Originally posted by doodinthemoodNice proof.
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 a ...[text shortened]... a number. You cannot have anything of radius infinity or volume infinity or anything like that.
Originally posted by doodinthemoodI remember having that as an interview question for University. That's pretty much the answer I gave.
[b]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 a ...[text shortened]... umbers, then they share a common factor, of 2. We have a contradiction, and so root 2 is irrational.
Originally posted by mtthwits one of the classic examples of proof by contradiction... i love this proof, and the other one usually seen that there is no largest prime number
I remember having that as an interview question for University. That's pretty much the answer I gave.
No largest prime number:
Suppose there is a largest prime number and we have a finite set of them. Then every number is the product of a certain number of primes, or is prime. If we multiply all the primes in that finite set of primes together, we get a number that has every prime as a factor. The next number to have any given prime as a factor would be this number plus whatever amount the prime is. However, as the smallest prime is 2, then adding one to this mass-prime-product will find a number that doesn't have any primes as a factor, so is a prime. This is a contradiction, so there is not a largest prime number.
All numbers (meaning 0, 1, 2, 3, ...) are interesting:
Suppose there are numbers that are not interesting.
Then there is also a smallest uninteresting number.
That makes that number interesting, wich is a contradiction.
Our initial assumption must be wrong, so no uninteresting numbers exist.
Originally posted by doodinthemoodInfinity is not a real number. Infinity is a transfinite number.
2) Infinity is not a number. You cannot have anything of radius infinity or volume infinity or anything like that.[/b]
Given:
a = "first ordinal transfinite number" = "number of integers"
b = "second ordinal transfinite number"
c = "number of reals"
r = "any real number greater than 1"
then:
a+r = a
a-r = a
a*r = a
a/r = a
a^r = a
r^a = b
so with the circle mentioned in the first post :
diameter = a
circumfrence = pi * a = a
area = pi * a^2 = a
Infinity is not a transfinite number either. The whole point of the words "transfinite" and "infinite" (and indeed "finite" ) is that they are not the same. Infinity can never be the value of anything. As the diameter of a circle tends to infinity, the circumfrence tends to infinity, and the area tends to infinity, but there is no real point in recognising this.
Originally posted by Swlabr1. Yes, pi*0 = 0, a rational number.
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.