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.

 

[agentredlum]

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).
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)

I am not disputing any of your points about keeping numbers as expressions in computer programs because writing it as a decimal will produce rounding errors. Example, replacing 1/7 with a decimal aapproximation is not wise for at least 2 reasons. 1) more memory is needed to store it as a decimal, depending on how much precision you want. 2) the decimal will produce rounding errors if the program uses it hundreds or even millions of times in the same calculation.

It is much better to keep it as 1/7 and after all the algebra is done and the final result stored and displayed on your screen, then you ask the computer to perform the long division of fractions in the final result. This will minimize errors and save memory. That is a good point, I agree with you on that.

Our difference of opinion boils down to this... you believe that in writing down 2/3 you have performed division (long division), I believe you have not.

Or maybe you believe fractions are more important than decimals?

I'm not trying to change your beliefs. I made what I thought were clever arguments to support my beliefs about the seemingly unimportant perceived inadequacies of long division.

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

If i write down 2÷3 as my answer you would accept it?

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?

When someone writes down 2/3 they haven't done a single calculation, how can they know the answer without calculating it?

Is the answer 2/3 ? Absolutely! Then the next question becomes 'what does 2/3 mean?'

You are right, it's not that important, but it is curious to me how one can start with the set of positive integers where addition and multiplication don't force you to make corrections however subtraction and division force you to make corrections. Subtraction forces you to extend the positive integers to include zero and the negatives, while division forces you (among other things) to accept a very non-intuitive result such as 1 = .9999... = 1.000...= 4.9999.../5.000...etc. Like micromass pointed out, the representations are infinite in number.

 

[agentredlum]

Well, you got lucky cause 12 is a multiple of 3. Try 1 divided by 5 in base 12.

http://www.wolframalpha.com/input/?i=1÷5+base+12&asynchronous=false&equal=Submit

Or better yet, try 1 divided by 7 in base 12.

http://www.wolframalpha.com/input/?i=1/7+base+12&asynchronous=false&equal=Submit

How about 1 divided by 13 in base 16.

http://www.wolframalpha.com/input/?i=1/13+base+16&asynchronous=false&equal=Submit

The point is that changing the base every time you need to do division is going to create a nightmare of trouble.

 

[pwsnafu]

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

If i write down 2÷3 as my answer you would accept it?

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?

When someone writes down 2/3 they haven't done a single calculation, how can they know the answer without calculating it?

Is the answer 2/3 ? Absolutely! Then the next question becomes 'what does 2/3 mean?'

You are not making sense. We define 2/3 to be the fraction corresponding to 2 divided by 3 so of course it is the correct answer. What else could it be? If I get a computer algebra system such as Maple or Mathematica and asked what is two divided by three it returns 2/3.

You are right, it's not that important, but it is curious to me how one can start with the set of positive integers where addition and multiplication don't force you to make corrections however subtraction and division force you to make corrections. Subtraction forces you to extend the positive integers to include zero and the negatives, while division forces you (among other things) to accept a very non-intuitive result such as 1 = .9999... = 1.000...= 4.9999.../5.000...etc.

No. Division creates the http://en.wikipedia.org/wiki/Field_of_fractions" of the integers, not decimal representations. That requires all of the real numbers.

 

[agentredlum]

You are not making sense. We define 2/3 to be the fraction corresponding to 2 divided by 3 so of course it is the correct answer. What else could it be? If I get a computer algebra system such as Maple or Mathematica and asked what is two divided by three it returns 2/3.
No. Division creates the http://en.wikipedia.org/wiki/Field_of_fractions" of the integers, not decimal representations. That requires all of the real numbers.

Listen, let me make this clear to you cause I don't think you understand what I'm talking about. You don't need all the real numbers to figure out that you got a problem when you try to divide 1 by 3.

Take a number like 6÷3 = 2 because 2*3 = 6 quotient times divisor gives you dividend, this is the way you check your work.

Or try 1.47÷7 = .21 because .21*7 = 1.4

Now to make my point try 1÷3 = .333333... but 3*.333333... = .9999... clearly this is so, there can be no mistake about it because the pattern is so obvious if you point it out to the average person on the street, they get it. Additionally, 1 and 3 are the smallest positive integers that exhibit this peculiar phenomenon. That's all i have been saying all along but I can't put too much detail in the post because it would turn into a book.

this leads one to make corrections and explore whether or not .9999... = 1

mathematicians prove that it does and then with more carefull considerations other non-intuitive results are found such as mentioned by micromass. Does it make a little sense now?

Why are you bringing abstract algebra into this mess?

Why point out the obvious? Do you think that I am not aware division creates fractions?

2/3 you have done NO CALCULATIONS!

Pick up pencil and paper and compute 97 divided by 23, try it, you might find it fun.

Relying on a machine to give you all the answers is not a good idea because it will never be as smart as you can be.

As for your statement that you need the real numbers in order to get decimal approximations to RATIONAL numbers is still under scrutiny because it implies that you somehow need the irrationals to complete the rationals. I don't believe that unless you provide evidence.

 

[micromass]

Listen, let me make this clear to you cause I don't think you understand what I'm talking about. You don't need all the real numbers to figure out that you got a problem when you try to divide 1 by 3.

Take a number like 6÷3 = 2 because 2*3 = 6 quotient times divisor gives you dividend, this is the way you check your work.

Or try 1.47÷7 = .21 because .21*7 = 1.4

Now to make my point try 1÷3 = .333333... but 3*.333333... = .9999... clearly this is so, there can be no mistake about it because the pattern is so obvious if you point it out to the average person on the street, they get it. Additionally, 1 and 3 are the smallest positive integers that exhibit this peculiar phenomenon. That's all i have been saying all along but I can't put too much detail in the post because it would turn into a book.

this leads one to make corrections and explore whether or not .9999... = 1

mathematicians prove that it does and then with more carefull considerations other non-intuitive results are found such as mentioned by micromass. Does it make a little sense now?

Why are you bringing abstract algebra into this mess?

Why point out the obvious? Do you think that I am not aware division creates fractions?

2/3 you have done NO CALCULATIONS!

Pick up pencil and paper and compute 97 divided by 23, try it, you might find it fun.

Relying on a machine to give you all the answers is not a good idea because it will never be as smart as you can be.

As for your statement that you need the real numbers in order to get decimal approximations to RATIONAL numbers is still under scrutiny because it implies that you somehow need the irrationals to complete the rationals. I don't believe that unless you provide evidence.

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.

 

[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