Prove that the numbers q and r are unique

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TSR360
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Q: http://prntscr.com/vkjz31

If q and r where both 0 then a = (0 x 0) + 0 where b = 0
This is a contradiction as b > 0.

Is that good enough or do I need show a contradiction for every instance where q = r?
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RDKGames
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(Original post by TSR360)
Q: http://prntscr.com/vkjz31

If q and r where both 0 then a = (0 x 0) + 0 where b = 0
This is a contradiction as b > 0.

Is that good enough or do I need show a contradiction for every instance where q = r?
Thats not good enough.

You need every instance.
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ghostwalker
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#3
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(Original post by TSR360)
Q: http://prntscr.com/vkjz31

If q and r where both 0 then a = (0 x 0) + 0 where b = 0
This is a contradiction as b > 0.

Is that good enough or do I need show a contradiction for every instance where q = r?
If q=r=0, then a=0, so you're attempting to show that zero can be written uniquely in the form bq+r. With this method you'll then need to do 1, 2, ..., and possibly the negative integers depending on what values a can take.

It also fails as you then assume b=0 and say this contradicts b>0. Well don't assume b=0 then - it's not a valid value.

I would suggest looking up the proof and fully understanding the logic of it, otherwise this thread is likely to run for a very long time.

PS: I am somewhat surprised this is A-level. The logic involved is only solidly taught in first year uni.
Last edited by ghostwalker; 8 months ago
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TSR360
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(Original post by RDKGames)
Thats not good enough.

You need every instance.
How about this http://prntscr.com/vkpmxg
I'm assuming q1 means q1 has the same properties as q but they're not identical.
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TSR360
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(Original post by ghostwalker)
If q=r=0, then a=0, so you're attempting to show that zero can be written uniquely in the form bq+r. With this method you'll then need to do 1, 2, ..., and possibly the negative integers depending on what values a can take.

It also fails as you then assume b=0 and say this contradicts b>0. Well don't assume b=0 then - it's not a valid value.

I would suggest looking up the proof and fully understanding the logic of it, otherwise this thread is likely to run for a very long time.

PS: I am somewhat surprised this is A-level. The logic involved is only solidly taught in first year uni.
There's questions a lot harder than this in my A level textboook. I'm using a beginner number theory textbook to help me understand the proofs more thoroughly, so there might be some overlap between a-level and uni maths but I don't see any other way of being able to do the proofs in a-level as it just isn't very logical.
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davros
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(Original post by TSR360)
There's questions a lot harder than this in my A level textboook. I'm using a beginner number theory textbook to help me understand the proofs more thoroughly, so there might be some overlap between a-level and uni maths but I don't see any other way of being able to do the proofs in a-level as it just isn't very logical.
The thing is, there are differences in the way in which proof is approached at uni from what is acceptable for A level (not to mention the dependence on intermediate results - or "lemmas" - plus the use of formal axioms), so you may confuse yourself even more by trying to use a uni textbook.

Can you give an example of an A level proof that you think is "more difficult" than this?
Last edited by davros; 8 months ago
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ghostwalker
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(Original post by TSR360)
There's questions a lot harder than this in my A level textboook. I'm using a beginner number theory textbook to help me understand the proofs more thoroughly, so there might be some overlap between a-level and uni maths but I don't see any other way of being able to do the proofs in a-level as it just isn't very logical.
I'm not the best person to analyse what your particular issue is and advise, but if you're struggling with the logic of A-level proofs, I seriously doubt looking at a number theory book is going to help you; if anything, it's likely to be a bit of a culture shock.
Last edited by ghostwalker; 8 months ago
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DFranklin
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(Original post by TSR360)
There's questions a lot harder than this in my A level textboook. I'm using a beginner number theory textbook to help me understand the proofs more thoroughly, so there might be some overlap between a-level and uni maths but I don't see any other way of being able to do the proofs in a-level as it just isn't very logical.
I'm going to add to the chorus of voices saying "mixing a university textbook and an A-level textbook is probably a mistake". Although it does depend a bit on the exact textbooks.

As far as this actual proof goes, you don't seem to understand what you're actually trying to prove. A proof of this would normally begin something like:

"Suppose that a = bq+r and a = bq_1 + r_1, with 0 \leq r < b and 0 \leq r_1 < b. Then ..."

(or alternatively, instead of "Then..." you might have "Suppose (for contradiction) that...").

You don't even mention q_1 and r_1, and instead talk about "if q and r are identical", which doesn't really make any sense at all. I think you're trying to talk about two representations

a = bq+r
and
a = bq+r

where the q, r in the 2nd representation are different from in the 1st representation. If so, all I can say is that using the same letter to mean two different things is going to be a fast track to madness, here.
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