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Proving the difference between odd numbers is even

I'm marking a primary school maths paper and they've been asked to prove that the difference between two odd numbers is even. They've gotten it right by choosing an example. Not very general :erm:.

I don't really know how to prove anything but I am curious to see how you'd properly prove it.

Would you say that odd numbers are 2n−12n-1 where n∈Zn \in \mathbb{Z} so the difference between them is 2n−1−(2m−1)2n-1-(2m-1) which is 2(n−m)⏟even\underbrace{2(n-m)}_{\text{even}}?
(edited 8 years ago)

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Original post by Kvothe the arcane

I'm marking a primary school maths paper and they've been asked to prove that the difference between two odd numbers is even. They've gotten it right by choosing an example. Not very general .

I don't really know how to prove anything but I am curious to see how you'd properly prove it.

Would you say that odd numbers are where so the difference between them is which is ?


Yeh, I would've done it the same way. How did the primary school children answer the question?
(edited 8 years ago)
Original post by ravioliyears
Yeh, I would've done it the same way. How did the primary school children answer the question?


I have edited my post as there were a couple errors. But something like "25-7=18" so it must be true. (it was a true/false question with a 2nd mark for explanation).
Original post by ravioliyears
Yeh, I would've done it the same way. How did the primary school children answer the question?


use numbers and examples as proof?
Original post by Kvothe the arcane
I have edited my post as there were a couple errors. But something like "25-7=18" so it must be true. (it was a true/false question with a 2nd mark for explanation).


Yeh, I see what you changed. Did they get the answer right?
Reply 5
Avatar for Zacken
Zacken
22
Or, for another option - odd numbers are congruent to 1 (mod2)1 \, \pmod{2} so the difference of two odd numbers is congruent to 0(mod2)0 \pmod{2} i.e; even.

Also, instead of ugly negative signs, you might as well just prove that the sum of any two odd numbers are even and that naturally extends to differences given that it is nothing but the additive inverse, so a slightly prettier proof could have been 2n+1+2m+1=2(m+n)2n+ 1 + 2m + 1 = 2(m+n) which also (for primary school kids) means that you avoid the error of distributing the negative into the bracket.

P.S: You might want to \text{even}. :tongue:
(edited 8 years ago)
Original post by ravioliyears
Yeh, I see what you changed. Did they get the answer right?


Official answer says "False. An odd number subtracted from another odd number will always given an even number, e.g. 17-5=12" so Yes.

I think things are taught a little differently from when I did my maths. For example, they aren't taught about the distributive, associative and commutative properties of numbers.
Thank you ^^ Zacken. Modular arithmetic is meaningless to me but it is on the list of things to learn/familiarise myself with next gap year :smile:.
Reply 8
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Zacken
22
Original post by Kvothe the arcane
Thank you ^^ Zacken. Modular arithmetic is meaningless to me but it is on the list of things to learn/familiarise myself with next gap year :smile:.


Probably not the most useful thing to you, given you're a natsci. :lol:
Reply 9
Avatar for Notnek
Notnek
21
Original post by Kvothe the arcane
I'm marking a primary school maths paper and they've been asked to prove that the difference between two odd numbers is even. They've gotten it right by choosing an example. Not very general :erm:.

I don't really know how to prove anything but I am curious to see how you'd properly prove it.

Would you say that odd numbers are 2n−12n-1 where n∈Zn \in \mathbb{Z} so the difference between them is 2n−1−(2m−1)2n-1-(2m-1) which is 2(n−m)⏟even\underbrace{2(n-m)}_{\text{even}}?

This is how I would "prove" it to a primary school pupil:


Example:

3, 4, 5, 6, 7, 8, 9

The difference between any two numbers is the number of steps you need to get from one number to the next, working right-to-left on the number line.

Now to get from one odd number to the next odd number working left e.g. 9 -> 7 you need to move 2 places. And if you want to get from any odd number to another e.g. 9 -> 3 you can get there by passing over all the odd numbers in between : 9 -> 7 -> 5 -> 3. Here we have moved 2 + 2 + 2 = 6 places.

