# Irrational Number Phenomenon

by cloud_sync
Tags: irrational, number, phenomenon
 P: 13 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?
P: 460
 Quote by Hurkyl 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'.
Mentor
P: 18,346
 Quote by agentredlum 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"
P: 460
 Quote by micromass 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...
Mentor
P: 18,346
 Quote by agentredlum 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.
P: 460
 Quote by micromass 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!
 P: 460 The decimal point was invented by this man. http://inventors.about.com/od/nstart...ohn_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.
 P: 828 Speaking of bases, why do mathematicians get Halloween and Christmas confused?
Mentor
P: 18,346
 Quote by robert1986 speaking of bases, why do mathematicians get halloween and christmas confused?
31oct=25dec?
P: 605
 Quote by agentredlum The decimal point was invented by this man. http://inventors.about.com/od/nstart...ohn_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.
http://en.wikipedia.org/wiki/Hindu-A...numeral_system. Check the paragraph about positional notation.
en.wikipedia.org/wiki/Counting_rods
Emeritus
PF Gold
P: 16,091
 Quote by agentredlum 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)
P: 2,470
 Quote by cloud_sync 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.
P: 460
 Quote by Hurkyl 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.
 P: 460 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...e&equal=Submit Or better yet, try 1 divided by 7 in base 12. http://www.wolframalpha.com/input/?i...e&equal=Submit How about 1 divided by 13 in base 16. http://www.wolframalpha.com/input/?i...e&equal=Submit The point is that changing the base every time you need to do division is going to create a nightmare of trouble.
P: 837
 Quote by agentredlum 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 field of fractions of the integers, not decimal representations. That requires all of the real numbers.
P: 460
 Quote by pwsnafu 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 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.
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P: 18,346
 Quote by agentredlum 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.
P: 460
 Quote by micromass 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?

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