How to Add
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- เผยแพร่เมื่อ 8 ก.ค. 2024
- Patreon: / anotherroof
Channel: / anotherroof
Website: anotherroof.top
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What are numbers? • What IS a Number? As E...
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.
Huge thank you to my Patrons. Without you, I wouldn't have the confidence to make TH-cam part of my career may not have continued making videos. If you'd like to support me and gain access to progress updates, bloopers, the Discord server where we can hang out, and have your name in the credits, please consider supporting me (link above)!
Join the subreddit! Thinking of doing a 10K subscriber Q&A video so head on over there and submit your questions, be they personal, mathematical, pedagogical, or whatever.
Reddit: / anotherroof
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. - บันเทิง
Thank you for watching! I recently hit 10K subscribers and planning a Q&A video. Head over to the Another Roof subreddit to ask your questions. If I get enough questions, I'll make the video -- should be a fun, less scripted one. www.reddit.com/r/anotherroof/comments/wj8hhn/10k_subscriber_qa/
If common questions arise related to this video, I'll respond here!
btw any reason you start induction at 1 instead of 0,
After all we built the natural numbers starting from 0 and so starting induction at 1 would mean that we would have to prove all those properties separately for 0. Also starting induction at 0 makes the computations less messy
Also I think you should have mentioned that one could define exponents in a very similar recursive way and prove their properties, again, by induction
I'm having trouble understanding how once could construct commutativity from 0. Specifically I'm stuck at showing how the general from a+(k+1)=(k+1)+a without having to specifically prove the case for 1. Am I stupid and you have to prove that 0 and 1 commute with every number?
@@ratatouille5172 It seams to me, yes, we do need to prove a+1=1+a for all a but we can still start the induction on a from zero and ,yes, we will need to prove a+0=0+a separately
"Be rational about it, keep things real and don't make things too complex"
It didn't go though my brain like nothing 😂
The inductive, deductive and abductive reasoning differences would make a GREAT video!
I would really like that kristian
totally agreed
Yes
is there a reductive?
he made this video
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!
Eventually you "forget" the specific constructions of natural numbers, integers, rationals, reals, etc. And just work with their properties (like we do in school). So "underneath" the integral notation you could think of countless nested empty sets, but of course that would be super impractical.
For 17+8, I did 16+8+1
Doing any integral on an empty set is easy, because Integrals are measures and one of the axioms of a measure m is that m({})=0. Without that axiom, you'd get inconsistent measures, because the measure of a countable union of disjunct sets is the sum of their individual measures. So you could just note that for an arbitrary number n, m({})=n*m({}) and that is unique if and only if m({})=0.
I'm reminded of Roger Penrose's The Road to Reality, which over a very long and very in-depth process, builds modern physics from the absolute basics.
@@simongunkel7457 I don't think that is what OP was referring to.
The only video with a ton of "Ads" that I Adequately Admire.
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
Just wait for my video set in the mathematical cinematic universe on the dispute over the validity of Cantor's Diagonal Argument, or as I call it, the Infinity War.
@@AnotherRoof If you ever you do a game theory set please use chess examples with video titles such as "The Two Towers", "The Return of The King" and I'm sure there's a maths problem called "The Fellowship of the Ring"
- A fellow maths teacher.
another cinematic universe
@@AnotherRoof If this isn't purely a joke, I'm quite interested; I've never heard of any dispute over the diagonal argument.
@@julianbushelli1331 It was a joke. However it is true that, at the time, Cantor's proofs and results were considered incorrect and laughable by many of his contemporaries. He became depressed as a result of the hostility and by the time he published his diagonal argument (which most mathematicians seemed to realise was an exceptional insight) his reputation was still shaky with people like Kronecker.
Giving a very brief overview here but it's worth reading about!
Never would have thought I'd be watching math proofs in my free time but here I am
That intro bit with all the words starting with "ad" was godlike writing
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.
Mate, thanks for reminding me -- I've been meaning to do this for days!
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!
nice pfp
Negatives are super easy with Church encodings. A negative is represented as a partial application of subtraction on a positive. So a negative integer is a function with one free variable. In this system not only is a number a function, but a negative can be both a number and a function that subtracts whatever you feed it.
