Where are the irrational numbers?

[Robert1986]

A few things.

First, the number 1/n does not tend to go anywhere: it just is. Only variables can "tend" to do anything. So, saying that 1/n (with n some fixed constant), 1.1, or 1.11 "tends" to go anywhere is just flat out wrong.

Cant Count is certainly correct in stating that there is an infinite number of rational numbers between any two rational numbers. Let a and b be two rationals. Now, since the rational numbers are countable, we can make a list of the rational numbers between a and b (though that fact isn't of critical importance in what follows). So, make the list. Each number on this list is a number that stays exactly as it is. It is NOT, in anyway, tending to be irrational. And it doesn't matter how difficult it is to compute a rational number, either.

If this is the first time you started dealing with this stuff, then (and I mean no offence) you don't really have intuition: this takes time and experience to build. Instead, you are using your "common sense." When I was first learning this stuff (and I still don't really have any intuition) it blew my mind. It gets really strange. Leave your common sense at the door in mathematics and trust proofs.

Cant Count, I would suggest googling the Cantor Set (this will really blow you away) and the countability and uncountability of the rationals and irrationals.

 

[agentredlum]

If you want to do something fun, pick up pencil and paper and compute 1/97.

It's very easy to show 1/2 is rational by computation. Not so easy to show 1/17 is rational. you would have to compute at least 16 decimal positions. Then you would see the pattern repeat and you can conclude that it is rational

 

[pessimist]

A few things.

First, the number 1/n does not tend to go anywhere: it just is. Only variables can "tend" to do anything. So, saying that 1/n (with n some fixed constant), 1.1, or 1.11 "tends" to go anywhere is just flat out wrong.

Cant Count is certainly correct in stating that there is an infinite number of rational numbers between any two rational numbers. Let a and b be two rationals. Now, since the rational numbers are countable, we can make a list of the rational numbers between a and b (though that fact isn't of critical importance in what follows). So, make the list. Each number on this list is a number that stays exactly as it is. It is NOT, in anyway, tending to be irrational. And it doesn't matter how difficult it is to compute a rational number, either.

If this is the first time you started dealing with this stuff, then (and I mean no offence) you don't really have intuition: this takes time and experience to build. Instead, you are using your "common sense." When I was first learning this stuff (and I still don't really have any intuition) it blew my mind. It gets really strange. Leave your common sense at the door in mathematics and trust proofs.

Cant Count, I would suggest googling the Cantor Set (this will really blow you away) and the countability and uncountability of the rationals and irrationals.

The cantor set is indeed one of the most remarkable things one encounters in mathematics. In my opinion it is the height of non intuitiveness.

 

[HallsofIvy]

If you want to do something fun, pick up pencil and paper and compute 1/97.

It's very easy to show 1/2 is rational by computation. Not so easy to show 1/17 is rational. you would have to compute at least 16 decimal positions. Then you would see the pattern repeat and you can conclude that it is rational

You understand, don't you that most people learn the definition of "rational number" as "can be written as a ratio (fraction) of integers". That is, 1/2 and 1/17 are rational (by that definition) just be looking at them, not by any computation. One can then show that every rational number can be written as a repeating decimal (thinking of 0.5 as "0.5000000" where "0" is the repeating part). You may well have learned to define a rational number as "a repeating decimal" and then learned to show that every fraction is a "rational number" and vice versa. If so, you are the second person in about twenty years that I have met who learned it that way! Where and when did you learn this definition of "rational number"?

 

[agentredlum]

You understand, don't you that most people learn the definition of "rational number" as "can be written as a ratio (fraction) of integers". That is, 1/2 and 1/17 are rational (by that definition) just be looking at them, not by any computation. One can then show that every rational number can be written as a repeating decimal (thinking of 0.5 as "0.5000000" where "0" is the repeating part). You may well have learned to define a rational number as "a repeating decimal" and then learned to show that every fraction is a "rational number" and vice versa. If so, you are the second person in about twenty years that I have met who learned it that way! Where and when did you learn this definition of "rational number"?

