Question one: Does these exist a real number 'a' such that a*Pi is a rational number? (NOTE: a rational number is a number that can be written in the form a/b, with a and b both integers, incase you were wondering...)
Question two: (my mind wandered onto this whilst contemplating the above question. I'm not too sure of the answer, but I have a hunch...)
The cardinality (essentially, size) of the real numers is "uncountably infinite", whilst the cardinality of the rational numbers is "countably infinite". That is to say, there does exist a bijection from the natural numbers into the rational numbers but not into the real numbers.
My question is this; imagine a circle of countably infinite diameter. Is it's circumference countably of uncountably infinite? Or is this a really stupid question without an answer? I feel it is the latter (really stupid...), but I'm not sure.
Originally posted by Swlabr Question one: Does these exist a real number 'a' such that a*Pi is a rational number? (NOTE: a rational number is a number that can be written in the form a/b, with a and b both integers, incase you were wondering...)
Question two: (my mind wandered onto this whilst contemplating the above question. I'm not too sure of the answer, but I have a hunch...)
T ...[text shortened]... estion without an answer? I feel it is the latter (really stupid...), but I'm not sure.
1) No. An irrational number cannot be multiplied to become a rational number.
2) I don't know what is a numer. But your question does require some consideration. Can you create uncountable possibilities with a countable number. I don't think you can, but I am sleepy and maybe a nights sleep might yield something more.
Originally posted by Gastel 1) No. An irrational number cannot be multiplied to become a rational number.
2) I don't know what is a numer. But your question does require some consideration. Can you create uncountable possibilities with a countable number. I don't think you can, but I am sleepy and maybe a nights sleep might yield something more.
an irrational number can most certainly be multiplied by another irrational to become rational [ex. sqrt(3) * sqrt(3) = 3]
in this particular case we seek an x such that x*pi = a/b ... let x be anything you want/pi (for example 6/pi) and it clearly equals a rational number (in this case [6/pi]*pi = 6).
clearly 6/pi is irrational, since pi is irrational, but it indeed is a real number.
Originally posted by Swlabr Question one: Does these exist a real number 'a' such that a*Pi is a rational number? (NOTE: a rational number is a number that can be written in the form a/b, with a and b both integers, incase you were wondering...)
Question two: (my mind wandered onto this whilst contemplating the above question. I'm not too sure of the answer, but I have a hunch...)
T ...[text shortened]... estion without an answer? I feel it is the latter (really stupid...), but I'm not sure.
How about 2/pi? would that work? 2/pi * pi = 2, but I can't remember if irrational numbers are real.
Originally posted by Swlabr Question one: Does these exist a real number 'a' such that a*Pi is a rational number? (NOTE: a rational number is a number that can be written in the form a/b, with a and b both integers, incase you were wondering...)
Question two: (my mind wandered onto this whilst contemplating the above question. I'm not too sure of the answer, but I have a hunch...)
T ...[text shortened]... estion without an answer? I feel it is the latter (really stupid...), but I'm not sure.
i wouldn't call the last part a "stupid" question, but i don't think it really makes sense... when mathematicians speak of cardinalities of infinity, they use terms like aleph-null (countably infinite) and c (the infinitude of the continuum) to describe discrete sets of values. The notion of a "countably infinite diameter," i think, makes very little sense... as a circle grew towards infinite size, its diameter's size would move through the continuum of the real line, not the "hole-ridden" lattice of the rationals...
so, the premise of your question doesn't make a whole lot of sense. however, if a circle has a diameter whose length is an element of the set of rational numbers, it's circumference indeed belongs to the set of irrationals. not sure if this is clear, or correct, or makes any sense either. but i hope it helps 🙂
Originally posted by smomofo How about 2/pi? would that work? 2/pi * pi = 2, but I can't remember if irrational numbers are real.
Ugh. I'm just happy the site is back.
yes this is what i was saying. the real numbers are made up of the set of rationals in union with the set of irrationals, so the irrationals are indeed real. basically anything without an imaginary part [involving sqrt(-1)]is a real number
Originally posted by Aetherael yes this is what i was saying. the real numbers are made up of the set of rationals in union with the set of irrationals, so the irrationals are indeed real. basically anything without an imaginary part [involving sqrt(-1)]is a real number
How do we know an irrational number is indeed irrational?
Let me propose two problems:
1- Proof that SQRT (2) is irrational.
2- Is the number 0.12345678910111213141516....irrational? Explain your answer.
Originally posted by Aetherael i wouldn't call the last part a "stupid" question, but i don't think it really makes sense... when mathematicians speak of cardinalities of infinity, they use terms like aleph-null (countably infinite) and c (the infinitude of the continuum) to describe discrete sets of values. The notion of a "countably infinite diameter," i think, makes very little sens sure if this is clear, or correct, or makes any sense either. but i hope it helps 🙂
I don't have much time at the moment (so not much time to post), but your thinking on it sounds pretty much like mine - that it really doesn't make too much sense as a question. I'll try and re-formulate my ideas over the weekend...
Proof that Root 2 is irrational:
Suppose, root 2 is rational. Then it could be written in the form A/B where A and B do not share any factors.
Root 2 = A/B
2 = (A/B)^2
2 = A^2/B^2
B^2x2 = A^2
So A^2 is an even number, and A is an even number. It can be written in the form 2C
2 = (2C)^2/B^2
2 = 4C^2/B^2
2B^2 = 4C^2
B^2 = 2C^2
So B^2 is also an even number, and B is an even number. But if B and A are both even numbers, then they share a common factor, of 2. We have a contradiction, and so root 2 is irrational.
As for the questions:
1) Yes. 1337/Pi works
2) Infinity is not a number. You cannot have anything of radius infinity or volume infinity or anything like that.
Originally posted by Aetherael an irrational number can most certainly be multiplied by another irrational to become rational [ex. sqrt(3) * sqrt(3) = 3]
in this particular case we seek an x such that x*pi = a/b ... let x be anything you want/pi (for example 6/pi) and it clearly equals a rational number (in this case [6/pi]*pi = 6).
clearly 6/pi is irrational, since pi is irrational, but it indeed is a real number.
I had read it as a rational number times an irrational to become a rational... now I reread it, I realize it says real.
Two questions involving Pi
22 Feb '08 22:38
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