This is what I really like about maths. Rigorously proving the things that seem so ‘obvious’. Thanks for this great video!
@sourabhsaha5773
5 жыл бұрын
That's actually the worst part of maths, the best part is being able rigorously formalize abstract stuff like n dimensional space and all
@gunhasirac
4 жыл бұрын
In my mind it actually goes the other way around. Have ring (integer, real number) first. Then abstratifies it and reduce axioms needed. In this process you have to make sure the axioms is sufficient and not redundant. In the other words, you should be able to reproduce what properties or results you already have (what this video is doing), and all axioms is necessary in the process. And then you proceed to work on this abstract framework to produce results that is not known so far. This helps because by the abstraction, the same concept requires less effort to process (need less RAM in your brain), and thus you have more room to come up with something new.
@rafaelmoura2103
4 жыл бұрын
except, in fact you use the "obvious" (axioms) to prove the not so obvious
@cube2fox
4 жыл бұрын
@@rafaelmoura2103 The axioms are not so obvious if you think of multiplication just as repeated addition. For example the axiom for the symmetry of multiplication would be obviously false since you can't add a number to itself a negative number of times. So if the multiplier can't be negative (while the multiplicand can), symmetry fails.
@cube2fox
4 жыл бұрын
@@Stefan-ls3pb That's what I'm saying
@guaperz
5 жыл бұрын
Great video! Now I know something for real. Now, could you do the integral of x^x? It would be awesome!
@pedrocusinato02
5 жыл бұрын
0*a is (n+(-n))*a then 0a=na+(-n)a=na-na=0
@danielsykes4251
5 жыл бұрын
Next video - Rigorous proof that 0 is equal to 0👌😂👌 [feat. vsauce]
@MartinPuskin
5 жыл бұрын
But 0!=1!
@danielsykes4251
5 жыл бұрын
Cancel the !'s, 0 = 1, QED.
@gian2kk
5 жыл бұрын
@@danielsykes4251 Divide both sides by !
@danielsykes4251
5 жыл бұрын
weird flex but ok
@avielberezin
5 жыл бұрын
You can prove that there is only 1 zero, with is sort of like saying 0 equals 0, but only that for every 0 you might find, its still the same 0. Suppose a and b are zeros, then a = a+b because every zero (b) we add to a, does not change the value of a. But also a+b = b because every zero (a) we add to b, does not change the value of b. Therefore: a = a+b = b. That is a proof that every two zeros, are actually the same zero. "0 equals 0"
@x15cyberrush9
5 жыл бұрын
Me: Very trivial Michael: *Or is it?* *Vsauce plays music*
@unoriginalusernameno999
4 жыл бұрын
Loss of Loss is Profit. Minus of Minus is Plus. I deserve a fields medal.
@neonblack211
4 жыл бұрын
By loss of loss I’ll assume you mean negative added to negative is negative, minus a minus is the same, but he’s talking about the product of two negatives here. I just straight up don’t care for what our saying... goddamn it
@livedandletdie
4 жыл бұрын
Neon Black he's talking about inverse functions, and the inverse function of an inverse function is the non-inverted function. However it only holds true for a few operations, the Successor Function, Addition and Multiplication. Exponentiation doesn't hold true to it, as there are 2 inversions of it, Log and Root. A^B where Log A = B and Root B = A. There are technically 2 inversions of Multiplication as well, but can anyone really tell the difference between ratio and division? They are after all almost the same thing.
@inyobill
4 жыл бұрын
@@neonblack211 more like: - (-a) = a
@charliesuarez1033
5 жыл бұрын
Now prove the Fundamental Theorem of Algebra Trivial... I know.
@emmanuelcelio6166
5 жыл бұрын
It's "easy" with complex analysis. There are two simple proof, one is using the Liouville theorem; the other one is using the open map theorem (of complex analysis) and the fact that any non-constant polynomial function from ℂ to ℂ is a closed map. :^)
@Brien831
4 жыл бұрын
Emmanuel Celio my agla prof proofed it in the 5th lecture using gaussian proof.
@joaodinisalvares2796
4 жыл бұрын
Gaussian proof is not fully correct tho
@spacejunk2186
5 жыл бұрын
We define a number A to have the property A*x=A, with x being an element of all sets ever. Now we define a number called 0 to equal to A. Thus, by definition, 0*Sigma=0. Perfect proof. No flaws whatsoever.
@connorhorman
5 жыл бұрын
Space junk Unless you take the limit of that expression as x increases without bound. In which case the expression has indeterminate form until you reciprocate x and factor in L’H. Would someone else like to deal with that calculus. Could probably do it if it wasn’t 1 in the morning rn.
@Brien831
4 жыл бұрын
connor horman infinity technically isnt elemtn of any field
@MegaMoh
4 жыл бұрын
you have to prove that such a number exists. you can't just define a number A such that A^x=0 and A belongs to all sets for example
@Chungeoisie
4 жыл бұрын
Is that really the case? I mean that's pretty much what i is. i is defined as one of the solutions to the equation x^2+1=0, and there's no way of proving that such a number exists.
@joda7697
3 жыл бұрын
@@MegaMoh Well of course you can define such a number, you just have to prove that: a) It's existence doesn't contradict previous Axioms. b) It can not be derived from previous Axioms _or_ it can be used to replace a previous Axiom, which now follows from this new one as a theorem
@sebastianhilscher8072
4 жыл бұрын
You actually didn't even use the special field axioms, specifically the property of being an abelian group under multiplication. With a minor correction, commutativity under addition also can be left out. So what you proved even holds for rings, requiring less axioms and being more general. Good job with the video tho.