It was a learning process, of negotiation between both notions. At some point after a few years I decided I liked the repeating decimal explanation better than the ratio explanation. I think Niven and Zuckerman Elementary Number Theory influenced me but I can't be sure cause i don't remember. Also i read a few history of mathematics books and the impression i got was great emphasis was placed on computation for thousands of years. It is only within the past 100 years that mathematics has moved toward abstractness more and more.

Personally I like and trust the computing part but I don't throw away the abstract part because it has many uses.

The examples i used above with 1/p i saw for the first time in 'Recreations in the Theory of Numbers, The Queen of Mathematics Entertains' by Bieler. Highly recommend this one is full of treasure.

I found this idea fascinating but i only remember a sketch, Beiler goes into it in much greater detail and the results are fascinating. Beiler definitely loves computation cause he does a lot in that book!

Also i worked in a math center for quite some time and professors would donate books so i pounced on everything. I came across a book on numerical analysis and some amazing things were done in there, that i had never seen or even imagined so my fondness for numerical computations grew.

I also saw formulas developed by Ramanujan that converge rapidly to higher precision. These formulas fascinated me and felt like having a glimpse at glorious possibilities that would take me a lifetime to understand. It's easy for others to say how he did what he did...AFTER HE DID IT! That man was self taught.

Very important to me increasing my respect for numerical calculations was how Euler noticed an agreement to some decimal approximation between 2 relations that led him to discover his famous e^(ix)=cos(x)+ isin(x)
I learned about that in 'The sqrt(-1), An Imaginary Tale' by Paul J, Nahin. Highly recommend this, there are many treasures, tricks, lots of Algebra and great historical account.

 

[agentredlum]

The book by Bieler 'Recreations in the Theory of Numbers, the Queen of Mathematics Entertains'

ALMOST EVERY SINGLE PAGE CONTAINS A TREASURE!

 

[agentredlum]

If you were to ask 'prove 1/17 is rational' most people would say 'it's the ratio of two integers, therefore it's rational' and that's fine by me. I would prefer to say 'it has a repeating decimal expansion' and then i would calculate it and show you.

Bielers example which i used above for the decimal expansion of 1/p, somebody at some time in the past did the calculations and noticed a pattern. Namely, some primes exhaust the possibilities, others do not. Years later others proved the results abstractly and fit them into a more general theory full of symbols and few numbers but they woudn't have been able to do that unless many before them picked up pencil and paper and did ARITHMETIC.

 

[Robert1986]

Hmmmm. Here's an easy proof that 1/17 is rational: it can be written as the ratio of two integers.

 

[Robert1986]

Despite the fact that you are using a (IMHO) a non-standard definition of rational, this doesn't change the fact that numbers are not tending anywhere.

 

[agentredlum]

Hmmmm. Here's an easy proof that 1/17 is rational: it can be written as the ratio of two integers.

Yes that's true but you miss all the fun of discovery about decimal expansions. Let's pretend repeating 0's don't count, after all, repeating 0's don't add anything to the VALUE of a number. An interesting question then would be 'why do some rational numbers have terminating decimal expansions and others do not?' Is there a way to tell which ones terminate, which ones repeat?

I ask... If the sequence of non repetition has googolplex^(googolplex) digits before it starts to repeat, does it tend toward irrationality.

Of course its rational. I know that, you know that, I'm just thinking here.

 

[agentredlum]

Despite the fact that you are using a (IMHO) a non-standard definition of rational, this doesn't change the fact that numbers are not tending anywhere.

The idea of a repeating decimal is not worthless...am i challenging your 'standard' notion that numbers must be well defined? That's not what I'm after.

 

[Hurkyl]

Also i read a few history of mathematics books and the impression i got was great emphasis was placed on computation for thousands of years.

