Breaking Math Podcast

How many piano tuners are there in New York City? How much cheese is there in Delaware? And how can you find out? All of this and more on this problem-episode of Breaking Math.This episode distributed under a Creative Commons Attribution-ShareAlike-Noncommercial 4.0 International License. For more information, visit creativecommons.orgFeaturing theme song and outro by Elliot Smith of Albuquerque.[Featuring: Sofía Baca, Meryl Flaherty]

i^2 = j^2 = k^2 = ijk = -1. This deceptively simple formula, discovered by Irish mathematician William Rowan Hamilton in 1843, led to a revolution in the way 19th century mathematicians and scientists thought about vectors and rotation. This formula, which extends the complex numbers, allows us to talk about certain three-dimensional problems with more ease. So what are quaternions? Where are they still used? And what is inscribed on Broom Bridge? All of this and more on this episode of Breaking Math.This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.The theme for this episode was written by Elliot Smith.[Featuring: Sofía Baca, Meryl Flaherty]

Mathematics is full of all sorts of objects that can be difficult to comprehend. For example, if we take a slip of paper and glue it to itself, we can get a ring. If we turn it a half turn before gluing it to itself, we get what's called a Möbius strip, which has only one side twice the length of the paper. If we glue the edges of the Möbius strip to each other, and make a tube, you'll run into trouble in three dimensions, because the object that this would make is called a Klein flask, and can only exist in four dimensions. So what is a fiber? What can fiber bundles teach us about higher dimensional objects?All of this, and more, on this episode of Breaking Math.[Featuring: Sofía Baca, Meryl Flaherty]

In introductory geometry classes, many of the objects dealt with can be considered 'elementary' in nature; things like tetrahedrons, spheres, cylinders, planes, triangles, lines, and other such concepts are common in these classes. However, we often have the need to describe more complex objects. These objects can often be quite organic, or even abstract in shape, and include things like spirals, flowery shapes, and other curved surfaces. These are often described better by differential geometry as opposed to the more elementary classical geometry. One helpful metric in describing these objects is how they are curved around a certain point. So how is curvature defined mathematically? What is the difference between negative and positive curvature? And what can Gauss' Theorema Egregium teach us about eating pizza?This episode distributed under a Creative Commons Attribution ShareAlike 4.0 International License. For more information, go to creativecommons.orgVisit our sponsor today at Brilliant.org/BreakingMath

Join Sofía and Gabriel as they talk about Morikawa's recently solved problem, first proposed in 1821 and not solved until last year!Also, if you haven't yet, check out our sponsor The Great Courses at thegreatcoursesplus.com/breakingmath for a free month! Learn basically anything there.The paper featured in this episode can be found at https://arxiv.org/abs/2008.00922This episode is distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. For more information, visit CreativeCommons.org![Featuring: Sofía Baca, Gabriel Hesch]

Join Sofía and Gabriel as they discuss an old but great proof of the irrationality of the square root of two.[Featuring: Sofía Baca, Gabriel Hesch]Patreon-Become a monthly supporter at patreon.com/breakingmathMerchandiseAd contained music track "Buffering" from Quiet Music for Tiny Robots.Distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. For more information, visit creativecommons.org.

If you are there, and I am here, we can measure the distance between us. If we are standing in a room, we can calculate the area of where we're standing; and, if we want, the volume. These are all examples of measures; which, essentially, tell us how much 'stuff' we have. So what is a measure? How are distance, area, and volume related? And how big is the Sierpinski triangle? All of this and more on this episode of Breaking Math.Ways to support the show:Patreon-Become a monthly supporter at patreon.com/breakingmathThe theme for this episode was written by Elliot Smith.Episode used in the ad was Buffering by Quiet Music for Tiny Robots.[Featuring: Sofía Baca; Meryl Flaherty]

Sofía and Gabriel discuss the question of "how many angles are there in a circle", and visit theorems from Euclid, as well as differential calculus.This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.Ways to support the show:Patreon-Become a monthly supporter at patreon.com/breakingmathThe theme for this episode was written by Elliot Smith.Music in the ad was Tiny Robot Armies by Quiet Music for Tiny Robots.[Featuring: Sofía Baca, Gabriel Hesch]

Look at all you phonies out there.You poseurs.All of you sheep. Counting 'til infinity. Counting sheep.*pff*What if I told you there were more there? Like, ... more than you can count?But what would a sheeple like you know about more than infinity that you can count?heh. *pff*So, like, what does it mean to count til infinity? What does it mean to count more? And, like, where do dimensions fall in all of this?Ways to support the show:Patreon-Become a monthly supporter at patreon.com/breakingmath(Correction: at 12:00, the paradox is actually due to Galileo Galilei)Distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. For more information, visit CreativeCommons.orgMusic used in the The Great Courses ad was Portal by Evan Shaeffer[Featuring: Sofía Baca, Gabriel Hesch]

As a child, did you ever have a conversation that went as follows:"When I grow up, I want to have a million cats""Well I'm gonna have a billion billion cats""Oh yeah? I'm gonna have infinity cats""Then I'm gonna have infinity plus one cats""That's nothing. I'm gonna have infinity infinity cats""I'm gonna have infinity infinity infinity infinity *gasp* infinity so many infinities that there are infinity infinities plus one cats"What if I told you that you were dabbling in the transfinite ordinal numbers? So what are ordinal numbers? What does "transfinite" mean? And what does it mean to have a number one larger than another infinite number?[Featuring: Sofía Baca; Diane Baca]Ways to support the show:PatreonBecome a monthly supporter at patreon.com/breakingmathThis episode is released under a Creative Commons attribution sharealike 4.0 international license. For more information, go to CreativeCommoms.orgThis episode features the song "Buffering" by "Quiet Music for Tiny Robots"

