# Proofs

## Euler’s Formula: Complex Numbers as Exponents

Last week we explored how the polar form of complex numbers gives multiplication a simple geometric meaning. Here we’ll go one more step, and express polar form exponentially, which makes DeMoivre’s theorem trivial, and gives us a simple notation to replace “cis”.

## Complex Numbers: Multiplication and Rotation

Having looked at the idea of complex numbers and how to perform basic operations on them, we are ready for one of the most important features for applications: their relationship to rotation. We’ll see this first in describing complex numbers by a length and an angle (polar form), then by discovering the meaning of multiplication …

## Negative x Negative = Positive? Abstract Proofs

Last time we looked at explanations for the product of negative numbers in terms of various concrete models or examples. But it really requires a mathematical proof, as we’ll explain and demonstrate here, first with a couple different proofs, then with the bigger picture, giving the context of such 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 …