"Computation" is not a synonym for "express in decimal form". In fact, I'm under the impression their pervasive use is a fairly recent phenomenon. (and possibly even that widespread knowledge of the existence of such a system is relatively recent)

Historically, people have liked expressing answers using ratios or radicals or geometric constructions; all are perfectly good ways of "computing" an answer. For a variable precision system, continued fractions were once a popular method. (and, I believe, even preferred to decimals)

 

[Robert1986]

The idea of a repeating decimal is not worthless...am i challenging your 'standard' notion that numbers must be well defined? That's not what I'm after.

Agreed. But the fact that 1/17 might be hard to compute by hand doesn't, in any way, mean that it is "tending" to be irrational. Numbers don't tend anywhere, they just are. And the fact that 1/17 is hard to compute by hand doesn't in any way, make it any less rational than 1/2.


So, I don't really know what your point was other than some rational numbers are difficult to compute by hand.

 

[agentredlum]

Agreed. But the fact that 1/17 might be hard to compute by hand doesn't, in any way, mean that it is "tending" to be irrational. Numbers don't tend anywhere, they just are. And the fact that 1/17 is hard to compute by hand doesn't in any way, make it any less rational than 1/2. So, I don't really know what your point was other than some rational numbers are difficult to compute by hand.

My point? (IMHO) The posters intuition about irrationals and infinity is not 100% incorrect. I knew there would probably be many others here who would say that it was in order to discourage that way of thinking, thus my attempt with the Examples using 1/p. No one has commented about that so maybe people don't find it interesting?

The 2 ideas of irrational, infinity, are connected at least by way of calculation with positive integers, and they are connected in many more ways. It appears to me the poster understands they are connected but is unsure why.

 

[agentredlum]

The fractions,

3/2,

17/12,

99/70,

577/408,

665857/470832

etc.

Are all approximations that give better and better precision to sqrt(2). Just consider the concept of a fraction as an idea. What is happening here? The numerator is going to infinity, the denominator is going to infinity and because of this you get what you want, sqrt(2), an irrational number.

Take a look at this. Please notice in particular the PRODUCT FORMULA for sqrt(2). It gives a fraction whose numerator tends to infinity and whose denominator tends to infinity and it does this by using positive integers.

http://en.wikipedia.org/wiki/Square_root_of_2

I really don't understand why this idea is so offensive?

 

[micromass]

The fractions,

3/2,

17/12,

99/70,

577/408,

665857/470832

etc.

Are all approximations that give better and better precision to sqrt(2). Just consider the concept of a fraction as an idea. What is happening here? The numerator is going to infinity, the denominator is going to infinity and because of this you get what you want, sqrt(2), an irrational number.

Take a look at this. Please notice in particular the PRODUCT FORMULA for sqrt(2). It gives a fraction whose numerator tends to infinity and whose denominator tends to infinity.

http://en.wikipedia.org/wiki/Square_root_of_2

I really don't understand why this idea is so offensive?

Take

9/10

99/100

999/1000

9999/10000

This sequence has numerators and denominators that grow to infinity, but still the sequence does not converge to an irrational. It converges to 1.

However, the idea to use fractions to approximate numbers is an interesting one. I'd suggest looking up "Cauchy sequences" in http://en.wikipedia.org/wiki/Construction_of_the_real_numbers
However, checking rationality is in general a very hard problem!

 

[agentredlum]

Take

9/10

99/100

999/1000

9999/10000

This sequence has numerators and denominators that grow to infinity, but still the sequence does not converge to an irrational. It converges to 1.

However, the idea to use fractions to approximate numbers is an interesting one. I'd suggest looking up "Cauchy sequences" in http://en.wikipedia.org/wiki/Construction_of_the_real_numbers
However, checking rationality is in general a very hard problem!

Good point.

Yes, i know that (infinity)/(infinity) is an indeterminate form and can represent any real number. And i certainly don't want to give the impression that all fractions are irrational! I merely wanted to put forth the idea that it is possible to construct fractions, in a simple way, that kinda, sort of 'blur' the distinction between rational and irrational. I say this with 'tongue in cheek' cause I know it's controversial and not standard.

