Proof involving rational and irrational numbers.

[Substance 0]

I need seriously need help with this one Let p, q∈R. Prove that if 1/p^2−p is irrational, then p is irrational. A similar example would also help a lot thx

 

[Deleted member 4993]

I need seriously need help with this one Let p, q∈R. Prove that if 1/p^2−p is irrational, then p is irrational. A similar example would also help a lot thx

Which one is your problem:

Let p, q∈R. Prove that if \(\displaystyle \frac{1}{p^2} − p\) is irrational, then p is irrational, or,​

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Let p, q∈R. Prove that if \(\displaystyle \frac{1}{p^2−p} \) is irrational, then p is irrational​

Do you know how to apply reductio ad absurdum ?
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[Substance 0]

2nd problem is my problem here, I really don't know where to begin here, could you give an example?

 

[pka]

I need seriously need help with this one Let p, q∈R. Prove that if 1/p^2−p is irrational, then p is irrational.

I am relatively sure that it is \(\dfrac{1}{p^2-p}\). And that is \(\dfrac{1}{p-1}-\dfrac{1}{p}\).
If \(a=\dfrac{3}{11}\) \(a\) rational and \(\dfrac{1}{a}=\dfrac{11}{3}\) which is rational also.
What \(a-1\)~? Rational? Thus if we suppose that \(p\) is rational does that lead to a contradiction?

 

[Dr.Peterson]

I need seriously need help with this one Let p, q∈R. Prove that if 1/(p^2−p) is irrational, then p is irrational. A similar example would also help a lot thx

I would try to prove the contrapositive. Can you state that? Can you then prove it?

 

[Substance 0]

I am relatively sure that it is \(\dfrac{1}{p^2-p}\). And that is \(\dfrac{1}{p-1}-\dfrac{1}{p}\).
If \(a=\dfrac{3}{11}\) \(a\) rational and \(\dfrac{1}{a}=\dfrac{11}{3}\) which is rational also.
What \(a-1\)~? Rational? Thus if we suppose that \(p\) is rational does that lead to a contradiction?

Would that prove that p is irrational?

 

[Substance 0]

I still don't quite understand it!

 

[Steven G]

If a/b is irrational will b/a be irrational?? Yes, this is true. can you state why??

So then p(p-1) is irrational. How can you conclude that p is irrational.

For the record, Dr Peterson suggestion to use the contrapositive will make this problem simple.

Please look at the contrapositive. Can you state it? In not, then please tell us and we will show you how to get the contrapositive.

 

[Substance 0]

If a/b is irrational will b/a be irrational?? Yes, this is true. can you state why??

So then p(p-1) is irrational. How can you conclude that p is irrational.

For the record, Dr Peterson suggestion to use the contrapositive will make this problem simple.

Please look at the contrapositive. Can you state it? In not, then please tell us and we will show you how to get the contrapositive.

I don't know how to

 

[Steven G]

The contrapositive of If A then B is If not B then not A.

Those two statement are logically the same. They are basically the same statement. So proving one is the same as proving the other.

Now can you state the contrapositive of if 1/p^2−p is irrational, then p is irrational?

 

[Dr.Peterson]

I don't know how to

I suggested using the contrapositive largely because I guessed that your problem would be in the context of a course in which you would have been taught such things (or will soon).

It will be very helpful if you tell us what your context actually is. What have you learned about proofs, and about irrational numbers? What do you know about logic? How about proof by contradiction, or "reductio ad absurdum"?

 

[Substance 0]

The contrapositive of If A then B is If not B then not A.

Those two statement are logically the same. They are basically the same statement. So proving one is the same as proving the other.

Now can you state the contrapositive of if 1/p^2−p is irrational, then p is irrational?

If p is not irrational then 1/p^2-p is not irrational

 

[Steven G]

You need to use parentheses!

Now NOT irrational means rational.

So prove that if p is rational, then 1/(p^2-p) is rational.

1/(p^2-p) = 1/[p(p-1)] = 1/(p-1) - 1/p.


Can you finish from here? If p is rational then p can be written as ....

 

[lookagain]

You need to use parentheses!
.
.
.

Jomo, even after you made an edit to your post #10, you were guilty of
(supposedly unintentionally) leaving off the grouping symbols in the
denominator for which you scolded the original poster, that your post in
this quote box refers to.

 

[Steven G]

Jomo, even after you made an edit to your post #10, you were guilty of
(supposedly unintentionally) leaving off the grouping symbols in the
denominator that you scolded the original poster, which your post in
this quote box refers to.

I noticed that. That is what I get for pasting what the OP wrote--after scolding them about it not being right. Ouch.