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Prove that for every real n > 0, n +(1/n) >= 2

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 91,691

Hi Agnishom;

This is one way:

Multiply both sides by n. This is okay because n is positive, we do not have to worry about reversing the <= .

Which is obviously greater than 0 for all n>0. We are done.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,797

The other would be just using the AM-GM inequality...

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,664

hi bobbym,

Did you mean, start with the true result

and work towards the required form

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 91,691

Hi Bob;

Unless I made some slip the other way would seem more esoteric. How do you know to start from that is an obvious reply and I have asked it to. Working it algebraically to the square rule is more natural. I see this method of producing squares on the LHS and 0 on the right done all the time in inequality books so that is why I used it.

**In mathematics, you don't understand things. You just get used to them.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,664

Trouble is: if you start by assuming what you are required to prove and show a true result it doesn't always follow that the proof is reversible. I have no problem with you thinking it through that way ... it's a good approach ... but you ought then to re-write it with the known truth first then leading to the thing you have to prove.

I'm trying to think of an example to illustrate when it might go wrong otherwise.

This is a bit contrived but it does show what I'm saying.

Prove that pi = 3

I'll assume pi = 3

I'll multiply both sides by zero

pi . 0 = 3 . 0

0 = 0

This is true therefore pi = 3 hhmmm.

I'll try to find a better example.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,664

I have one.

Prove that 30 = 150

I'll start with 30 = 150

=> sin 30 = sin 150

=> 0.5 = 0.5

This is true so 30 = 150.

Conclusion: assuming what you have to prove and showing it leads to a true result is valid only if all the steps are still ok when the proof is written in reverse. To be rigorous you should therefore write the steps the right way round. Then you know you haven't 'fiddled it'.

here's another one.

3 = -3

=> 3^2 = (-3)^2

=> 9 = 9

etc etc.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 91,691

Hi Bob;

I understand what you are saying but take a look at this pdf. You will see that he proves all of them by working in the forward direction to a true statement. In these cases the rule that squares are always positive or the product or sum of them is always positive. The first couple are the same as I did.

http://onlinemathcircle.com/wp-content/ … ture-7.pdf

What is the difference between your examples and these book proofs that are allowing this?

**In mathematics, you don't understand things. You just get used to them.**

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 91,691

Hi Agnishom;

Sorry for the confusion caused by the first proof but Bob's comments had to be considered. So I revised the proof to be more rigorous.

Start with the identity

Now we can say

And we are done.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,797

bob bundy wrote:

I have one.

Prove that 30 = 150

I'll start with 30 = 150

=> sin 30 = sin 150

=> 0.5 = 0.5

This is true so 30 = 150.

Conclusion: assuming what you have to prove and showing it leads to a true result is valid only if all the steps are still ok when the proof is written in reverse. To be rigorous you should therefore write the steps the right way round. Then you know you haven't 'fiddled it'.

here's another one.

3 = -3

=> 3^2 = (-3)^2

=> 9 = 9

etc etc.

Bob

Those are different from bobbym's proof because bobbym's steps are equivalent. Yours are not e.g. from x^2=y^2 doesn't follow x=y...

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,797

Hmmm... Why not use the Preview button?

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 91,691

On this browser it is often off to the side and I can not easily see it.

**In mathematics, you don't understand things. You just get used to them.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,797

Ok.

In post #10 is the explanation of why your proof is correct and the quasiproof examples aren't.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 91,691

Hi anonimnystefy;

I think so too but hopefully I can convince Bob after he reads the pdf in post #8.

His input was gratefully received, I am not an expert on inequalities, not even close.

I have revised the proof anyway to be on the safe side. It took me all night to come up with that identity. I am looking at it now in a book that I did not read or know about.

**In mathematics, you don't understand things. You just get used to them.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,664

hi bobbym,

Ok, I see. he does it to. If I met him I'd still make the same point.

http://en.wikipedia.org/wiki/Propositional_calculus

If you are asked to prove B, and you know A is true and that you can create a series of mathematically valid steps that lead from A to B, then that constitutes your proof.

But A => B is not the same as B => A ( all cows are mammals but all mammals are not cows )

If you find a series of mathematically valid steps that leads from B to A (and A is true) then you only have a valid proof that A => B if every step on the way is reversible.

Now all yours are (and so are his) so maybe you think it is trivial to have to say so. But what worries me is that others who are not so good at maths may see this and think you can always do this ( start with B and get A which is true )

My examples show that it doesn't always work so I'd prefer that, once you have the steps, you then write them in the A leads to B order. That way everyone san see you have proved what was required without doubt.

Am I being pedantic ? Probably, but then I'm a teacher. I've seen too many students make silly mistakes because they weren't clear about what they were doing. So I try to instill what I think are correct habits, right from the beginning.

