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samspotting

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In summary, the conversation discussed presenting a proof of dedekind cuts for an undergrad talk and how to make it more interesting. The suggestion was to provide motivation by explaining why decimals are not sufficient and then starting from the rationals. It was also mentioned that dedekind cuts allow for the manipulation of irrational numbers, such as pi + pi, without a defined algorithm. This is where they become useful.

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samspotting

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Tac-Tics

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samspotting said:

You might want to give motivation for it.

I would start with explaining why decimals don't cut it (non-unique representation). Then, start from the rationals. Remind your audience that given ANY two rationals, they know an algorithm to add, subtract, and multiply them. But they have no such algorithm for irrational numbers. Even something simple as pi + pi... how can you show that this is equal to 2*pi? "Obviously", it's a rule from algebra. But how do you know if this "rule" holds without knowing precisely what a real number IS?

Answer: dedekind cuts let you do just this.

Dedekind cuts are a mathematical concept created by German mathematician Richard Dedekind. They are a way to represent real numbers by dividing the number line into two sets, with one set containing all numbers less than the chosen number and the other set containing all numbers greater than or equal to the chosen number.

Dedekind cuts are used in mathematics as a way to define real numbers without relying on a specific numerical representation. They are particularly useful in the construction of the real numbers and in rigorous proofs involving real numbers.

Dedekind cuts have several important properties, including the existence of a least upper bound and greatest lower bound for each set, the completeness property (meaning every non-empty set of Dedekind cuts has a least upper bound), and the ability to perform arithmetic operations such as addition and multiplication.

Dedekind cuts are closely related to other mathematical concepts, such as rational numbers, irrational numbers, and the real numbers. They are also used in the construction of other mathematical structures, such as the complex numbers and the p-adic numbers.

Dedekind cuts have applications in various areas of mathematics, including analysis, number theory, and topology. They are also used in computer science and physics, particularly in the study of continuous and infinite structures.

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