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Previous videos!
What are numbers?    • What IS a Number? As Explained by a M...  
How to count:    • How to Count  

As previously explored, we can define the natural numbers in terms of sets. We can even use them to count. But how are operations like addition and multiplication defined? Along with a crash-course in proof by induction, we'll define these operations in this video and prove that they possess all the lovely properties with which we are familiar.

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00:00 - Intro
04:02 - Axiom Recap
05:28 - First Attempt at Addition
09:05 - Addition Definition
15:30 - Inductive Principle
21:45 - Associativity of Addition
25:25 - Commutativity of Addition
30:30 - Multiplication Definition
37:50 - Distributivity
43:09 - Subtraction?
44:25 - Closing Remarks

*It is sufficient to define addition without the n+1:=S(n) part. However, I made the pedagogical choice to include this -- it's less efficient but more intuitive for the uninitiated, in my opinion.

**The base cases of our inductions should deal with the n=0 case, as then the n=1 case is true by virtue of "true for k ⇒ true for (k+1)" where k=0. However, the n=0 case is always completely trivial so I decided to make my base cases at n=1 so that viewers less experienced with proof by induction get to see more "work" being done and get a feel for how to complete proofs by appealing to axioms. Let's just go ahead and assume we've proved the n=0 case separately in all proofs!

**I misspeak here: We are proving that if a given number k commutes with *a then the next number commutes with *a*. Apologies!

****Psst, it's me, Yellow T-Shirt Alex, put the solution in the website slug. You'll also need those numbers I told you to keep safe!

All music by Danijel Zambo.