How about if you say "Obviously, every step can be reversed, so a true statement leads to the required result" or "The proof that the logic for this can be written <true statement> leads to <that which was required> is left as an exercise to the reader."

Bob

ps. to Stefy. "equivalence" yes! That's my whole point. You can do this because the steps are reversible. I am trying to say that because this doesn't always hold you have to make some comment that it is Ok to do it here.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 91,691

Hi Bob;

I notice that in many books they start these proofs with the phrase "first observe that such and such identity," so I rewrote the the proof in post #9.

so maybe you think it is trivial to have to say so.

Truthfully, I did not know enough to call it trivial. I was oblivious.

Pedantic is what gets the money in math. I appreciated your comments. I truly did not know that.

**In mathematics, you don't understand things. You just get used to them.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,664

hi bobbym and all

Pedantic is what gets the money in math

Metaphorically ... maybe. Literally ... not in my world. I'd be a rich man.

I've got another , even better example of why you shouldn't start with what you have to prove.

example.

Suppose 1 = 2

Then 2 = 1

adding

3 = 3

This is true. So does that mean 1 = 2 ?

Bob

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**zetafunc.****Guest**

It's a very helpful tactic for exams -- in one of my mechanics papers (M5) I worked backwards from a 'show that', then just rewrote what I did in reverse and crossed out the original. I think the exam board do not like this however.

**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,664

hi zetafunc

It is a very sensible tactic and I would give you whatever marks were going for it. What makes you think the boards don't like it ?

Bob

Actually, just read your post again. When they say 'show that' they are deliberately giving you the result to make it easier and so that you can do the next step even if you cannot do the 'show'. You may think examiners are cruel *****!## but mostly they are quite nice folk who want to give you all the marks they can. And it is much easier to mark a question where the student clearly knows what they are doing, so they are already sympathetic.

In the current English GCSE debacle, the examiners wanted to award the higher grades and it was Ofqual that put the boot in.

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**zetafunc.****Guest**

Well, it makes some problems ridiculously easy... there was a problem in my M5 (June 2012) paper which gave some scenario involving angular motion and asked you to show that the braking force is equal to _____ (or something along those lines). But you could just work backwards and it became a lot easier.

**zetafunc.****Guest**

But if they want to be helpful and prove that you know what you're doing so everyone is on the same level for the next part of the question, why not cut out unintended solutions?

Also, regarding the English GCSE debacle, do you know if it affected other subjects, such as A-level History? My friends ended up getting Bs and some even Ds, though they were seen as potential Oxbridge applicants. I was also curious to know if it affected any maths exams -- my D1 exam for instance was marked to 85 UMS, but I have no clue where I lost that 15%.

**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,664

Exam design has come on a long way in the last 50 years. You used to get questions where you had to work through all the parts with no prompts and if you couldn't do part (a) then the rest couldn't even be attempted. The trouble with that sort of question is that it doesn't necessarily pick out the strongest candidates and give them the highest marks. Let's say there are 10 things you have to know, A,B,C ..., J

If a question contains elements of A,B and C and a strong candidate knows nine out of the ten topics really well you'd expect them to get high marks. After all, they know 90% of the work. But if, say C, is their weak point then they cannot do a whole question, even though they know A and B. So luck plays too big a part in the results.

I'll give you a real example that come up on a paper about 40 years ago. The candidates were given a diagram of a rhombus with the lengths of both diagonals. They had to calculate the length of a side, the perimeter, the area and the angles between the sides. That tests pythag, area of triangles, and trig but you have to know one crucial fact before you can do any of it. If you know that the diagonals bisect each other at right angles then you're away but without that fact you cannot do any of it!

Now certainly a candidate who doesn't know the properties of a rhombus deserves to be penalised, but maybe 1 mark, not 4. It puts too much weight on a single part of the syllabus which distorts the correlation between how much the students know and the mark they come out with.

Nowadays they try to set question parts which are independant of each other so that you are only penalised once for any lack of knowledge/skill. That's why they say 'show that'. It means you can still tackle the rest of the question and should do so! (using the thing you had to show)

I haven't heard that other subjects were affected. The media has been red hot on this so I think we would have heard if there was the slightest thing wrong elsewhere. You can ask for a photocopy of your paper but it costs money. And I think the deadline for this may have gone.

Your teacher will have the <real mark> to <UMS> conversion so you can see exactly how many marks you lost. Then you might be able to work out where you lost them.

Bob

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Sorry for not being able to be online tomorrow

As for me post #2 was a hazy one, and I had doubts about it.

But from #9 I am now sure about the proof

Thanks all

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 91,691

Hi Agnishom;

I rewrote the proof from post #2 in post #9 to make it more rigorous. Use that one.

**In mathematics, you don't understand things. You just get used to them.**

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Hi bobbym;

I will.

Hi anonynistefy;

Would you please explain the AM-GM ethod?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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