There are a lot of parallels here with array indexing in computer science. An array of size 5 would be indexed using the integers 0 through 4. In a way, we're doing the same thing here, but we're referring to each array by its size and assuming a standard set contained within.
this channel is very *add* ictive
Oh dear. I love it!
Your videos help me fall asleep, thanks
No wait that came out wrong
Can't wait to see the video. Really love you way of presenting information. This channel, though small, is already a part of mathematical TH-cam for me, alongside Numberphile and Mathologer.
Don't forget 3Blue1Brown.
Also I agree!
yes, i agree. but i like to add that this channel certainly will not stay small for long.
@@cashkurtz5780 I dont think he needs to list every math youtuber
You should check out black pen red pen too
This channel might actually become one of my favorite TH-cam channels. I'm really curious which way we go next. There are so many possibilities.
I have so many ideas, I'm also curious which way I'll go next!
I have a hunch it might involve subtraction. Or maybe negative numbers.
@@AnotherRoof Build integers, rationals, etc, would be a nice path imo
Equivalence classes, surely.
Ordinal infinities maybe
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.
It is! Oh I loved graph theory when learning discrete math
@@MisterIncog it single handedly fixed my focus to discrete hehe
Really impressive how few cuts there are in your videos even though they are pretty long, makes it very nice to watch
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.
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.
A fresh, fun style of presentation with amazing clarity and detail. Perfect *addition* to the maths TH-cam community!
I genuinely love watching these videos. You caught my attention with "what is a number." I always liked math because I understood how to do what the teacher told me to do. But watching the most basic principles be broken down in ways I didn't know existed is honestly fascinating.
It would be nice if as you went on with this series, on the closing cards, we have a picture of all the bricks, ideally (if possible) laid out in such a way that one brick is positioned above other bricks if it is derived from those bricks (if it is a theorem), whereby all the axioms would be the literal foundation of the brick wall, that is, on the very bottom.
Kind of funny that I am suggesting a channel named Another Roof lay literal bricks to build a wall.
I just realized that there would probably be vastly more theorems than axioms, so I'm not sure how such a wall would hold physically, under the influence of a gravitational force. Answer: the mathematical force of logical proof.
Can always lay them on the ground flat so that it's not a problem while still showing that visual.
Can't wait when you talk about all the non-natural numbers! I'm really enjoying these videos which look at the root of numbers and their interactions.
44:14
"I hate to end on a NEGATIVE, but that will have to wait until next time. Don't give me that look. Be RATIONAL about it. I wanna keep things REAL in this video and not make it too COMPLEX."
Oh man, I love that sentence containing all the sets of numbers.
N, Z, Q, R, C
And the way he said it was very natural.
The quality of your videos is outstanding being better than most of the big channels
This is giving me a lot of Lambda Calculus vibes. It’s really cool to see how much it (especially recursion) and set theory are intertwined.
You're the solution for the problem, that we've never had time in school for the ground of mathematics. The "What are we even doing here". Love it. Thanks for not leaving out any detail. 💚💚
Me: "Explain like I'm 5..."
Another Roof: "About that..."
I love the wordplay jokes :)
not that I don't love everything else.. but the math is to be expected.. The wordplay is sweetener!
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
Whenever you make a video, I'm always excited for the next one. I was excited for this one, therefore I am excited for the induction video
Mate, these are incredible. You must be putting so much effort into creating this series that it boggles my mind.
I did physics in uni first (highly recommended if your favourite sensations in life involve headaches), but moving to IT after a long break. We didn't really learn why and how the numbers worked in the three semesters I did, we just worked with them. I love catching up on the things I missed to get back into thinking logically to start into a new attempt at a degree.
i have been looking for this kida content for years,thank you:
PLEAS keep going
This is actually a lesson in number theory, but more fun. My teacher made a really good work explaining this, but I like how you did it. Thanks for the video.
thank you so much for this series! it reminds me a lot of "Good Math" by Mark C Chu-Carroll, but the visuals make it easier to understand. This is a great series for adults wanting to understand fundamentals! There are very few channels where I don't mind watching 45min videos (15min is usually my max), but I find myself engrossed and looking forward to the time well spent!
Looking forward to this.
