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?”

The Realm of Integers (Types of Numbers #3)

To kick off the tower of numbers, so to speak, the last article in this series discussed the “basic” numbers – mainly focusing on the positive whole numbers but also including zero. Now, we haven’t quite explored every aspect of the number zero yet, and we ran into problems with subtraction within the so-called “naturalContinue reading “The Realm of Integers (Types of Numbers #3)”

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)”

Why Ask Questions About Numbers? (Types of Numbers #1)

This is a first post in a series in which I’d like to discuss various different types of numbers that mathematicians study. This may sound strange – and that is fair. After all, aren’t all numbers just… numbers? Why would some numbers be so different from other numbers? What sense does that make? I’m honestlyContinue reading “Why Ask Questions About Numbers? (Types of Numbers #1)”

The Importance of Derivatives and What Comes Next (Explaining Calculus #13)

This post doesn’t exactly introduce anything new about calculus. But, we must remember, it is important to reflect whenever we learn. Even in mathematics. In this post, I plan on achieving two main goals. One, we will reflect on what we’ve done so far by introducing the new ideas of limits and derivatives to ourContinue reading “The Importance of Derivatives and What Comes Next (Explaining Calculus #13)”

Critical Thinking Toolkit: The Correlation-Causation Fallacy

Here, I’d like to discuss two interconnected tendencies we human beings have. We like looking for patterns, and we like explaining things. These are both incredibly important features of the way we think as humans. But they are not identical. It is easy to get them confused, and we often do get them confused. ThisContinue reading “Critical Thinking Toolkit: The Correlation-Causation Fallacy”

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”