Quaternion

Mathematicians create entire worlds from a small set of rules. This can be seen with the advent of real numbers. The world around the real numbers is still growing even today. However, there are always roadblocks that stop mathematicians dead in their tracks. For example, in the real numbers, there is no solution to because […]

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Topological Twists: The Möbius Strip

The Möbius strip is a curious object.The strip was discovered simultaneously and independently by German mathematicians August Ferdinand Möbius and Johann Benedict Listing. Möbius took the concept farther than Listing by exploring the properties of this strip. Let us discuss some of these properties.     If you happen to have a ribbon or something […]

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Topological Data Analysis part 1

Data is a large part of our everyday lives. We personally take in all kinds of data everyday. Things such as weather, political polls, meal costs, and much much more make up most of our daily lives. Companies collect data as well. Medical, consumer, weather, and other kinds of data have been collected for years. […]

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The Shape of Data

Data is a large part of our everyday lives. We personally take in all kinds of data everyday. Things such as weather, political polls, meal costs, and much much more make up most of our daily lives. Companies collect data as well. Medical, consumer, weather, and other kinds of data have been collected for years. […]

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Axioms: Sets (ZF and ZFC)

In a previous post Axioms: Sets (Russell’s Paradox) we discussed axiomatic systems and we explored an example of why we might need one. Here we will view the axiomatic system that “fixed” naive set theory. This material will be a little heavy so in between each axiom We will see a little intuition to better help understand the […]

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Axioms: Sets (Russell’s Paradox)

Bertrand Russell (1872-1970) It isn’t too often that we think about or even explore the fundamental building blocks of mathematics. These building blocks are called axioms. Axioms are statements taken to be true, i.e. they cannot be proven. This causes mathematicians, whether they know it or not, to take a lot of things on faith. All of […]

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Banach-Tarski Paradox

It is a common occurrence in mathematics that when something does go wrong, it goes terribly wrong. This exact phenomenon occurs with the Banach-Tarski Paradox. Informally, it says that one can take a sphere (in 3 or more dimensional space)  can be split into finitely many pieces and, using only rigid motions, can be rearranged […]

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