The presentation of counting in Godel Escher Bach was my first introduction into this really primitive type of math, but I'm liking your videos better.
we've spent so much time, over 2 hours (summing the time from the previous videos in this set) to learn how to add 1 + 1, im crying here, on to the next one
That sentence at the end... the way it was said was so natural
Yes, i would like to see the inductive reasoning video. But i'd watch anything from you lol
I find interesting that in terms of elements of a set the order doesn't matter and we need to use some other sets to encode a situation in which order matters, but in the case of addition and multiplication, we go on our way to prove we can change the order, instead of assuming it just can.
It does makes sense overall
What really brings it home is when you start trying to mess around with subtraction the same way: (5-3)-2 is different both from (3-5)-2 and from 5-(3-2).
Or there are systems where the equivalent of multiplication in that system doesn't commute - where the order matters.
@@rmsgrey well that wouldn't be an equivalent anymore, would it?
I think we already have this in the form of function composition.
f(x) = 3x+1
g(x) = 2x+2
f(g(x)) = f(2x+2) = 3(2x+2)+1 = 6x+6+1 = 6x+7
g(f(x)) = g(3x+1) = 2(3x+1)+2 = 6x+2+2 = 6x+4
6x+7 ≠ 6x+4
f(g(x)) ≠ g(f(x))
Yeah, function composition is an example of a thing that resembles multiplication, but doesn't commute. Other common examples include the vector cross product (which anti-commutes: ab=-ba and isn't even associative) and matrix multiplication.
I like that you actually construct math rather than introducing everything axiomatically. But let’s face it: the reals as the equivalence classes of rational Cauchy sequences or Dedekind cuts will be a mess a few episodes in the future, and proving the properties you need to do anything interesting will be … long
Challenge accepted.
@@AnotherRoof As someone who took real analysis and could never really wrap my head around either of these constructions, this is something I would love to see! The clarity of your explanations and the way you build up all definitions from intuition is just fantastic. No doubt you'll be able to make a great video on the topic.
Yeah, the jump from rationals to reals is a pretty big one - you're suddenly introducing uncountably many new numbers, most of which no-one will ever hear of...
And no-one ever bothers with the algebraics...
@@rmsgrey The problem is that when you are working with real numbers or the algebraic numbers, it becomes impractical and conceptually useless to work with explicit constructions and encodings. To properly give a formal introduction to the algebraic numbers, you need ring theory, and to properly give a formal introduction to the real numbers, you need lattice theory on top of ring theory. Set-theoretic constructions are not appropriate when dealing with these higher-level mathematical objects.
For instance, axiomatically, it is very easy to write a list of simple axioms that uniquely define what the real numbers are. Talking about the algebraic numbers is even easier: the field of algebraic numbers is the algebraic closure of the field of rational numbers. However, while it is very easy to understand the axioms, actually constructing these objects using nothing but sets is complicated, and to be honest, a waste of time. That is not to say that it cannot be done, but rather, that it should not be done.
These videos can branch into WAY more abstract areas of mathematics like Group Theory.
Group theory starts easy enough and only uses a small number of the blocks we have so far.
My PhD was in group theory so I'm all about it -- I want to wait until I've made a lot of videos before I venture into this topic because I want much more experience in order to ensure I do justice to the topic I love the most!
@@AnotherRoof yeah, i love Group Theory too! Keep going!!!
Watched until the end instead of going to sleep when I should have. Got several math puns, totally worth it.
Very instructive, even though my own date nights rarely progress to the multiplication stage.
I've always viewed adding as an extension of counting. if 8 and 17 represent groups of items, then whatever path you walk to count the total of both groups added together in terms of ordering, will always lead to the same result. Multiplication is a series of self-additions, so this video makes a ton of sense to me.
This leaves so many questions, like how do you define adding irrational numbers? What is their set definition, and how does their successor work? Really hyped for next video :D
spoiler alert: all the different types of numbers are actually different. The above-mentioned definitions of addition and multiplication only work for natural numbers. To get a new number system, you'll have to define the new number system (usually in terms of different number system), define new arithmetic, and prove all the properties all over again.
Actually defining what "real numbers" are is a very controversial topic. They require defining results of infinite chains of operations. Which, according to some, constitutes a proof by contradiction that they don't exist. They certainly don't exist, as far as computation and arithmetic is concerned.