I agree with you about using fractions to approximate numbers is interesting, but i would like to take it one step further, nobody knows what 1/17 represents until the decimal expansion is given. Of course algebra can be done with 1/17 and you can get correct results, but IMHO 1/17 is similar to a variable x whose value is yet to be determined in a linear equation.

I am not saying replace 1/17 with a decimal in all calculations. Not at all! This would be a computational nightmare!

I am putting forth the idea that dividing 1 by 17 can sometimes provide insight.

 

[agentredlum]

However, the idea to use fractions to approximate numbers is an interesting one. I'd suggest looking up "Cauchy sequences" in http://en.wikipedia.org/wiki/Construction_of_the_real_numbers
However, checking rationality is in general a very hard problem!

The link you provided is very helpful and interesting so thanx!

In particular, i never heard of surreal numbers, hyperreal nymbers before today.

 

[I_am_learning]

Good point.

Yes, i know that (infinity)/(infinity) is an indeterminate form and can represent any real number. And i certainly don't want to give the impression that all fractions are irrational! I merely wanted to put forth the idea that it is possible to construct fractions, in a simple way, that kinda, sort of 'blur' the distinction between rational and irrational. I say this with 'tongue in cheek' cause I know it's controversial and not standard.

I agree with you about using fractions to approximate numbers is interesting, but i would like to take it one step further, nobody knows what 1/17 represents until the decimal expansion is given. Of course algebra can be done with 1/17 and you can get correct results, but IMHO 1/17 is similar to a variable x whose value is yet to be determined in a linear equation.

I am not saying replace 1/17 with a decimal in all calculations. Not at all! This would be a computational nightmare!

I am putting forth the idea that dividing 1 by 17 can sometimes provide insight.

I like your explanation that rational fraction tends to irrational fraction when we increase the numerator and denomarator towards infinity, but following some rule.
In your increasingly more accurate apporximation to sqrt(2) series, suppose that the series can be written as
sqrt(2) =~ x(n)/y(n)
=~ means approximately equal to
x(1) = 3, y(1) = 2
x(2) = 17, y(2) = 12
and so on.
Don't we all agree that, when n tends to infinity
sqrt(2) = x(n)/y(n) (n tends to infinity)

For every finitie n, x(n)/y(n) is rational, however for n tends to infinity, x(n)/y(n) is irrational.
Nice thing. Don't this prove your point that rational faction can tend to irrational.


And for your 1/17 thing, I feel just the opposite way. 1/17 clearly speaks to me of 1 part in 17 however, 0.05882... don't much make sense to me. :)

 

[agentredlum]

I like your explanation that rational fraction tends to irrational fraction when we increase the numerator and denomarator towards infinity, but following some rule.
In your increasingly more accurate apporximation to sqrt(2) series, suppose that the series can be written as
sqrt(2) =~ x(n)/y(n)
=~ means approximately equal to
x(1) = 3, y(1) = 2
x(2) = 17, y(2) = 12
and so on.
Don't we all agree that, when n tends to infinity
sqrt(2) = x(n)/y(n) (n tends to infinity)

For every finitie n, x(n)/y(n) is rational, however for n tends to infinity, x(n)/y(n) is irrational.
Nice thing. Don't this prove your point that rational faction can tend to irrational.And for your 1/17 thing, I feel just the opposite way. 1/17 clearly speaks to me of 1 part in 17 however, 0.05882... don't much make sense to me. :)

Thanx for giving me hope that i am not a total fool.

well for me, i can more easily understand 6 parts in 100 or 59 parts in 1000 or 588 parts in 10000 than i can understand 1 part in 17 because of the use of powers of ten which are easy to visualize for me.

IMHO there is nothing wrong with your visualization but if 0.05882... doesn't make much sense, i hope my way of visualizing it helps.

Would you have any trouble visualizing

(1/17)^2,

3/(17)^3,

7/37

19/53

23/127 ?



