That alerts the reader that you are using proof by contradiction and will plug away at the proof until it collapses logically. You work until you find the contradiction. A rational number can be written as a ratio, or a fraction numerator over denominator.

Any fraction can be simplified to its irreducible form , so 2 6 can simplify to 1 3 but can be simplified no further. Notice in its simplified form at least one term of the fraction is odd. An irrational number cannot be expressed as a fraction or ratio. Here we can see that a 2 has to be an even number, because the square of every even number is even, and the square of every odd number is odd.

Recall that a and b cannot both be even, so b must be odd. The contradiction emerges: b 2 is even, so b is even, but we just got through showing it was odd. It is also contradicted because if a is even and b is even, the fraction is not in simplest form, but we started by saying it was irreducible.

The 2 cannot be rational, so it must be irrational. At the contradiction, you should stop your work. You have proven the truth of the statement by showing that a claim that it is false cannot hold up to logic.

This was a challenging lesson. You may well benefit from rereading it several times, but once you do, you should feel more confident in your understanding of proof by contradiction.

Now you are able to recognize and apply proof by contradiction in proofs, develop a logical case to show that the premise is false, until your argument fails by contradiction, and recognize the contradiction in your argument that demonstrates the validity of the original premise. Square Matrix. Get better grades with tutoring from top-rated professional tutors. Get help fast. Want to see the math tutors near you? Proof by Contradiction One of the most powerful types of proof in mathematics is proof by contradiction or an indirect proof.

Video Definition Steps Examples Example 2 Proof By Contradiction Definition Proof by contradiction in logic and mathematics is a proof that determines the truth of a statement by assuming the proposition is false, then working to show its falsity until the result of that assumption is a contradiction. The Mathematician's Toolbox The metaphor of a toolbox only takes you so far in mathematics; what you really have is a powerful mind, and one of the best strategies you can store in that wonderful brain of yours is proof by contradiction.

Proof By Contradiction: Steps Lets break it down into steps to clarify the process of proof by contradiction. We follow these steps when using proof by contradiction: Assume your statement to be false. Proceed as you would with a direct proof. We then see that. It is also important to realize that every integer is a rational number since any integer can be written as a fraction. Because the rational numbers are closed under the standard operations and the definition of an irrational number simply says that the number is not rational, we often use a proof by contradiction to prove that a number is irrational.

This is illustrated in the next proposition. The proof that the square root of 2 is an irrational number is one of the classic proofs in mathematics, and every mathematics student should know this proof.

This is why we will be doing some preliminary work with rational numbers and integers before completing the proof. The theorem we will be proving can be stated as follows:. In order to complete this proof, we need to be able to work with some basic facts that follow about rational numbers and even integers.

So in a proof by contradiction of Theorem 3. So we assume that the statement of the theorem is false. That is, we assume that. We can now substitute this into equation 1 , which gives. Prove that the following 4 by 4 square cannot be completed to form a magic square.

Hint : Assign each of the six blank cells in the square a name. Either construct such a magic square or prove that it is not possible. A proof by contradiction will be used. So we assume the proposition is false. The last inequality is clearly a contradiction and so we have proved the proposition. A Proof by Contradiction. Consider the following proposition:. So we assume that the statement is false. Exploring a Quadratic Equation.

Writing Guidelines: Keep the Reader Informed A very important piece of information about a proof is the method of proof to be used. We will prove this result by proving the contrapositive of the statement. We will prove this statement using a proof by contradiction. Proposition 3.

Progress Check 3. Answer Add texts here. Do not delete this text first. Determine at least five different integers that are congruent to 2 modulo 4, and determine at least five different integers that are congruent to 3 modulo 6. Are there any integers that are in both of these lists? For this proposition, why does it seem reasonable to try a proof by contradiction? For this proposition, state clearly the assumptions that need to be made at the beginning of a proof by contradiction, and then use a proof by contradiction to prove this proposition.

Proving that Something Does Not Exist In mathematics, we sometimes need to prove that something does not exist or that something is not possible. That is just the state of affairs. Most mathematicians won't give this a second thought since the circumstantial evidence for lack of contradictions suffices to put any serious doubts to rest. Moreover, since we choose the axioms we work with, if a contradiction with the currently more or less accepted choice of axioms is found, we'll simply change the axioms so the found contradiction disappears.

It is unlikely that will ever happen, but if it does it probably won't be a big deal and most of mathematics will survive intact.

As for your friend's confusion with the two results you mention, s he is just taking you about years back in time when people had all sorts of weird ideas about infinity and before rigorous definitions for dealing with infinite series were laid down.

The apparent contradictions you see are nothing but the result of carelessly playing with undefined concepts. These problems were immediately solved with the advent of rigorous calculus. The series you give simply does not converges, not to any number. As for the concept of infinity, it can be defined rigorously in many different ways and can be manipulated without causing any contradiction, as long as one understands the context.

There is already great general answer by Ittay Weiss, so I will try a different approach. In fact, I will try to explain a bit the infinite sum you stated. As for the infinity, one could write a lot about it mainly because there are multiple infinities, each with different properties , and I couldn't even hope to fit it here. Unfortunately, I don't know any sources or text about infinity that I would like to recommend.

We have no idea what such a sum might mean without any proper definition, which wasn't stated. However, suppose that we could manipulate it in a way similar to numbers. We got the same expression! Nevertheless, there is a way of dealing with this e. However, note that we didn't argue whether there actually is such an interpretation. If you would like to know more about it, you can start here. The most common appearance of contradictions in mathematics is when one inserts their own ideas about a mathematical concept or object that aren't actually true.

There are a number of other summation methods for series of numbers. There are unfortunately too many things mathematicians want to calculate to have an unambiguous notion for all of them. In reality, there are many different sorts of mathematical structures that work in different ways.