Here you can see that you can get from one odd number to another by moving by some multiple of 2.

Hence even yo.
Original post by notnek
...


Thank you :smile:. Lovely explanation.

How would you expect them to prove it on a paper? While your meaning is clear, your explanation is probably a bit verbose for the exam.
Reply 11
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Notnek
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Original post by Kvothe the arcane
Thank you :smile:. Lovely explanation.

How would you expect them to prove it on a paper? While your meaning is clear, your explanation is probably a bit verbose for the exam.

What was the actual question?

The official answer that you gave doesn't match with your original question.
Original post by notnek
What was the actual question?

The official answer that you gave doesn't match with your original question.


Winston says, "when you subtract an odd number from another odd number the answer will always be odd.'

Is this true or false? Tick the correct answer

True False \text{True} \ \fbox{} \qquad \text{False} \ \fbox{}

Explain your answer.
(edited 8 years ago)
Reply 13
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Notnek
21
Original post by Kvothe the arcane
....

Ah now that's quite different to your original question.

Their statement says that the result is always odd when you subtract two odd numbers.

To prove that this is false then all you need is give one example that shows the difference to be even.

This doesn't prove that the difference is always even but it proves that the difference is not always odd.
(edited 8 years ago)
Original post by notnek
Ah now that's quite different to your original question.

Their statement says that the result is always odd when you subtract two odd numbers.

To prove that this is false then all you need is give one example that shows the difference to be even.

This doesn't prove that the difference is always even but it proves that the difference is not always odd.


Ah, of course. You only need one contradiction. Good thing you asked.

As my old teacher would say, I should have RTQ (read the question)!
(edited 8 years ago)
Reply 15
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Zacken
22
Original post by Kvothe the arcane
Ah, of course. You only need one contradiction. Good thing you asked.


In this case, it's called a counterexample. A contradiction is something else entirely.
Original post by Zacken
In this case, it's called a counterexample. A contradiction is something else entirely.


A contradiction would have been to say that as it is always even, it couldn't be odd?
Reply 17
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Zacken
22
Original post by Kvothe the arcane
A contradiction would have been to say that as it is always even, it couldn't be odd?


Not quite. If you assume something and then working off the basis of that assumption, you get to something that's obviously false, then that something is called a contradiction. Say the only thing you know isn't true is that 2=32 = 3.

Then, we want to see whether 1=21=2 is true or not. We add 1 to both sides, that's a valid mathematical step so we get 2=32=3 but we know that that's not true, and this is called our contradiction. So since adding 1 to both sides was a valid mathematical step, the only thing suspect with out entire proof was our assumption, so that assumption must be wrong and hence, we've proved that 1≠21 \neq 2. This is called a proof by contradiction.
Original post by Zacken
Not quite. If you assume something and then working off the basis of that assumption, you get to something that's obviously false, then that something is called a contradiction. Say the only thing you know isn't true is that 2=32 = 3.

Then, we want to see whether 1=21=2 is true or not. We add 1 to both sides, that's a valid mathematical step so we get 2=32=3 but we know that that's not true, and this is called our contradiction. So since adding 1 to both sides was a valid mathematical step, the only thing suspect with out entire proof was our assumption, so that assumption must be wrong and hence, we've proved that 1≠21 \neq 2. This is called a proof by contradiction.


I see. But numbers can't be both even and odd. Numbers only have one parity. They are mutually contradictory attributes.
So if I propose that "the difference is odd" but assume it false and prove that the "difference is even", doesn't this imply that my original proposition is false? As the difference can't the both odd and even?

I apologise for the questioning.
(edited 8 years ago)
Reply 19
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Zacken
22
Original post by Kvothe the arcane
I see. But numbers can't be both even and odd. Numbers only have one parity. They are mutually contradictory attributes.
So if I propose that "the difference is odd" but assume it false and prove that the "difference is even", doesn't this imply that my original proposition is false? As the difference can't the both odd and even?

I apologise for the questioning.


You seem to have assumed what you proved? Also if you assume it false then you're assuming that the difference is even so the "prove that the difference is even" bit becomes redundant?

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