Where are the irrational numbers?

[HallsofIvy]

Yes, every number, rational or irrational, can be represented as a limit of a sequence of rational numbers. In fact, you make use of that when you represent an irrational number in "decimal form". Saying that \pi= 3.1415926... means precisely that the sequence of rational numbers (any terminating decimal is a rational number) 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, ... has \pi as limit.

One way of defining the real numbers, in terms of the rational numbers, is to use equivalence classes of sequences of rational numbers:

Let S be the set of all increasing, unbounded (equivalently "Cauchy") sequences of rational numbers. (Such sequences do not necessarily converge in the rational numbers.) We say that two such sequences, {a_n} and {b_n} are "equivalent" if and only if the sequence {a_n- b_n} converges to 0.

A "real number" is an equivalence class of such sequences.

 

[HallsofIvy]

Is there a simple logical explanation for that? (Countable/uncountable don't appear to have enough logic)
Sorry if I am having you to repeat.

They do if you understand the technical meaning of "countable" and "uncountable".
An infinite set is said to be "countable" if and only if there is a one to one mapping of the set onto the natural numbers, 1, 2, 3, ... An infinite set is said to be "uncountable" if and only if it is not countable.

A simple illustration that the set of all rational numbers is countable is given here:
http://www.homeschoolmath.net/teaching/rational-numbers-countable.php

A discussion of Cantor's proof that the set of all real numbers (and hence the set of all irrational numbers) is uncountable is given here:
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

 

[agentredlum]

Although, I haven't followed whole of the thread, I came across the thought experiment, where you randomly place the tip of your pencil on a line marked ----------> 0--------1.
Why do you say that the pencil always lands at irrational number? Because there are just as many rationals as irrationals (both infinite), the chances must be equal.
I know I am wrong (because you appear to be great mathmatician :) ), but I would like to learn. :]

The thought experiment is designed to be simple enough that any non-expert can understand and to boggle the mind. If it has done this for you then it is a success. It is designed to bring into question the pedestrian understanding of infinity and invite your curiosity to learn more.

Georg Cantor was the first to show that some infinities are equal, some infinities are larger than others. Then he showed that if you consider any infinity, there is always another infinity larger. That is MIND-BOGGLING!

Cantor did not do this by counting numbers. He did it by establishing a one-to-one correspondence between members of sets.

Here is a simplified example of his technique. Suppose there is a classroom with chairs and students outside in the hallway. Someone asks, 'which set is larger, the set of chairs, or the set of students? Somebody else says, 'I know what to do, count the number of chairs and count the number of students, then we will know which set is greater.' 'WAIT!' says Cantor, 'ask the students to sit down, if there are chairs left over then the set of chairs is larger, if there are students left over then the set of students is larger, if all students are seated and all chairs have 1 student then the sets are equal.'

This simplified version shows it is possible to determine which set is larger WITHOUT COUNTING the members of any two sets and without knowledge of the magnitude of each set by using a one-to-one correspondence, to every student is assigned 1 chair, and to every chair is assigned 1 student.

This is very helpfull when considering infinite sets because no one can actually count up to infinity but after Cantor things you can count and things you can't count were forced to have a different meaning. That is where the terms 'countable infinity' and 'uncountable infinity' originated.

 

[Robert1986]

It's just the arctan function. The way I like to think of this bijection is by imagining a horizontal line in the plane through the origin. It makes an angle of zero with the x-axis and its slope is zero.


As you rotate the line counterclockwise, as the line goes from horizontal to vertical the angle goes from 0 to pi/2; and the slope goes from 0 to +infinity.

Likewise as you rotate a horizontal line clockwise, the angle goes from from 0 to -pi/2 (= 3*pi/2), and the slope goes from 0 to -infinity.

Ahhhh. Now I see what you are saying. My apologies; my interpretation of your post was, in fact wrong. When you started off by saying "It's just the arctan function." I thought that what you described next was how you constructed the graph of the arctan function in your mind.



Now that I see what you mean, I see that it is actually a pretty interesting way of looking at it.

My apologies.


EDIT: If this post sounds sarchastic, I don't meant it to be; it is sincere.

 

[I_am_learning]

They do if you understand the technical meaning of "countable" and "uncountable".
An infinite set is said to be "countable" if and only if there is a one to one mapping of the set onto the natural numbers, 1, 2, 3, ... An infinite set is said to be "uncountable" if and only if it is not countable.

A simple illustration that the set of all rational numbers is countable is given here:
http://www.homeschoolmath.net/teaching/rational-numbers-countable.php

A discussion of Cantor's proof that the set of all real numbers (and hence the set of all irrational numbers) is uncountable is given here:
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

Nice.
So, rational numbers is countable because it can be precisely located in the infinity X infinity (2-dimensional) matrix of that homeschoolmath site.

Natural number is countable because it can be precisely located in 1x inifinity matrix (1-dimensional) (1 2 3 4 5 ... inf)

Irrational numbers are uncountable because no such matrix exists. Infact, we would have to develop a infinite dimensional matrix (infinity X infinity X infinity X infinity ...) so as to make a matrix that would contain all the Irrational numbers. So, that makes it un-countable and infinitely many than the rational number.
Pretty satisfied now.
However, I am sure you folks have much better grasp of the concept than this.
Thanks.

