Proof help please

Original post by CatInTheCorner

Hi, for proof questions, they always have prove bla bla bla is a real number, or an irrational number. I know you write even numbers as 2k and odd numbers as 2k+1, but for the other ones I never know how you represent them. What are the different types of numbers I need to know for a level edexcel and how do I write them down?

List:
1)rational
2)irrational
3)real
4)integer

Rational numbers can be represented as a/b where a and b are integers (b is non-zero).

For the other types there's no way to represent them apart from just using a single variable like x. How you deal with those types of numbers depends on the question. E.g. for proving that something is irrational, you can often use contradiction by assuming that it is rational i.e. it's of the form a/b.

(edited 1 year ago)

Reply 2

OP

also what's the difference between proof by contradiction or by counter example? I have a question and I'm a bit lost.

Screenshot 2023-04-10 155408.jpg

if a squared minus b squared is greater than 0. and a and b are reak, then a - b is greater than 0. I showed that if a squared minus b squared is greater than 0 then a - b must also be greater than 0, but is that contradiction?

Reply 3

OP

Original post by Notnek

Rational numbers can be represented as a/b where a and b are integers (b is non-zero).

For the other types there's no way to represent them apart from just using a single variable like x. How you deal with those types of numbers depends on the question. E.g. for proving that something is irrational, you can often use contradiction by assuming that it is rational i.e. it's of the form a/b.

oh wonderful thank you. i just thought there was some invisible language I didn't know

Reply 4

Original post by CatInTheCorner

also what's the difference between proof by contradiction or by counter example? I have a question and I'm a bit lost.

Screenshot 2023-04-10 155408.jpg

if a squared minus b squared is greater than 0. and a and b are reak, then a - b is greater than 0. I showed that if a squared minus b squared is greater than 0 then a - b must also be greater than 0, but is that contradiction?

A counter-example is really simple. You just need to come up with any example where the statement doesn't hold true.

If you find one example that doesn't hold then you've disproved the statement.

Reply 5

OP

Original post by Notnek

A counter-example is really simple. You just need to come up with any example where the statement doesn't hold true.

If you find one example that doesn't hold then you've disproved the statement.

So i've done proof by contradiction, damn. Ok thank you, thanks so much

Reply 6

Muttley79

20

Original post by CatInTheCorner

also what's the difference between proof by contradiction or by counter example? I have a question and I'm a bit lost.

Screenshot 2023-04-10 155408.jpg

if a squared minus b squared is greater than 0. and a and b are reak, then a - b is greater than 0. I showed that if a squared minus b squared is greater than 0 then a - b must also be greater than 0, but is that contradiction?

A counter-example is just an example that doesn't work so if a = -5 and b = what? it isn't true

Reply 7

Original post by CatInTheCorner

So i've done proof by contradiction, damn. Ok thank you, thanks so much

Yes it looks like that's what you've attempted. Contradiction can be used to prove something (one of many methods). Counter-examples can be used to disprove something, which is what this question is asking for.

Reply 8

OP

Original post by Notnek

Yes it looks like that's what you've attempted. Contradiction can be used to prove something (one of many methods). Counter-examples can be used to disprove something, which is what this question is asking for.

thanks guys. I love TSR when it isn't wierd incels. Ya'll are amazing :adore:

Reply 9

Original post by CatInTheCorner

thanks guys. I love TSR when it isn't wierd incels. Ya'll are amazing :adore:

The maths forum has always been the best part of TSR. I used it myself when doing my A Levels years back and it continues to have so many experienced helpers.

Reply 10

OP

hiii mi amigos im back,

"if x and y are irrational, and x does not equal y, then xy is irrational"

Is a counter example x = root 9 and y = root 16? Or because they can be written as whole numbers they aren't irrational? How do I approach this?

Reply 11

tonyiptony

11

Original post by CatInTheCorner

also what's the difference between proof by contradiction or by counter example? I have a question and I'm a bit lost.

Screenshot 2023-04-10 155408.jpg

if a squared minus b squared is greater than 0. and a and b are reak, then a - b is greater than 0. I showed that if a squared minus b squared is greater than 0 then a - b must also be greater than 0, but is that contradiction?

Reiterating previous posters' comments, the main difference is that proof by contradiction (By Way Of Contradiction) is a strategy to show a claim is TRUE, while counterexample is a strategy to show a claim is FALSE.

So put it in some context,
(i) We want to show that the claim "all horses are brown" is FALSE. All I need to do is to find one horse that is NOT brown - there's a counter-example. It's often easy.
(ii) We want to show that the claim "sqrt(2) is irrational" is TRUE. Then we first assume the contrary is true, that "sqrt(2) is NOT irrational", then through deduction we end up with something absurd like "sqrt(2) is both rational and irrational" (see standard proof in your textbook), so our assumption is false - that's BWOC.

I don't know why maths people have a weird obsession with horses sometimes...

(edited 1 year ago)

Reply 12

OP

Original post by Notnek

The maths forum has always been the best part of TSR. I used it myself when doing my A Levels years back and it continues to have so many experienced helpers.

Love it, this and the medicine forums ( if you take away the I'm in year 10 and I ONLY want to go to oxbridge, I am going to take 4 a levels and also will save the world. What subjects do I pick?" are superb

Reply 13

tonyiptony

11

Original post by CatInTheCorner

hiii mi amigos im back,

"if x and y are irrational, and x does not equal y, then xy is irrational"

Is a counter example x = root 9 and y = root 16? Or because they can be written as whole numbers they aren't irrational? How do I approach this?

That's not a valid counter example, as neither sqrt(9) nor sqrt(16) are irrational.
That said, the statement is in fact FALSE, and the counter example is quite easy.

Useful tip: Usually when showing something related to ir/rational numbers, all you need is sqrt(2).

EDIT: It is related to the logic that "you can show ANYTHING IS TRUE if you assume something that's false". A funny or rather grim example would be to say "if 1+1=3, then the world ends tomorrow". It is (vacuously) true, to get a bit technical.

(edited 1 year ago)

Reply 14

Original post by CatInTheCorner

hiii mi amigos im back,

"if x and y are irrational, and x does not equal y, then xy is irrational"

Is a counter example x = root 9 and y = root 16? Or because they can be written as whole numbers they aren't irrational? How do I approach this?

root 9 and root 16 are rational numbers since they are equal to 3 and 4 respectively. So those don't work.

What irrational numbers can you think of? Experiment with some.

Reply 15

OP

thanks babes it's what i thought :bawling:

so if I use root 2 as x, let's say

then root 2 timesed by something irrational has to make something rational.

I would have said root 2 but x cannot equal y. I'm so lost????

Reply 16

Original post by CatInTheCorner

thanks babes it's what i thought :bawling:

so if I use root 2 as x, let's say

then root 2 timesed by something irrational has to make something rational.

I would have said root 2 but x cannot equal y. I'm so lost????

Other square roots are irrational...

Just not root 9 and root 16 :smile:

Reply 17

OP