Recent content by basil32

Thanks everybody!

yeah, the observation make sense but I need a lemma which proves that \frac{p^2}{q^2} is reduced form whenever \frac{p}{q} is reduced. How do you do that?

ok. x^{2} = \frac{p^{2}}{q^{2}} . Now q^{2}x^{2} = p^{2} \Rightarrow x^{2} \mid p^{2} \Rightarrow x \mid p but I can't arrive at x \mid q for the contradiction (when I replace p =xk in the q^{2}x^{2} = (xk)^{2} the x^{2} on both side cancel)

Homework Statement Prove If x^2 is irrational then x is irrational. I can find for example π^2 which is irrational and then π is irrational but I don't know how to approach the proof. Any hint?

a = \vert x \vert and b = x^{2} a^{2} = \vert x \vert ^{2} = x ^ {2} = b a = \vert x \vert which is nonnegative. correct?

That x^{2} \geq 0 and \vert x \vert \geq 0 ?

Homework Statement Prove that |x| = sqrt(x^2) The Attempt at a Solution I've written two proofs but I don't know if they can be justified as real proofs or whether they are valid or not. Proof 1: \surd x^{2} = \surd \vert x \vert ^{2} = \vert x \vert Proof 2: First Case ) Suppose...

A bit of a blow. I thought I finished but apparently it requires much more careful thinking. The choice of n is the smallest positive integer that xn=e. So any m < n would contradict this. What's to prove?

g-1xng = e xng=ge xn = g.g-1 xn = e Thanks.

(g-1xg)2 = (g-1xg)(g-1xg) => g-1x2g so by induction (g-1xg)n = g-1xng and?

Homework Statement If x and g are elements of the group G, prove that |x|=|g-1xg|. Deduce that |ab| =|ba| for all a,b \in GHomework Equations The Attempt at a Solution I've written a proof but does not seem quite right: if (g-1xg)n= xm then n must be equal m g-nxngn = xm then xngn =gnxm xn...