The Student Room Group
Competition - post a photo you've taken of nature >>
Competition - post a photo you've taken of nature >>
Competition - post a photo you've taken of nature >>

This discussion is now closed.

Check out other Related discussions

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
Avatar for TheWolf
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
Avatar for samd
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
proof.jpg9.5KB
Reply 3
Avatar for dvs
dvs
10
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
Avatar for TheWolf
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
Avatar for samd
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?
Reply 6
Avatar for samd
samd
OP
10
TheWolf
thanx didn tknow that =o abit sneaky the way its put in the spec tho


It also says tan(θ/2) is not required!

See attachment.
tanhalf.jpg8.3KB
Reply 7
Avatar for dvs
dvs
10
Raise both sides to the power of q, since [2^(p/q)]^q = 2^p.
Reply 8
Avatar for samd
samd
OP
10
Thanks.
Reply 9
Avatar for TheWolf
TheWolf
15
samdavyson
It also says tan(θ/2) is not required!

See attachment.


Looks like they changed the spec recently? If so, that's v.gay. :mad:

Related discussions

Latest

Trending

Trending

Articles for you