My point is that as numerators and denominators become larger, the decimal representation becomes more useful in understanding the actual value of the fraction.

 

[I_am_learning]

Thanx for giving me hope that i am not a total fool.

well for me, i can more easily understand 6 parts in 100 or 59 parts in 1000 or 588 parts in 10000 than i can understand 1 part in 17 because of the use of powers of ten which are easy to visualize for me.

IMHO there is nothing wrong with your visualization but if 0.05882... doesn't make much sense, i hope my way of visualizing it helps.

Well that was really silly of me to not realize that 0.05822 is just around 5.822/100 ( 5.8 part in 100) or even better, 5822 part in 100000 (don't count the 0's because, I didn't .

However, after all its just a crude guessing. I can't really find any difference in my visualization of either 5 part in 237 or 37 part in 806 or 41 part in 1000. In all case I visualize the part being a tiny fraction of the whole.
Do you have any such sharp visualization?

 

[Hurkyl]

Are all approximations that give better and better precision to sqrt(2). Just consider the concept of a fraction as an idea. What is happening here? The numerator is going to infinity, the denominator is going to infinity and because of this you get what you want, sqrt(2), an irrational number.

Ah, you conceive wrongly! It's not the concept of a fraction you're thinking of, it's the concept of a sequence of fractions. (And specifically, of a sequence that converges to something)

 

[agentredlum]

Ah, you conceive wrongly! It's not the concept of a fraction you're thinking of, it's the concept of a sequence of fractions. (And specifically, of a sequence that converges to something)

Not at all. I talked about a single fraction, you mentioned sequence. Can you think of a fraction as an abstract idea or do you have to quantify it with numbers?

In the product formula for sqrt(2) how many fractions do you see?

 

[agentredlum]

Look, you can think of it as a limit if you want. Take the limit as x and y go to infinity of x/y. This is an indeterminate form (infinity)/(infinity) which means it could be any real number, including an irrational number.

 

[agentredlum]

Well that was really silly of me to not realize that 0.05822 is just around 5.822/100 ( 5.8 part in 100) or even better, 5822 part in 100000 (don't count the 0's because, I didn't .

However, after all its just a crude guessing. I can't really find any difference in my visualization of either 5 part in 237 or 37 part in 806 or 41 part in 1000. In all case I visualize the part being a tiny fraction of the whole.
Do you have any such sharp visualization?

Not beyond the first few decimal points. I can get a rough estimation at a glance using notion of distance but then i zoom in, in my minds eye and start comparing to other objects that i am familiar with, such as cells bacteria, DNA strands, molecules, atoms, protons, electrons. Then I start thinking in terms of wavelength of light. Ultraviolet, x-ray, etc. Particularly useful is an ANGSTROM because it's 10^(-10). This is not easy and takes great effort but you get better as you practice.

Something that helped me get an understanding about decreasing quantities was a video by Arthur C. Clarke, may he rest in peace.



I'm going to watch it again now that your question reminded me of it.

Here is a great example of 'zooming in'. As far as fractals are concerned you can do this forever because they have infinite complexity.

http://www.youtube.com/watch?v=0jGaio87u3A&NR=1

 

[Hurkyl]

Not at all. I talked about a single fraction, you mentioned sequence.

Eh? You were not talking about a single fraction -- you fairly explicitly listed a family of fractions.

The inability to tell the difference between the idea of "a number" and "a family of numbers" is a rather serious liability in calculus. It is, for example, one of the major causes that lead people to insist that "$0.999\ldots \neq 1$" -- e.g. "0.999... is a number that tends to 1, but isn't actually 1".

I honestly can't tell if you are just in love with using "fraction" to describe "family of fractions", or if you are starting down a path that will have you coming back here in three months insisting that $\sqrt{2}$ is a rational number.

Yes, I know you might think that last comment silly -- but there have been at least two people who have visited this forum who have insisted exactly that, using arguments that resemble what you are arguing.

In the product formula for sqrt(2) how many fractions do you see?