There are a lot of things in the universe, but no matter how you break them down, you will still have far fewer particles than even some of the smaller of what we're calling the 'very large numbers'. Many people have a fascination with these numbers, and perhaps it is because their sheer scale can boggle the mind. So what numbers can be called 'large'? When are they useful? And what is the Ackermann function? All of this and more on this episode of Breaking Math[Featuring: Sofía Baca; Diane Baca]Ways to support the show:PatreonBecome a monthly supporter at patreon.com/breakingmathMerchandisePurchase a Math Poster on Tensor Calculus at our facebook store at facebook.com/breakingmathpodcast

Neuroscience is a topic that, in many ways, is in its infancy. The tools that are being used in this field are constantly being honed and reevaluated as our understanding of the brain and mind increase. And it's no surprise: the brain is responsible for the way we interact with the world, and the idea that ideas hone one another is not new to anyone who possesses a mind. But how can the tools that we use to study the brain and the mind be linked? How do the mind and the brain encode one another? And what does Bayes have to do with this? All of this and more on this episode of Breaking Math.[Featuring: Sofía Baca, Gabriel Hesch; Peter Zeidman]PatreonBecome a monthly supporter at patreon.com/breakingmathThis episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.

Spheres and circles are simple objects. They are objects that are uniformly curved throughout in some way or another. They can also be defined as objects which have a boundary that is uniformly distant from some point, using some definition of distance. Circles and spheres were integral to the study of mathematics at least from the days of Euclid, being the objects generated by tracing the ends of idealized compasses. However, these objects have many wonderful and often surprising mathematical properties. To this point, a circle's circumference divided by its diameter is the mathematical constant pi, which has been a topic of fascination for mathematicians for as long as circles have been considered.[Featuring Sofía Baca; Meryl Flaherty]Patreon Become a monthly supporter at patreon.com/breakingmath

Join Sofia and Gabriel on this problem episode where we explore "base 3-to-2" — a base system we explored on the last podcast — and how it relates to "base 3/2" from last episode.[Featuring: Sofía Baca; Gabriel Hesch]

A numerical base is a system of representing numbers using a sequence of symbols. However, like any mathematical concept, it can be extended and re-imagined in many different forms. A term used occasionally in mathematics is the term 'exotic', which just means 'different than usual in an odd or quirky way'. In this episode we are covering exotic bases. We will start with something very familiar (viz., decimal points) as a continuation of our previous episode, and then progress to the more odd, such as non-integer and complex bases. So how can the base systems we covered last time be extended to represent fractional numbers? How can fractional numbers be used as a base for integers? And what is pi plus e times i in base i + 1?This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.[Featuring: Sofía Baca; Merryl Flaherty]Ways to support the show:Patreon Become a monthly supporter at patreon.com/breakingmath

Numbering was originally done with tally marks: the number of tally marks indicated the number of items being counted, and they were grouped together by fives. A little later, people wrote numbers down by chunking the number in a similar way into larger numbers: there were symbols for ten, ten times that, and so forth, for example, in ancient Egypt; and we are all familiar with the Is, Vs, Xs, Ls, Cs, and Ds, at least, of Roman numerals. However, over time, several peoples, including the Inuit, Indians, Sumerians, and Mayans, had figured out how to chunk numbers indefinitely, and make numbers to count seemingly uncountable quantities using the mind, and write them down in a few easily mastered motions. These are known as place-value systems, and the study of bases has its root in them: talking about bases helps us talk about what is happening when we use these magical symbols.

Machines have been used to simplify labor since time immemorial, and simplify thought in the last few hundred years. We are at a point now where we have the electronic computer to aid us in our endeavor, which allows us to build hypothetical thinking machines by simply writing their blueprints — namely, the code that represents their function — in a general way that can be easily reproduced by others. This has given rise to an astonishing array of techniques used to process data, and in recent years, much focus has been given to methods that are used to answer questions where the question or answer is not always black and white. So what is machine learning? What problems can it be used to solve? And what strategies are used in developing novel approaches to machine learning problems? This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org. For more Breaking Math info, visit BreakingMathPodcast.app [Featuring: Sofía Baca, Gabriel Hesch] References: https://spec

Machines, during the lifetime of anyone who is listening to this, have advanced and revolutionized the way that we live our lives. Many listening to this, for example, have lived through the rise of smart phones, 3d printing, massive advancements in lithium ion batteries, the Internet, robotics, and some have even lived through the introduction of cable TV, color television, and computers as an appliance. All advances in machinery, however, since the beginning of time have one thing in common: they make what we want to do easier. One of the great tragedies of being imperfect entities, however, is that we make mistakes. Sometimes those mistakes can lead to war, famine, blood feuds, miscalculation, the punishment of the innocent, and other terrible things. It has, thus, been the goal of many, for a very long time, to come up with a system for not making these mistakes in the first place: a thinking machine, which would help eliminate bias in situations. Such a fantastic machine is looking like it's becoming clo

Join Gabriel and Sofía as they delve into some introductory calculus concepts.[Featuring: Sofía Baca, Gabriel Hesch]Ways to support the show:Patreon Become a monthly supporter at patreon.com/breakingmath

Time is something that everyone has an idea of, but is hard to describe. Roughly, the arrow of time is the same as the arrow of causality. However, what happens when that is not the case? It is so often the case in our experience that this possibility brings not only scientific and mathematic, but ontological difficulties. So what is retrocausality? What are closed timelike curves? And how does this all relate to entanglement?This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.[Featuring: Sofía Baca, Gabriel Hesch]