Math Is Fun Forum

jay17

Member

Registered: 2009-06-21

Posts: 4

help with real analysis

i really need help with this pls..thanks

1.    Prove: If a is irrational,  1/a is irrational.
2.    Prove that  √2 + √3 is irrational.
3.    Prove that if  a is irrational,  √a is irrational.
4.    Prove:  log 2 is irrational.
5.    Prove: If cos2θ  is irrational, cosθ  is irrational.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: help with real analysis

For #2

See Jane Fairfax's post #3 on this page.

http://www.mathisfunforum.com/viewtopic.php?pid=114051

For #3

We are given that a is irrational so it cannot be expressed as m/n when m,n ∈ Integers with (m,n)=1.

We want to prove that √a is irrational.

Lets assume √a is rational with (m,n)=1.Then it can be expressed like this:

square both sides and we get

so a is clearly rational but this contradicts what we are given. So √a is irrational.

Last edited by bobbym (2009-06-22 03:34:12)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

jay17

Member

Registered: 2009-06-21

Posts: 4

Re: help with real analysis

how did jane arrived at

?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: help with real analysis

Hi jayk;

Her step 2

to prove this just multiply both sides by √ 2 +√ 3

Hard to say maybe from the identity (b+a)-(a-b) = 2b with b = √2 and a = √3

Last edited by bobbym (2009-06-21 20:27:28)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

jay17

Member

Registered: 2009-06-21

Posts: 4

Re: help with real analysis

i get it now..thank you so much bobbym