# Proofs

## Why Does a² + b² = c² in a Hyperbola?

(A new question of the week) In an ellipse, $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ with focal distance c, parameters a, b, and c all make natural sense, and it is easy enough to see why $$a^2 = b^2 + c^2$$. But in the hyperbola, $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$, the equivalent relationship, $$a^2 + b^2 = c^2$$, is not nearly as natural, nor …

## Average Distance Between Two Sets of Points

(A new question of the week) Here we have a different kind of question than usual: A conjecture about distances between points, with a request for confirmation. Normally we like to just give hints to help a student figure something out; this was a request for a theorem that ought to exist, and trying to …

## Equivalent Definitions of e

(A new question of the week) It is not unusual for mathematicians to define a concept in multiple ways, which can be proved to be equivalent. One definition may lead to a theorem, which another presentation uses as the definition, from which the original definition can be proved as a theorem. Here, in yet another …

## Proving Certain Polynomials Form a Group

Abstract algebra can be a huge leap for many students, who may know algebra well, but are not used to abstraction – generalizing the concept of numbers so we can invent new kinds of “numbers” and “operations” and comparing their properties. Here we will look at a question from a student beginning the study of …

## Summing Squares: Finding or Proving a Formula

Last week we looked at problems about counting the squares of all sizes in a checkerboard. Some solutions required finding the sum of consecutive squares, $$1^2+2^2+3^2+\dots+n^2$$, for which we used a formula whose derivation I deferred to this week. Here we’ll see a couple proofs that require knowing the formula ahead of time, and a …

## Distances on Earth 2: The Haversine Formula

Last week we started a series about finding distances on a sphere (which approximates the shape of the earth), using a straightforward formula from spherical geometry. But in practice, that formula turns out not to be ideal, so a different formula is used when accuracy in all circumstances matters. That is this week’s topic: first …

## Fibonacci, Pascal, and Induction

A couple weeks ago, while looking at word problems involving the Fibonacci sequence, we saw two answers to the same problem, one involving Fibonacci and the other using combinations that formed an interesting pattern in Pascal’s Triangle. I promised a proof of the relationship, and it’s time to do that. And while we’re there, since …

## Generalizing and Summing the Fibonacci Sequence

Continuing our look at the Fibonacci sequence, we’ll extend the idea to “generalized Fibonacci sequences” (with different starting numbers), and see that the ratio of consecutive terms is the same in general as in the usual special case. Then we’ll look at the sum of terms of both the special and general sequence, turning it …

## A Few Inductive Fibonacci Proofs

Having studied proof by induction and met the Fibonacci sequence, it’s time to do a few proofs of facts about the sequence. We’ll see three quite different kinds of facts, and five different proofs, most of them by induction. We’ll also see repeatedly that the statement of the problem may need correction or clarification, so …

## Graph Coloring: Working Through a Proof

(A new question of the week) The Math Doctors have different levels of knowledge in various fields; I myself tend to focus on topics through calculus, which I know best, and leave the higher-level questions to others who are more recently familiar with them. But sometimes, both here and in my tutoring at a community …