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The Real Analysis Minute! The rationals are gappy.
Root 2 is missing from the rationals. Course site — read more
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The Real Analysis Minute! Rational numbers form a gappy ordered field.
Here we look at the rationals and the idea of a field. Videos organized into lessons: https://bit.ly/raminnotion — read more
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The Real Analysis Minute! Some exercises on ring theory.
A few exercises on rings before moving on to other topics. Videos organized into lessons here: https://www.notion.so/axiomtutor/The-Real-Analysis-Minute-2154b66370fd806a84a0d4dc893f6575?source=copy_link — read more
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The Real Analysis Minute! The integers mod n form a ring.
A quick explanation of the integers mod n, and a challenge to prove that they form a ring. Videos organized into lessons here: https://www.notion.so/axiomtutor/The-Real-Analysis-Minute-2154b66370fd806a84a0d4dc893f6575?source=copy_link — read more
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The Real Analysis Minute! The integers form a ring.
Since integer multiplication is an associative operation with identity, it distributes over addition, and addition is a commutative group, then the integers form a ring. Videos organized into lessons: https://www.notion.so/axiomtutor/The-Real-Analysis-Minute-2154b66370fd806a84a0d4dc893f6575?source=copy_link — read more
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The Real Analysis Minute! A concrete example of an abstract group.
The Klein four-group, in a Cayley table. Videos organized into lessons here: https://www.notion.so/axiomtutor/The-Real-Analysis-Minute-2154b66370fd806a84a0d4dc893f6575?source=copy_link — read more
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The Real Analysis Minute! Commutative groups.
Definition of commutative groups. Then an exercise about the exponent laws, and then an exercise showing that a group is commutative. Web page organizing videos into lessons: https://www.notion.so/axiomtutor/The-Real-Analysis-Minute-2154b66370fd806a84a0d4dc893f6575?source=copy_link — read more
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The group exponent laws
Definition of group exponents and two basic laws familiar from algebra. Videos organized into lessons: https://www.notion.so/axiomtutor/The-Real-Analysis-Minute-2154b66370fd806a84a0d4dc893f6575?source=copy_link — read more
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<1min proof, group inverses are unique
The Real Analysis Minute!, each lesson is at most a minute. Today we prove that group inverses are unique. Link to the web page:https://www.notion.so/axiomtutor/The-Real-Analysis-Minute-2154b66370fd806a84a0d4dc893f6575?source=copy_link — read more
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The (mathematical) Josephus problem:
Finding insights into the Josephus problem by employing base-2 expansions. This leads to a case-study in using the repertoire method, for proving a hypothesized solution of a recurrence relation. — read more
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Axiom Courses 
