Okay I don't get this =( I think I've made a decent attempt but I need some help getting there all the way. This is due tomorrow. can someone help me? =(
a. I think I have this first part. You can prove it by contradiction.a. If a is rational and b is irrational, is a+b necessarily irrational? What if a and b are both irrational?
b. If a is rational and b is irrational, is ab necessarily irrational?
c. Is there a number of a such that a^2 is irrational, but a^4 is rational?
R= some rational number
a+b = R
b = R-a
A rational number minus a rational number is a rational number. This would mean b = rational, which is not true. therefore a+b is irrational.
This second part, if both are irrational? I was thinking:
a+b = R
a = R-b, or b = R-a. I'm not sure how this helps me x(
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b. If a is rational and b is irrational, is ab necessarily irrational?
No idea, but here's my attempt:
Two cases. If a = 0, ab = 0, and 0 is rational so ab is rational.
if a =/= 0...proof by contradiction maybe?
a*b = rational
a*b = a*b
b = a*b*a^-1?
But then b=b? And that doesn't help me.
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c Is there a number a such that a^2 is irrational, but a^4 is rational?
Well again I have no idea but here's my attempt:
a=b
a^2 = ab
ab = x
b = x/a
b = x * a^-1
can someone help me finish these? =( thanks in advance.