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

2 years ago

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Niko Lockman


26 Answers

Joseph B Profile Picture
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A proof by contradiction starts by assuming the opposite of what you're trying to prove. You then work by logical steps until you arrive at a statement that is clearly incorrect (like 1 = 0 or the like), i.e. "a contradiction". If your steps have all been logical then the fault can only be with the assumption at the start, so the original statement must be true.


If x is a particular value it cannot be both rational and irrational. If it is a variable (like in y=mx+c) it can take both rational and irrational values, but not at the same time.

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Domagoj Bagic

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

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Adefuye Adetayo Olugbenga

A proof by contradiction is one whose end results are false. X can be rational or irrational if x is a real or natural number

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Paula Mpembe Franco

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.





Graham S Profile Picture
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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.

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William G Profile Picture
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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.

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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.

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James Donlevy

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.

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Eduardo

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."

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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.

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Ahmed Mehrez

Hi Niko.


Proof by contradiction is a method of proof where instead of directly proving some statement, you show that if the statement were false then you hit a contradiction, which tells you that the statement cannot be false, so it must be true.


You are asking how x can be both irrational and rational. Well, it cannot be. A number is exactly one of rational or irrational, never both. One is the opposite of the other.


When proving by contradiction that some number x is irrational, we do indeed assume that it is rational. We are doing this in order to see that if x were rational by making logical deductions we can arrive at a contradiction, which tells us that the assumption that x is rational is false, and hence x is not rational, i.e. x is irrational.


As an example, consider proving that √2 is irrational. Here is how the proof goes.

Assume that √2 is rational. Then it can be written as a fraction, so let √2 = a/b where a and b have no common factors other than 1 (this just means that the fraction is written in simplest form, something that we can do to any fraction).

Now squaring both sides we have 2 = (a/b)^2 = a^2/b^2.

Multiplying by b^2 we have 2b^2 = a^2. Thus a^2 is even, since it equals 2 times some integer.

Therefore a must be even (the square of an even is even, and the square of an odd is odd. think about why this is). We can write a = 2k for some integer k since a is even.

We then have 2b^2 = (2k)^2 = 2^2 k^2 = 4k^2 and dividing by 2 we have b^2 = 2k^2, so b^2 is even and hence b is even.

But then both a and b are even which means the fraction is not in simplest form since we assumed they have no common factors other than 1. This is a contradiction. Therefore √2 is not rational.

Hence √2 is irrational.


Hope this helps!

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Boris Heijliger-Krogulski

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!

See the source image

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Duncan Torbet

Proof by contradiction works by assuming some mathematical statement to be false, but then after following mathematically logical steps, one reaches a conclusion that we know to be false. This means that, since our intermediate steps in the proof were all true, our original assumption must be incorrect - the original statement must be true!


Regarding rational and irrational numbers; a rational number cannot also be an irrational number. The way that irrational numbers can be defined is as a real number that is not rational.


Hope that helps :)

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A proof by contradiction works as follows: If we want to prove a statement, X, is true, start by assuming X is false. We then derive the consequences of X being false. If any of these consequences contradict a result we know to be true, then the only possible problem can be our initial assumption that X is false. Therefore, X must be true.


As an example, let's say we want to prove by contradiction that if n^2 is an odd integer, then n must be odd. We begin by assuming that this statement is not true, and that n is even. If n is even, we can write n = 2k, for any integer k. Then,


n^2 = (2k)^2 = 4k^2 = 2(2k^2),


which is even, since it is 2 multiplied by a number. However, this contradicts our assumption that n^2 is odd. Therefore, if n^2 is odd, then n must also be odd, which proves the statement.

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Proof by contradiction is a method of proving where you first assume your statement to be false. Then use logical steps to show that indeed the statement is true. As a statement can not be true and false at the same time, we deduce that the statement is true. A classic example is proving that the square root of 2 is irrational. So first you assume it to be rational and use logical steps to deduce that actually, it can't be rational. Therefore we conclude that it is irrational.

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