This channel is one of my favourites already, there are only 2 channels that I'm eagerly waiting to see the next video of. Those being ur channel and Jet Lag: The Game. Will be waiting for ur next video. :)
This channel is gonna blow at some point
Loving these videos. So interesting! And the perfect amount of depth
if he can explain such easy looking concepts in such an intuitive and comprehensive way, I can inly imagine what will his Real numbers and Complex numbers video will gonna be!
never stop uploading
I know you've said you're curious to see what will you do on this channel later, but I'm really enjoying this structured series of fundamental concepts and definitions and I wish it keeps going. I'm hooked for the negatives now.
Thanks for your comment :) My plan is for video 4 to be the fourth (and last, for now) in this series of building mathematics from the ground up. From there, I have many ideas, but as a purist you can be sure I'll use a similar approach to whatever subject I tackle.
@@AnotherRoof Hey, thanks for the response. I'm sure I will enjoy whatever subject you tackle.
@@AnotherRoof Given your obvious love of paper craft, geometry fundamentals might be a good fit!
Excellent tutorial! I would point out that the "Axiom of Induction", which plays a crucial role in these proofs, is, I recall, not allowed in "First Order Logic"; thus these proofs make use of "Second Order Logic". This is significant because "First Order Logic" is well understood and has very nice properties. "Second Order Logic" is a bit "wilder", and Logicians don't have as clear an understanding of higher order Logic. Most practical mathematics requires "Second Order Logic" for stating and proving theorems. There may be ways of proving these basic Arithmetic theorems using only "First Order Logic", without the "Axiom of Induction", but this might require the introduction of additional axioms. Actually, there is the famous "Skolem Lowenheim Paradox" that shows that "First Order Logic" can not uniquely describe infinite number systems like the integers.
Wait, you're just tricking me into taking Real Analysis again with this series of videos! I just know this'll eventually land on sequences and lebesgue integration
I have a graduate degree in "math", so I completely follow this and it's utterly familiar. I'm curious if anyone with no more than high school algebra is following this with understanding. I think it's amazing that Alex is taking on such a formal and structured approach to "This is maths." But there's also a reason it hasn't really been done (this well) on TH-cam to date. Hoping this breaks through.
I just graduated high school and am following nicely, but I have taken more than algebra. I have taken calculus and statistics, and I have explored many math TH-cam videos.
The occasional jokes and references really take these videos to a whole new level. I'm held captive waiting for the abductive reasoning video.
Gotta love those paper dolls :D
Even all theree video topics are somewhat familiar, I really enjoy these videos. They are very nicely thought trough and conducted. Very clear road forward.
Thanks and keep posting.
Oh yeah! I'm one of the 3 who stuck with you till the end. I'm feeling so much closer to understanding options pricing! 👍
Not sure I ever saw/heard a better explanation of Induction such that I understood what was going on.
I would really love if you would keep going with the concept of building up the "mathematical universe" so to speak.
I've been doing some looking into category theory, and think it's neat that you had to define the identity functions of addition and multiplication so it has certain properties. (It's A+0=A and A*1=A on the bricks. Basically you need some way to do "nothing" with a function or it doesn't necessarily have certain useful properties)
You technically don't have to define it explicitly. It's sufficient that the definition implies it.
I love this channel so much, another great video
Yesss I was looking forward to your next video
what a video man! So informative and so many aha! moments! so good! would love to see more videos in this series!
Really loving this series, and can't wait to see what you have in store. This is one of the best explanations of induction I've heard yet. Wonder if you'll make it to transfinite induction. Proving facts about finite numbers using infinities is just SO COOL.
I love this page so much
The production value is so high on these they are really fun to watch
This is also the best video on induction, I have ever seen. Well done!
Love this. Surprised you haven't brought out a layer cake to demonstrate...
Really enjoying these videos. Really easy to understand how these concepts actually work. Looking forward to the next one!
Each video is better than the previous :)
I loved the rythme & pace. Love the subject. KTGW!
These small concrete bricks are really cute!