Or think like old-school video games where you go off one end of the map and reappear on the other side. This may be unfamiliar to the lay person, who kept learning of new mathematical structures as extending previous ones; one learns about natural numbers and then about zero if you used the convention that natural numbers are positive rather than nonnegative , then about negative numbers, rational numbers, real numbers and complex numbers.

The lay person naturally builds up the idea there's "one true universe" of numbers and the numbers they learn in school just make up more and more of this universe. But unfortunately, that's really a very incorrect view of mathematics. Mathematicians invent new structures all the time to quantify or otherwise describe the kinds of things they're studying, and being motivated by and used for different purposes, they don't often fit together.

For example, the extended real lines and projective real lines are incompatible with each other. And they are quite incompatible with other contexts put more emphasis on the algebraic laws rather than geometric picture.

Before Russell Paradox It was generally believed that there were not contradictions. Better said, the matter hadn't been spotted yet. After Bertrand Russell the first crisis in maths appeared. And, from my point of view, the issue wasn't solved and it is not solved yet. I know there is the Sets and Classes Theory, all supported by some set of axioms, from which we can deduce all known proved theorems.

But for me that didn't solved the issue. The axioms were chosen precisely to be able to derive all known maths to that time. Thus we still don't know exactly what is a contradiction in maths or why they appear.

We only can do maths that derives from the axioms, so contradictions are impossible by definition at the axioms range. And that is, in my opinion, awful, cause we have killed Mathatics.

We cannot go beyond axioms, and those are not perfect there are known writings that point out that we lack axioms to prove Goldbach Conjecture, to find Prime List general term, o get the Well Ordered formula for Real Numbers. I think we are in a real Maths crisis we cannot face because we aren't aware of it, or don't want to be aware of.

I haven't found writings on this subject and would love someone could give some clue were to find or at least look for it. Mathematics is obviously contradictory. Predicativist systems disagree with classical systems on the allowable scope of quantification in definitions.

Constructivists disagree on the use of excluded middle. Ultrafinitists disagree on induction principles. Not only that, there are mathematical systems that are intentionally contradictory. Paraconsistent logics show that not all contradictions in all systems are reductionary and trivial.

Complex and useful logical theories may be built using contradictory premises. Dialetheists will even argue that this is the natural state of our actual reasoning, and accepting this psychologically can be the first step to avoiding troubling dilemmas.

Mathematics is where contradiction is formalised. You can't be explicit about a contradiction without using the abstraction and logical apparatus of mathematics.

Otherwise it is just "that doesn't seem right". Mathematics deals with contradictions all the time. In model theory, you will often see proofs that adding axioms X and Y to system Z is contradictory and has no models.

The 20th century, with the rise of model theory and the formalization of metamathematics, has revolutionised the study of contradiction in math. And major advances in rewriting logics and dynamic epistemic logics all formalise the response to contradictions in their calculi. It is troubling to me to see such highly voted answers claiming that mathematics has no contradictions.

That is completely false, though. Mathematics has many foundational theories. It is the process of formalization in general. Contradictions breed healthy foundational debate, as they have from very early on.

Many logical schools have found contradiction to offer important foundational guidance, and many revered it. Nagarjuna and the Buddhist logicians extended Catuskoti to Rg Veda ontological interpretation. Stoic logicians wrote greatly on contradiction and accepted that some contradictions could not be rejected on logical grounds, which informed their entire approach to empiricism.

There is no one mathematics. There are many schools from around the world and across time that contribute to an ever growing collective investigation of formalism and it's consequences. Mathematics does not need fundamentalists. It needs people who can honestly recognise the scope and beauty of formalism as a device to reason about the world, it's limits, it's contradictions, and every wonderful thing it does for us.

Many objects in mathematics even have different syntactic definitions! Looking for a rigorous understanding of the process of meaning has been a goal of philosophy for millennia, but only really in the past century was a useful formalisation of this finally made in the works of Tarski and others, following the strongly phenomenology based logical research of the Lvov-Warsaw school.

What came out of this work was the distinction between syntax with definitions and semantics with meaning. What does this have to do with anything? The OP asks if things in mathematics are "well-defined".

This is a tricky term. In common usage, it means that the object is "unambiguous" and "unique", that taking the definition gives a very clear and specific object that we all can agree on at least those who understand the definition! But there are a few problems here:.

Why does any of this matter? Others say it is "loose" talking. Remember this: "false" is a semantic assertion. And in this case, such an assertion can be incorrect. The symbols may have a variety of interpretations, and in some of those, that statement is entirely correct. I am not trying to say that mathematics is a mess or makes mistakes or anything negative with any of this.

Actually, quite the opposite seems to be the case: mathematics is where messes and mistakes can be looked at formally and this process seems to be a great source for human advancement. I am simply trying to be precise about what we actually see in mathematics. As the other answers clearly show, even mathematicians can harbor fuzzy understandings of the extent of contradiction in mathematics, and clearly it may be argued that even "Mathematics" itself is not fully "well-defined".

Actually in the first development of set theory a very intereseting contradiction was introduced. It is the so called Russel Paradox. I say it is interesting because is one of the very few cases where our intuition on what should be considered a founding truth might fail.

Also it is quite interesting the fact that such a paradox is a mathematical translation of the famous Liar paradox which was already taken into account by greek philosophers. The contradiction was discovered by Russel, in , while he was reading a manuscript by Frege about the formalization of Cantor set theory. In mathematics contradiction is a useful tool to prove some theorems. If you want prove some statement, sometimes its easier to disprove that the statement is not true.

So you assume its not true and derive a contradiction, which proves that the statement is in fact true. Apart from that, mathematics is build on a few fundamental concepts that don't need a formal proof called Axioms.