## Why Prove a Theorem Twice?

I have written before on the problem and solution of the problem of “Pythagorean Triples”. The problem, based on the Pythagorean theorem for right triangles, asks for all possible solutions to this equation which have all of whole numbers. Not just any right triangle works – for instance if then , which is definitely notContinue reading “Why Prove a Theorem Twice?”

## Finding the Fibonacci Numbers: The Formula

This is the third post in a series about an exact formula for the Fibonacci numbers, , which are defined by the initial values and the recurrence relation . We have made a lot of progress towards our goal. We discovered a connection between , the golden ratio , and the Fibonacci sequence by findingContinue reading “Finding the Fibonacci Numbers: The Formula”

## Proof by Infinite Descent

We have previously discussed proof by contradiction . 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”

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”

## The Foundation of Logic

There are a lot of different ways to approach understanding what mathematics truly is in modern times. And these different approaches are fundamentally different. Does one come from the angle of discussing the beauty of patterns? Or perhaps of the power of the human mind to move from specific examples to general conclusions (similar toContinue reading “The Foundation of Logic”