Topics from “Pre-Calculus” (Explaining Calculus #1)

Before we can go into a discussion of calculus itself, it is important to set up some of the underlying concepts that calculus uses. You might view this as something analogous to learning the letters of the alphabet before you can start learning to read words and sentences. Without letters, you aren’t going to beContinue reading “Topics from “Pre-Calculus” (Explaining Calculus #1)”

Explaining Calculus (Intro)

In our day and age, this word strikes fear into many hearts. Much like ‘rocket science,’ it is used in our culture to represent anything that is extremely difficult or even something incomprehensible. I am talking about calculus. Those who have taken a course in this topic will have a good idea of what kindsContinue reading “Explaining Calculus (Intro)”

Finding Patterns in the Fibonacci Sequence

This is the final post (at least for now) in a series on the Fibonacci numbers. We’ve gone through a proof of how to find an exact formula for all Fibonacci numbers, and how to find exact formulas for sequences of numbers that have a similar definition to the Fibonacci numbers. But, the fact thatContinue reading “Finding Patterns in the Fibonacci Sequence”

Finding the Fibonacci Numbers: A Similar Formula

In this series of posts about the Fibonacci sequence , a very famous sequence of numbers within mathematics, we have just concluded showing how you can take the recursive formula (which uses previous values of to compute the next values) and turn that formula into an exact formula that can skip right over the previousContinue reading “Finding the Fibonacci Numbers: A Similar Formula”

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”

Finding the Fibonacci Numbers: Getting Our Bearings

This is the second in a series of posts discussing a quite elegant and interesting problem in the history of mathematics. We have previously defined the Fibonacci numbers using the starting point and defining for every larger value of . This set up is often called a recursive formula, since we use the same processContinue reading “Finding the Fibonacci Numbers: Getting Our Bearings”

Finding the Fibonacci Numbers: The Problem

The 1202 mathematics textbook Liber Abaci is arguably one of the most important contributions to the development of the scientific and cultural systems we have in the world today, especially for Western society. Written by Leonardo of Pisa, known colloquially by the name Fibonacci, this text introduced to the Western world important notations and mathematicalContinue reading “Finding the Fibonacci Numbers: The Problem”

The Quadratic Formula (Solution)

Having discussed the motivation of why something like a “quadratic formula” is a useful thing to discover and understand, I’d like to work through some of the ideas that might lead one to discover a quadratic formula. Reducing the Number of Unknowns Remember that the equation we care about is , with (so that thisContinue reading “The Quadratic Formula (Solution)”

The Quadratic Formula (Problem)

This problem is something that is pretty well known from our high school education – the quadratic formula. In a follow-up post, I will present the solution through a shortened version of the process by which a mathematician might have gone about finding a solution. Equations are among the most important objects in mathematics. InContinue reading “The Quadratic Formula (Problem)”

Critical Thinking Toolkit: Weighing Evidence with Bayes’ Theorem

When we are involved in important discussions, it is important to not take pivotal claims of others – or ourselves – merely at face value. Instead, we debate the various available ideas. That is, we bring forth evidence that we have thought about and that we believe supports our position, and we listen to theContinue reading “Critical Thinking Toolkit: Weighing Evidence with Bayes’ Theorem”