Quick Math Proof

[History_Buff]

Been doing this assignment, can't seem to find a solution to it, though I'm sure it's a pretty basic proof So, if someone could, prove or give a counter-example to:

For all real numbers r, if r is rational and r does not equal 0, then r * (2)^(1/2) is irrational.

 

[Warman17]

2^(1/2) is an irrational number I believe. Any rational number mutiplied by an irrational number produces an irrational number.

 

[Irish Caesar]

True. Do you need to prove that sqrt(2) is irrational?

 

[History_Buff]

I know root 2 is irrational, but we aren't allowed to just assume the product of an irrational and a rational is irrational.

But I know how to do it now I think. Just assume the product to be rational, express r as x/y, and the product as n/m. Then sqrt(2) would be expressed as yn/xm, which can't be true, so sqrt(2) must be irrational.

Don't know why it didn't occur to me before.

 

[solar]

That is a good proof if you are allowed to assume a priori that sqrt(2) is irrational. To prove that it actually is is a little more involved. Assume x=sqrt(2) is rational. Let x=a/b where a/b is a fraction in lowest terms.

Square both sides 2=a^2/b^2
2*b^2=a^2 so a^2 is even (since b is an integer) so a is also even. Now let c=a/2 c is also an integer since a is even. c/b=sqrt(2)/2=1/sqrt(2) so inverting gives b/c=sqrt(2)

b^2/c^2=2 b^2=2*c^2, so b^2 and by extension b must also be even.

But since a and b are both even a/b cannot possibly be in lowest terms. Since this was assumed sqrt(2) must be irrational by contradiction

 

[Chieftess]

Discrete Math.

I think you need to say, "Suppose not. V r * (2)^(1/2) is rational" (V = the "for all numbers..." symbol - I think it's upside down.), and solve for r showing that the result is not a rational number. You might wind up with something like, "XYZ by definition is not a rational number. Therefor, r * (2)^(1/2) is irratational..."