Someone: okay little Timmy 1+1=2
This guy: YOU NEED PROOF
I really Like the props. They really Help
I love the analogy for inductive reasoning
Man, what a great channel have I found! your counting and numbers was like discovering mathematics for the first time.
would love to see more such fundamental math proof vidoes
I am so glad I am here and see the beginning of this channels carrier!
This is just a fantastic video. The speed that you explain everything is so easily comprehensible. I feel like I'm actually learning
YaY!!! Another proof!! I love your take on teaching maths. Thank you for the vids.
This series is straight up amazing
Please make the video about the deductive, inductive and abductive reasoning!
These are wonderful. You're doing great!
Aw, I could have sworn I was already subscribed to your channel months ago... Sad that I missed this 5 months ago but happy to be seeing it now! I love the feeling of gradually grokking more and more as the video continues! Wonderful work!
I showed this and your previous two videos to the 1st graders that I tutor, and now they all understand how to add. Thanks!
Infinity is greater than any number you could arbitrarily choose to compare to it. So in this way, it's more of a function than a number.
A function that gives you the successor of its input is one example of infinity. So asking "how big is infinity?" is like asking "how big is addition?" The questions don't apply to non-numbers.
Similarly, asking which of two infinities is larger is like asking "Is multiplication bigger than addition?" The answer is undefined until you choose inputs to those functions and compare the results.
Luckily, any application of the concept of infinity only needs to use a finite amount of precision to be a useful tool. For example, we only know or use so many digits of pi. Yet pi is still extremely useful.
Great vids, I enjoy the new perspective. I've learned a lot.
As Always, An Admirable Attempt At Advocating Arithmetics! 👍
obsessed with this channel, even though i mostly know this things i always come out with a better understanding
The explanation for induction was very intuitive and pleasing, really good
Man I love these videos you explain these concepts very well
I really like your videos. Important, clear, funny and with masterful cardboard and colouring-in skillz.
Can't wait to read your doubtless equally colourful and box-filled thesis.
loving these videos, you're a really great presenter. other people have commented but i'd also like a video on types of reasoning :D
Little late to influence the algorithm, but commenting anyway. Love your stuff, man! Keep up the great work!
p.s. Absolutely love the use of stone blocks for axioms/important theorems. They're a great practical reminder as you explain things and they're metaphorically evocative. Just something very simple and genius that work perfectly for your videos.
I love you V for Vendetta style intro speech. Your humor is engaging with out overshadowing the education.
I mean, this is really wonderful stuff. I'm so glad the almighty algorithm chose to show me the beginning of this series a month ago. I'd love to eventually see you do the Banach-Tarski Paradox. Or Gödel's incompleteness theorems! Or Cantor's diagonal argument!! Even though a lot of this stuff is familiar, it's like hearing an old familiar tale told by a master storyteller - you realize there's a lot more to the story than you first thought.
Interestingly, while the diagonalizaion is relatively simple, incompleteness is quite a bit more advanced - requiring an introduction to mathematical logic and proof systems (and very technical). And compared to the other two Banach-Tarski is extremely advanced (I think it requires at the very least a lengthy introduction about groups and a lengthy introduction about geometry, which in itself contains multiple topics to cover) so presenting even a sketch of the proof is probably unfeasible (though I would be happy to be proven wrong :))
Thanks for your comments! As I say at the end of the video, I only want to cover topics that I don't feel have been covered in the TH-cam maths community, or, topics in which I think I can add a new perspective. Ori is right about Banach-Tarski, I think vsauce's video is excellent in terms of balancing depth and accessibility. I do have some things I'd love to talk about, quite advanced stuff which would require a multi-video approach, but we'll see if my channel grows big enough that I can justify doing those!
@@AnotherRoof Yeah this is why I should watch the whole video before commenting... I had to pause and go to sleep but I had to share my excitement. Anyway, wherever you go next is great with me! I'm just excited for whatever that happens to be.
2:22, V for vendetta momment. Im loving the channel btw.
Another great video, love the style. Looking forward to negatives
Another wonderful video! I'm hoping this series is leading up to the full construction of the real numbers, which very easily extends to the complex numbers.
I absolutely love those videos of yours
Great video, you're an excellent professor :)
Well that is the first time that it's ever made sense to me why proof by induction actually works. Great explanation
It was very nice how you worked all the sets of numbers at the end. :)