- #1
Sven
- 6
- 0
Rational and irrational numbers. (semi-urgent)
I need to figure this out by tomorrow =/
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?
none.
a. I think I have this first part. You can prove it by contradiction.
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(
-----
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.
-----
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
...again no idea, please help. x(
I need to figure this out by tomorrow =/
Homework Statement
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?
Homework Equations
none.
The Attempt at a Solution
a. I think I have this first part. You can prove it by contradiction.
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(
-----
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.
-----
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
...again no idea, please help. x(