 

[spamiam]

First, forget about the semi-circle thing for now. This doesn't have anything to do with the arctan bijection. You asked for a bijection and micromass (or someone) gave you one. It is just a bijective function from the entire real line to to the interval (-a,a). (Graph it in WolframAlpha.)

Actually, if you write out the function for my semicircle bijection, you'll see it's more or less the inverse to the one given previously. If you work it out, the semicircle method gives g : (-c, c) \to \mathbb{R} by
g(x) = \frac{2c}{\pi}\tan\left(\frac{\pi}{2c}x\right)<br />
and the one given before was f : \mathbb{R}\rightarrow (-a,a) by f(x) = \frac{2a}{\pi} \arctan (x) \, .

But I like describing the bijection I gave pictorially because I think it's much more intuitive.

 

[Robert1986]

Actually, if you write out the function for my semicircle bijection, you'll see it's more or less the inverse to the one given previously. If you work it out, the semicircle method gives g : (-c, c) \to \mathbb{R} by
g(x) = \frac{2c}{\pi}\tan\left(\frac{\pi}{2c}x\right)<br />
and the one given before was f : \mathbb{R}\rightarrow (-a,a) by f(x) = \frac{2a}{\pi} \arctan (x) \, .

But I like describing the bijection I gave pictorially because I think it's much more intuitive.

Correct. I meant to mention that he should also consider your example.

Your example is certainly more intuitive and should be what he is looking for given he didn't like the more technical stuff. I also like your description, as well, for the same reason that you like it.

 

[cant_count]

Not a mathematician. I've tried to do some reading on this. My intuitive thought is this: between any two rational numbers there's an infinite number of rational numbers because you can invent any fraction (1.1, 1.11, 1.111, ...). I guess everybody would agree on that. However, when you "hit" infinity in your inventing, then you get an irrational number. Physically impossible, but in theory, if there really is such thing as infinity, then you could. Is that a paradox? Does it mean rational numbers tend towards continuity? - Does it mean anything at all? (I think I've blown a few brain-cells.)

It's a bit like: what's the definition of random? If the digits of Pi are random (and I haven't yet understood whether they are or not) - then does that not mean this: that any sequence of digits you can invent will occur somewhere in it ... including Pi "eventually" repeating itself .. ?

I guess it's like saying you're trying to reach infinity.

(This is the trouble with letting anybody into your forums ! )

 

[pessimist]

your intuitive thought is absolutely not the way to go. The notion of infinity in itself is a counter intuitive notion + your thought about hitting infinity doesn't really make much sense.

 

[agentredlum]

Not a mathematician. I've tried to do some reading on this. My intuitive thought is this: between any two rational numbers there's an infinite number of rational numbers because you can invent any fraction (1.1, 1.11, 1.111, ...). I guess everybody would agree on that. However, when you "hit" infinity in your inventing, then you get an irrational number. Physically impossible, but in theory, if there really is such thing as infinity, then you could. Is that a paradox? Does it mean rational numbers tend towards continuity? - Does it mean anything at all? (I think I've blown a few brain-cells.)

It's a bit like: what's the definition of random? If the digits of Pi are random (and I haven't yet understood whether they are or not) - then does that not mean this: that any sequence of digits you can invent will occur somewhere in it ... including Pi "eventually" repeating itself .. ?

I guess it's like saying you're trying to reach infinity.

(This is the trouble with letting anybody into your forums ! )

I like your idea about rational numbers 1/n as n goes to infinity. Yes, i believe some of them tend toward becoming irrational. Never thought of it that way until you mentioned it so thank you.

Heres what i mean, for a prime p, 1/p can have at most p-1 digits after the decimal before it starts to repeat.

Some primes do not exhaust the possibilities, some primes do.

Example for the prime 3, 1/3 can have at most 2 digits after the decimal before it starts to repeat but
1/3 = .3333... repetition begins after 1 digit so 3 does not exhaust the possibilities

Example for the prime 7, 1/7 can have at most 6 digits after the decimal before it starts to repeat
1/7 = .142857142857142857... repetition begins after the 6th decimal position so 7 exhausts the possibilities.

It's interesting to look at primes in this way.

1/11 = .090909... repetition begins after the 2nd position so 11 does not exhaust the possibilities.

1/13 = .076923 076923 076923... repetition begins after the 6th decimal position so 13 does not exhaust the possibilities.

1/17 = .0588235294117647 0588235294117647 0588235294117647 ... repetition begins after the 16th decimal position so 17 exhausts all the possibilities.

Now, we know that the primes are infinite in number, this means they get bigger and bigger. There may be many primes out there larger than 10^1000000 that have somewhere in the neighborhood of 10^1000000 - 1 non repeating digits, and then repetition begins. As far as calculation is concerned this 1/p would be as difficult to calculate as sqrt(2) to the 10^1000000 - 1 decimal position. So in some sense, there are rational numbers that tend to be irrational.

This idea is going to generate a lot of controversy here so pretend that i am joking.

There is a method similar to long division for finding square roots without using a calculator, you can find it toward the bottom of the following page

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

 

[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.