Explicitly, I see contained in the notation one fraction-valued expression in the variable n. Implicitly there are two related sequences of fractions: the infinite sequence of terms, and the infinite sequence of partial products.

And the product formula itself is, of course, not a fraction at all, e.g. because the outermost verb is "The infinite product of..." and not "The quotient of..."

 

[agentredlum]

Eh? You were not talking about a single fraction -- you fairly explicitly listed a family of fractions.

The inability to tell the difference between the idea of "a number" and "a family of numbers" is a rather serious liability in calculus. It is, for example, one of the major causes that lead people to insist that "$0.999\ldots \neq 1$" -- e.g. "0.999... is a number that tends to 1, but isn't actually 1".

I honestly can't tell if you are just in love with using "fraction" to describe "family of fractions", or if you are starting down a path that will have you coming back here in three months insisting that $\sqrt{2}$ is a rational number.

Yes, I know you might think that last comment silly -- but there have been at least two people who have visited this forum who have insisted exactly that, using arguments that resemble what you are arguing.

Explicitly, I see contained in the notation one fraction-valued expression in the variable n. Implicitly there are two related sequences of fractions: the infinite sequence of terms, and the infinite sequence of partial products.

And the product formula itself is, of course, not a fraction at all, e.g. because the outermost verb is "The infinite product of..." and not "The quotient of..."

Well, i guess people see what they want to see depending on the point they want to make. Yes, i listed fractions that converge to sqrt(2), but i am not interested in ANY intermediate fractions. I am only interested in the fraction whose numerator and denominator have gone to infinity by some rule, as I_AM_LEARNING pointed out in post # 89. This is a fraction I CANNOT list in the normal sense so I am asking for a little lattitude here, and for people to use the power of their imagination. This is not a rigorous approach, i understand that, but my original observation was not intended to be unquestionable truth. I used the words 'in some sense'

When I look at the product formula for sqrt(2), I see all those things you mentioned, no question, but i also see a single fraction whose numerator and denominator have gone to infinity by some rule.

I know what a sequence is. I know what partial products are. I'm not sure if you know what 'taking a step back' and' looking at the big picture' means. That's what I'm trying to do here, in a way that makes a little sense, not perfect sense.

I think sqrt(2) is irrational, no question, but I am open to the possibility that sqrt(2) can be thought about as being rational 'in some sense'

I think .999... = 1 no question, but i am open to the possibility that this can create other problems 'in some sense'

 

[agentredlum]

How many fractions do you see when you look at 2/3 ? 'In some sense' I see an infinite product of fractions.

...(-3/-3)(-2/-2)(-1/-1)(2/3)(1/1)(2/2)(3/3)...

You can get creative.

...(-pi/-e)(-e/-pi)(2/3)(e/pi)(pi/e)...

you can fill in the dots however you like, just make sure it works.

The possibilities are only limited by a persons imagination.

 

[micromass]

How many fractions do you see when you look at 2/3 ? 'In some sense' I see an infinite product of fractions.

...(-3/-3)(-2/-2)(-1/-1)(2/3)(1/1)(2/2)(3/3)...

You can get creative.

...(-pi/-e)(-e/-pi)(2/3)(e/pi)(pi/e)...

you can fill in the dots however you like, just make sure it works.

The possibilities are only limited by a persons imagination.

This is very dangerous to do. There are many traps when dealing with infinite products, so you better say exactly what you mean with it. How do you evaluate such an infinite fraction??

 

[agentredlum]

This is very dangerous to do. There are many traps when dealing with infinite products, so you better say exactly what you mean with it. How do you evaluate such an infinite fraction??

I disregard everything and pick the number in the 'middle'.



Yeeess, I know it's dangerous but IMHO so are the subtle points of arithmetic.

 

[micromass]

I disregard everything and pick the number in the 'middle'.

OK, so what is the point in the middle?? You have an infinite product which extends to both sides, there is no middle...

Yeeess, I know it's dangerous but IMHO so are the subtle points of arithmetic.

