Fundamental Theorem of Calculus, Part 1 (Explaining Calculus #16)

We have recently talked about two concepts that appear rather disparate. We have discussed antiderivatives – which is the idea of reversing all the rules for taking derivatives, and Riemann sums, which are used to calculate the areas underneath complicated shapes. What do these have to do with each other? It does not look likeContinue reading “Fundamental Theorem of Calculus, Part 1 (Explaining Calculus #16)”

Proof by Infinite Descent

We have previously discussed proof by contradiction [1]. Here, we will be describing what can be viewed as a specialized version of this method. This method, however, is sufficiently specialized that is it worth discussing separately. As a mathematician, I find this idea absolutely brilliant. Even though it isn’t terribly difficult to explain, only anContinue reading “Proof by Infinite Descent”

Proof by Contrapositive

Sometimes when you are trying to solve a problem, you realize you really don’t have a lot of information to start with. One piece of good advice in problem-solving is to try to work backwards. That is, sometimes if you know what you want your solution to look like, you can backtrack to learn somethingContinue reading “Proof by Contrapositive”

Proof by Contradiction

The proof method that we will talk about here is quite different than many others. In his famous book A Mathematician’s Apology, the great mathematician G.H. Hardy made an analogy between this proof style, which we call a proof by contradiction, to a gambit in chess. So before I try to analyze what this proofContinue reading “Proof by Contradiction”

Proof by Induction

The word induction refers to any thought pattern which moves from specific examples to more general principles. The best known example of induction is probably the scientific method – we collect data by repeating the same specific experiments multiple times, find regularities and make guesses about a potential overarching framework into which the data mightContinue reading “Proof by Induction”

Types of Proofs in Math

Previously, I have talked about logic and some of the most important rules of logic. These are quite important and useful in doing mathematics. However, it is necessary to go further, because logic is not specific enough. Mathematics analyzes patterns that involve concepts like shape, number, repetition, and symmetry. Pure logic does not adequately handleContinue reading “Types of Proofs in Math”

Pythagorean Triples? (Solution #1, Part 2)

In Part 1, we have begun discussing primitive Pythagorean triples, and thought a little bit about them. Now, we want to try to characterize all primitive triples. That is now our goal. Limiting The Possibilities Suppose we are given a primitive triple (a,b,c). Recall that this means that the three positive whole numbers a, b,Continue reading “Pythagorean Triples? (Solution #1, Part 2)”

Pythagorean Triples? (Solution #1, Part 1)

(If you haven’t read the “Problem” post with the same title, go there first. This will make more sense if you do.) We want to find all the Pythagorean triples (a,b,c). The first thing a mathematician would probably do is to try some small examples, gather some information, and then look for patterns within thatContinue reading “Pythagorean Triples? (Solution #1, Part 1)”

What Is a Proof?

I’ve talked a fair amount in some of the earlier posts about the idea of a proof. Now that we’ve developed a conceptual underpinning of what that means, we can see one in action. I hope my readers enjoy this as much as I do, as what we will discuss here is among my favorites.Continue reading “What Is a Proof?”