base N for numbering. In fact, that's basically what you do with rational numbers to begin with. You just use a different notation, calling it 1/N.