If we gave you a bunch of positive whole numbers and asked what you get when you add them all up, you'd have a pretty easy time of it... you are The Math Club, after all. But what if we gave you an infinite series of positive whole numbers and asked you what you get when you add them all up? In this episode, Pete and Noah talk about how mathematicians go about assigning a value in this seemingly impossible situation. Join us and learn about divergent series, and be ready to be surprised by some of the results.

• The Numberphile video

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Questions Answered in the Episode: • What is the difference between a convergent series and a divergent series? • How can the sum of all positive integers be negative one-twelfth?

If you ever took a geometry class, you probably had to do some mathematical proofs. From the ancient Greeks to modern day mathematicians, proofs have played a vital role in advancing our understanding of the world around us. In this episode, Pete and Noah talk about their own experiences with proofs and look at several different kinds of proof along the way. Hop aboard the math train, and join us in an exploration of the fascinating world of mathematical proofs.

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Questions Answered in the Episode: • What are the different types of mathematical proofs? • Why are geometric proofs often taught in a two column format?

This week Pete and Noah team up with Shane and Mealey from the Unprofessional Development podcast, a show that features guests from the world of education sharing their stories about teaching. This joint episode finds the four hosts discussing teaching, math, teaching math, and spicy foods.

• Unprofessional Development Podcast • Follow @unprocast on Twitter

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When you think about parts of a whole, you probably picture fractions that consist of a numerator and a denominator. But that's not how all cultures throughout history worked with fractional parts. In this episode, Pete and Noah visit ancient Egypt and discuss the completely different way their mathematicians showed parts of a whole. Plus, there's a puzzle about camels!

• Rhind Mathematical Papyrus

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Questions Answered in the Episode: • What is an Egyptian fraction and how were they written in ancient Egypt? • How did the ancient Egyptians use unit fractions to solve practical problems?

In September of 1990, Parade Magazine columnist Marilyn vos Savant answered a reader's question about a mathematical brain teaser, and received quite the backlash when most of the math community thought her solution was incorrect. Just about three years later, two new friends (and future Math Club hosts) bonded over math for the first time, discussing and debating this exact same puzzle. In this episode, Pete and Noah weave the two stories together into an interesting discussion about conditional probability, counterintuitive results, and being gracious when confronted with, what you believe is, someone else's error.

• Try our Scratch simulation
• Try our faster Scratch simulation
• Monty Hall interview about the problem• Leave us a voice message • Email us • Follow us on Twitter

Questions Answered in the Episode: • What is the Monty Hall problem? • Why is it better to switch doors in the Monty Hall Problem? • How does conditional probability explain the Monty Hall problem?

This episode is a special math adventure for younger listeners. Pete and Noah are called into school to help with all of the mathy details around a class pizza party, and quickly discover that math is all around us all of the time. Chock full of interesting characters and math challenges for listeners to solve, this story is sure to delight listeners from third grade on up.

Interactive website that goes with the story: https://bit.ly/mathclubjunior

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In this episode, Noah comes to Pete for some help with a really big paint job. After Pete explains just how big (and impossible) the job really is, they have a conversation about surfaces of revolution, and the seeming paradox of Gabriel's Horn.

• Hyundai's 35 meter vuvuzela
• Leave us a voice message with your favorite math joke

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Questions Answered in the Episode: • What is Gabriel's Horn? • How can a shape have infinite surface area but finite volume? • What is a surface of revolution?

Season's Greetings, Math Club! With the holidays on the horizon, a famous song leads Pete and Noah to talk about strategies for adding up long sequences of numbers. Come join the fun, and learn how a young Carl Gauss may (or may not) have done it. And if you're up for a challenge, we even "sum it up" with a special holiday puzzle for you to try.

• Brian Hayes's article in American Scientist

• Submit your puzzle solution here - Google Form

• Submit your puzzle solution here - Twitter

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Questions Answered in the Episode: • What was the method Carl Friedrich Gauss used to quickly sum the numbers one through 100? • What is the general formula used to find the sum of an arithmetic series for the first n integers?

In our last episode, Pete explained to Noah that imaginary numbers aren't so imaginary after all. This time around, he shows Noah how all the other numbers kinda are. So, if you thought there was something less than real about the square root of negative one, wait 'til you hear about the square root of positive two! Buckle up, Math Club, it's time to let our imagination run wild!!

3Blue1Brown's video on visualizing quaternions

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Questions Answered in the Episode: • What is the mathematical definition of an irrational number? • How do quaternions expand upon the complex number system?

Imaginary numbers. We've all heard about them, but… what are they? Why are they imaginary? And what can we do with them? In this episode, Pete and Noah take on these questions and explore a bit of the history and practical applications of these unusual and complex numbers. • Leave us a voice message • Email us • Follow us on Twitter

Questions Answered in the Episode: • Who coined the term imaginary numbers and why? • What is the geometric interpretation of a complex number?