What is proof by contradiction? How can x be both rational and irrational, please help

Hey Niko! Basically with a proof of contradiction you start with the original statement, lets say "The square root of 2 is irrational". you start by taking the CONTRADICTION of that, meaning you are going to (try to) prove that the square root of 2 is actually rational. When you try to solve this, you will in fact turn out with the conclusion that the square root of 2 cannot be rationally written down, contradicting your statement of root 2 being rational, hence the original statement of it being irrational is true! Here is a picture, see if this helps!

Proof by contradiction is a form of mathematical proof. In this specific type of proof, you are required to prove that a statement is true by showing that there would need to be contradictions amongst your initial assumptions for the statement to be false. Therefore, the initial statement must be true.

The case of x being "both rational and irrational" is something that comes up in multiple different proof by contradiction questions so I am not sure what the context is here?

However, I can still explain why it seems x is considered to be two opposite things. If the statement you want to prove says that x is irrational then you will need to show that other assumptions from the statement would be contradicted if you were to assume that x is instead rational. If you write x as a fraction of two integers which do not have common factors (making x rational) then you will see that some of your assumptions will be contradicted by this, suggesting that the original statement was true.

I hope this helps!

Suppose I want to prove that "flowers can be yellow". Either "flowers can be yellow" is true, or "flowers cannot be yellow" is true, but not both. So proving by contradiction looks for a contradiction in the false statement to show that it is impossible. In this case the false statement is "flowers cannot be yellow" so let's suppose that flowers can't be yellow; this would mean that I would never find a yellow flower, but that is a contradiction since I HAVE found a yellow flower (a sunflower) therefore it can't possibly be true that "flowers cannot be yellow" so that must mean that "flowers can be yellow", hence we have proven that our initial statement ("flowers can be yellow") is true.

In your case, either "x is rational" OR "x is irrational" but not both, so there will be a contradiction in your reasoning for the false statement showing that it is impossible for that false statement to be true, hence the other statement is true.

The idea is to see what else must be true if "x is rational". If this statement is false, then there will be a contradiction somewhere which shows that it is impossible.

Firstly, an irrational number is defined as a real number that is not rational. Therefore, if x is a real number, then x cannot be rational and irrational at the same time.

A contradiction is defined by the dictionary as "a combination of statements which are opposed to one another." Examples of contradictions:

1. All plants are green. Not all plants are green.
2. Mike is 6 foot tall. Mike is not 6 foot tall.
3. 10 is an even number. 10 is an odd number.

Therefore the combination of statements "x is rational and x is not rational" is a contradiction, or in other words, "x is rational and irrational."

Say now that we wanted to prove that x is irrational by contradiction. Then we assume first that the opposite statement is true, that is, we assume that x is rational. This will derive in a series of logical consequences that might end up in a contradiction. If this is the case, then our initial assumption must be wrong, since contradictions are never true. They are impossible.

There is no way whatsoever that x can be rational. Because this would derive in a contradiction. Therefore x must be irrational.

You can find many examples of proofs by contradiction online. For example: "Prove that the square root of 2 is irrational."

Proof by contradiction is a way of proving a statement by showing that the 'opposite' of the statement can not be possible.

So for example, if you were asked to prove that 'x' is rational, you can do this by proving that 'x' cannot be irrational. The way in which you achieve this is by proving that for 'x' to be irrational, you reach a contradiction and thus 'x' cannot be irrational, proving 'x' is rational.

Proof by contradiction works by starting assuming something is true, and then by logically handling the statement you eventually show that it doesn't work, i.e. the statement was actually false all along. In your example, you'd prove something is irrational by assuming it's rational, doing maths to it until you make a statement that can only work if the number was irrational.

Proof by contradiction is one of the simpler methods of proof available. It essentially boils down to proving that two conditions cannot exist simultaneously assuming that a condition is true, working through what happens if that condition is true and showing that this leads to an impossible situation. Of course a chosen number cannot be rational and irrational, so if a proof had been conducted assuming that x is rational, and x is later found to be irrational, the situation we are describing is impossible.

Hi Niko! A proof by contradiction does not directly prove something to be true, but rather proves that it is false. We can use it to show something else is true, though, if that's the only other option. One famous example that you are perhaps referring to in your question is used to show that the square root of two is an irrational number.

As you rightly point out, a number cannot be both rational and irrational – so to show that a number is irrational, we can instead show that it's definitely not rational. We can do that with a proof by contradiction: we start by assuming it is rational, then work through some equations based on that assumption until we prove two conclusions that cannot both be true. Then, if we've reached an impossible conclusion, we know that our initial assumption must have been wrong – the number can't be rational after all, so it must be irrational!

A proof by contradiction is any proof that works in this way – starting by assuming the opposite of what you are trying to prove, and showing that that assumption leads to an impossible (that is, self-contradictory) result.

Proof by contradiction can only be used to validate a proposition as false.

The idea is to confront the proposition with contradiction and in this case, it is safe to conclude that the initial proposition is wrong.

For example:

Proposition A: Anna has no siblings.

If any evidence is presented showing that Anna has any number of siblings, we conclude that proposition A is wrong and we verify its validity by contradiction.

The contradiction: Anna has no siblings and a sibling at the same time.

(This proof is very useful because it is only necessary to have one contradictory situation to have a verdict. Therefore it is used in subjects like physics. )

In this context, no x can not be both rational and irrational.

Well, x can not be both at the same time, that is the contradiction. Proof by contradiction begins assuming that the statement you want to proof is false, then you follow logical valid steps until you get a contradiction. Since all the steps have been correct, then your initial hypothesis, which consisted in denying the statement you want to prove, is false, therefore your statement must be true.

Hello Niko,

that is indeed right. A number can NOT be both rational and irrational at the same time! What are we trying to do with the proof of contradiction is create a premise and then find a contradiction in it which would imply our premise wasn't correct.

In your specific example, what we are trying to do and achieve is we are supposing some irrational number is rational and then we do some computing with it and we find a contradiction. Bare with me!

let's work on one example; the square root of 2

suppose that the square root of 2 is rational Which means it can be written as the ratio of two integers p and q

sqrt(2)=p/q

we picked p and q such that they have no common factors. If we square that whole expression we get

2=p~2/q~2

which implies:

2q~2=p~2

thus p~2 is even which means p is even as well. Thus p~2 is dividable by 4. Hence q~2 must be even and q must be even as well(they are on opposite side of equation). Both q and p are even which is a contradiction with the initial statement that there are no common factors between p and q. We can conclude that our premise is wrong and therefore we conclude that sqrt(2) is irrational.

Really hope that helped and please give me good review since it took me almost an hour to make the sound argument out of this.

Kind regards,

Domagoj