How to Add

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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.

[AnotherRoof]

If common questions arise related to this video, I'll respond here!

[_abdul]

The only video with a ton of "Ads" that I Adequately Admire.

[thalt]

That intro bit with all the words starting with "ad" was godlike writing

[kristianmarinov]

The inductive, deductive and abductive reasoning differences would make a GREAT video!

[radqnico]

"Be rational about it, keep things real and don't make things too complex"
It didn't go though my brain like nothing 😂

[kenet]

I love how sequential these are. At first we learned numbers, then we learned counting, and then we learn arithmetic.

Can't wait to do insanely difficult integrals on empty sets!

[isaackromer]

Your videos help me fall asleep, thanks



No wait that came out wrong

[simonthelen]

This channel might actually become one of my favorite YouTube channels. I'm really curious which way we go next. There are so many possibilities.

[a_commenter]

Before watching this series, I had known that numbers could be represented as sets, but I'd never really understood how. I'm super excited to find out how things like negatives and fractions are encoded!

[RickyRatte]

While probably making it harder for new viewers, I like how the videos don't stand on their own, but slowly evolve into a mathematics cinematic universe

[stephengray]

I'm loving this series. It's way better taught than any of my university maths lectures were. I would like to suggest that you put this series in a playlist, though.

[eonstar]

Never would have thought I'd be watching math proofs in my free time but here I am

[Hi_Brien]

I always wanted a youtube series that hit our entire wall of mathematics. I realize that graph theory is more and more fundamental by the day.

[rufa_avis]

Can't wait to see the video. Really love you way of presenting information. This channel, though small, is already a part of mathematical YouTube for me, alongside Numberphile and Mathologer.

[vladmunteanu]

Really impressive how few cuts there are in your videos even though they are pretty long, makes it very nice to watch

[klemo]

as a kid i always thought of how could civilization advance from nothing to computers, and how could computers advance from 0 and 1 to all of the complicated computation
My answer to that question was just like this videos, you start defining little "bricks" of proved truth and use it to define a new "brick"; you could spend so much time proving a brick, but once you prove it, you can now freely use it to build very complicated stuff
in these videos you illustrate very well this concept and i just wanted to say thank you for making this so well and rigorous

[susmitislam]

A fresh, fun style of presentation with amazing clarity and detail. Perfect *addition* to the maths YouTube community!

[wiadroman]

I love the purity of the recursive definition of addition. From now on, in my code I will be implementing addition of two numbers as a loop finding the successors of successors.

[TechyBen]

Me: "Explain like I'm 5..."
Another Roof: "About that..."

[AdmiralJota]

When you were showing your thought process for induction, you wrote down the goal you were aiming for (at 29:09). I think it might be really useful if you did that every time you used induction, since knowing what you're aiming for makes it a lot easier to follow how the last step actually proves the thing you're trying to prove.