Hi Agnishom;

This is one way:

hi bobbym,

Did you mean,   start with the true result

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

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?

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...

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

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.

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.

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.

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.

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

Hi Agnishom;

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