Originally posted by doodinthemood 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 doodinthemood [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.
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.
Originally posted by doodinthemood 2) Infinity is not a number. You cannot have anything of radius infinity or volume infinity or anything like that.[/b]
Infinity is not a real number. Infinity is a transfinite number.
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 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.