Arithmetic is not dangerous. It's very well defined. You just need to follow the rules.

 

[agentredlum]

OK, so what is the point in the middle?? You have an infinite product which extends to both sides, there is no middle...
Arithmetic is not dangerous. It's very well defined. You just need to follow the rules.

I thought that it was obvious i was using middle sarcastially bcause i put quotes around it. Then there was the biggrin...

Just because its not easy to FIND 2/3 in that infinite product doesn't mean it isn't THERE. You can write it down first THEN surround it however you wish, just make sure it works.

 

[micromass]

I thought that it was obvious i was using middle sarcastially bcause i put quotes around it. Then there was the biggrin...

Just because its not easy to FIND 2/3 in that infinite product doesn't mean it isn't THERE. You can write it down first THEN surround it however you wish, just make sure it works.

You can write everything you want to. But it's useless if you can't evaluate it properly...

 

[Hurkyl]

I'm not sure if you know what 'taking a step back' and' looking at the big picture' means.

It means, among other things, to stop focusing on minor details and evaluate what you're doing and why.

So let's try it. You are very fixated on the idea of trying to apply the word "fraction" to a lot of situations where it doesn't fit.

Can you explain to me your motivation for this fixation? Can you describe what you are having to do in order to follow this fixation? Is this approach really a good way to achieve whatever goals you have?



I hate to say things that way, but it seriously looks like you are being self-destructive -- you're not only resisting attempts to examine the idea in your head, but you are actively rejecting knowledge that could be useful for the purpose.

This latest post looks like you are trying to wrap your head around the fact that the rationals are dense in the reals, or possibly that the reals are the Cauchy completion of the rational numbers.

Are you going to continue crippling yourself by refusing to move beyond thoughts like "$\sqrt{2}$ is a fraction in some sense" -- or are you going to start examining just what that "some sense" really is, and try to explain it in terms of existing mathematical ideas or even to devise new mathematical ideas invented just for this purpose, should no existing ones apply?




I'm reminded of the time I first digested the notion of "tangent vector to a manifold". I remember actively disliking the definition I had seen. But rather than just mentally rejecting the notion I had seen, I set out trying to define what I thought a tangent vector ought to be. By the time I had worked out all the difficulties and had something I was happy with, I had pretty much written down exactly what I had originally rejected.

 

[agentredlum]

It means, among other things, to stop focusing on minor details and evaluate what you're doing and why.

So let's try it. You are very fixated on the idea of trying to apply the word "fraction" to a lot of situations where it doesn't fit.

Can you explain to me your motivation for this fixation? Can you describe what you are having to do in order to follow this fixation? Is this approach really a good way to achieve whatever goals you have?
I hate to say things that way, but it seriously looks like you are being self-destructive -- you're not only resisting attempts to examine the idea in your head, but you are actively rejecting knowledge that could be useful for the purpose.

This latest post looks like you are trying to wrap your head around the fact that the rationals are dense in the reals, or possibly that the reals are the Cauchy completion of the rational numbers.

Are you going to continue crippling yourself by refusing to move beyond thoughts like "$\sqrt{2}$ is a fraction in some sense" -- or are you going to start examining just what that "some sense" really is, and try to explain it in terms of existing mathematical ideas or even to devise new mathematical ideas invented just for this purpose, should no existing ones apply?

I'm reminded of the time I first digested the notion of "tangent vector to a manifold". I remember actively disliking the definition I had seen. But rather than just mentally rejecting the notion I had seen, I set out trying to define what I thought a tangent vector ought to be. By the time I had worked out all the difficulties and had something I was happy with, I had pretty much written down exactly what I had originally rejected.

Oh man...

I REJECT NOTHING, I QUESTION EVERYTHING. Questioning everything is not the same as rejecting anything. Wouldn't you think I was a fool if I accepted everything without question? Making my point does not mean i have to reject yours. You think it does but I am not responsible for that. I can see it your way and agree it's useful. You can't see it my way even after i ask for a little latitude. That's not fair.