Edexcel C3: Proof by Contradiction

I have only just realised that this is on the spec. (and according to the spec it is guaranteed that 1 question will involve proof).

I have looked it up on Google and I follow the idea:

Take something that you want to disprove and show that it leads to a contradiction.

Many of the proofs involve rational/irrational numbers and showing that such and such is irrational.

Could someone go through the steps to show that log(2)5 is irrational?

Thank you.

Reply 1

TheWolf

15

samdavyson

I have only just realised that this is on the spec. (and according to the spec it is guaranteed that 1 question will involve proof).

Any links?

Reply 2

samd

OP

10

For that on the spec or for information on proof by contradiction?

The spec is at: http://www.edexcel.org.uk/VirtualContent/83441/GCE_Mathematics_Specification.pdf

And the bit of interest is page 33, but I have attached a screen shot of the paragraph of importance.

As for proof by contradiction try: http://zimmer.csufresno.edu/~larryc/proofs/proofs.contradict.html
http://www.delphiforfun.org/Programs/Math_Topics/proof_by_contradiction.htm
http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node24.html

and loads more here

Could someone go through the steps to show that log(2)5 is irrational?

Assume that it's rational, i.e.
log(2) 5 = p/q, where both p & q are non-zero integers.
5 = 2^(p/q)
5^q = 2^p

The LHS is odd for all integral q, while the RHS is even for all integral p. Contradiction.

Reply 4

TheWolf

15

samdavyson

For that on the spec or for information on proof by contradiction?

The spec is at: http://www.edexcel.org.uk/VirtualContent/83441/GCE_Mathematics_Specification.pdf

And the bit of interest is page 33, but I have attached a screen shot of the paragraph of importance.

As for proof by contradiction try: http://zimmer.csufresno.edu/~larryc/proofs/proofs.contradict.html
http://www.delphiforfun.org/Programs/Math_Topics/proof_by_contradiction.htm
http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node24.html

and loads more here

thanx didn tknow that =o abit sneaky the way its put in the spec tho

Reply 5

samd

OP

10

dvs

Assume that it's rational, i.e.
log(2) 5 = p/q, where both p & q are non-zero integers.
5 = 2^(p/q)
5^q = 2^p

The LHS is odd for all integral q, while the RHS is even for all integral p. Contradiction.

That is good.

How do you go from the secon dfrom bottom line to the bottom line so fast?

I would go:
5 = 2^(p/q)
ln5 = (p/q)ln2
qln5 = pln2
ln5^q = ln2^p
=> 5^q = 2^p

This may be what you did but you just did it in one step, or is there a good quick way?