@semiawesomatic6064
5 жыл бұрын
I've never seen math done like this. I'm taking my first proofs course next term, and if it's like this, I'm probably gonna like it.
@breeeesh
5 жыл бұрын
Fields aren't very ideal
@patricksalhany8787
5 жыл бұрын
🤣
@Zzznmop
5 жыл бұрын
I hav no permission slip signed but will sneak onto bus papa
@TheNiTeMaR3
5 жыл бұрын
Since I’m a maths & physics boi also, I never chose to do much abstract maths like fields & groups, and to be honest, I’m still not the biggest fan of it. However watching you do stuff like this on your channel does make me dislike it less and sometimes its even enjoyable 😄 keep rocking flammy
@zoltankurti
5 жыл бұрын
Groups are vital in physics.
@martinshoosterman
4 жыл бұрын
Couldn't you have done this in an arbitrary ring? Or at the very least, in an integral domain?
@MegaMoh
4 жыл бұрын
How about a*0=a(1-1)=a-a=0 (-a)b+ab=(-a+a)b=(a-a)b=0 which implies -(a)b=-ab [although commutativity and associativity of multiplication is in the field axioms and I think that -x=(-1)x by definition which means we can just say that (-a)b=((-1)a)b=(-1)(ab)=-ab but not sure whether that is true by definition] (-a)(-b)+(-a)b=(-a)(-b+b)=(-a)*0=0 and since (-a)b=-ab and the additive inverse of -ab is ab, then (-a)(-b)=ab I think these are simplier idk
@levitheentity4000
4 жыл бұрын
what? i searched for negative time like... in relativity, quantum physics, and stuff
@jamesyeung3286
4 жыл бұрын
quantum physics?
@schaz7563
5 жыл бұрын
It seemed pretty normie but then I realized I would've not been able to prove shiet 😭
@skylerbowerbank5847
4 жыл бұрын
Not the most entertaining, but i did enjoy this video Love the energy
@brodyscarlett5527
5 жыл бұрын
Hottest proofs around
@dariopl8664
5 жыл бұрын
So if I say 2*3 I'm either describing 3 times the number 2 or 2 times the number 3. But, if I say 2*0 it'll be like 2 times the 0 value or 0 times the 2 number. Thus, you'll get nothing at last! Por cierto excelente video, siempre es bueno aprender una forma de corroborar un hecho matemático mediante el propio uso de él, saludos desde Ecuador! :)
@Davidamp
5 жыл бұрын
Saludos, hermano ecuatoriano
@nj8245
5 жыл бұрын
Boi, you're getting faster with these outros...
@ahlpym
4 жыл бұрын
A common sense approach to this question, before I watch the video (I don't imagine "So that arithmetic works properly" is going to satisfy a layman): Why "- x - = +"? Because "- x + = -" and we can rearrange things. But why that? Because owing dept to multiple people doesn't suddenly create profit. If I owe $2 to 5 people, that's "-2x5" and obviously that should still be negative. In this case, it's -10. Rearranging these, we get "-(1/2) x -10 = 5" Why "0 x sum = 0"? Because if you have no boxes with 10 things in each, you have nothing. It doesn't matter how many things are in each box if you don't have any boxes to begin with. Edit: Oh, this was a rigorous approach, not a common sense reason. All right, that's important to have. This was "What rules make it so?", not "Why do we want it so?"
@oraviram
5 жыл бұрын
Fields are my favorite types of groups under multiplication. I like 'em best when they don't satisfy the axiom with the inverse! ;) But more seriously, I have a suggestion for you: You seem to start proving a bunch of trivial facts without really making the context clear. I think you should make a series constructing all the number systems up to the reals in a very clear way, starting with the axioms of sets (or the natural numbers, if you are too lazy to do that). Along the way you could also introduce fields, ordering and other things for abstract algebra that makes those things more general. Then instead of proving random obvious properties (let's face it, those proofs are usually listed as the super easy exercises you do just to make sure you get the concept in books on those things), you could...wait, explaining it will take a few sentences: For each system, you could make the motivation in one video (for example, for the real numbers it will be the lack of a rational number whose square is 2, and then you show how the least upper bound property fixes this). Then in the next video, or if the motivation is short (like the "next" concept for the natural numbers) even in the same video, construct the numbers formally and prove that they do what you want (for the real numbers, for example, prove that they are indeed a complete ordered field). Then you could make a video about some trivial consequences of the definitions (like that zero times any number is zero), and say the best phrase ever: Proving the other properties is left as an exercise to you. Then if a system has some major theorem related to it you could make a video proving that. For example, I would love to see someone finally treating decimal expansions, or expansions of any base, formally. Just if you are planning to actually listen to this crazy and unrealistic idea: Make sure your starting point is very clear. Sets are definitely needed, so first make a video about them. You don't have to make multiple videos with rigorous axiomatization, since I think intuitive notions for sets usually suffice (you could maybe come back to it after you are done with the real numbers), but everything else must come from a clear definition based only on sets. For example, terms like "ordered pair", "function" and "sequence" are often treated intuitively when doing things like that, but all three must be defined in terms of sets, maybe proving one important property from the definition (like that (a, b) = (c, d) if and only if a = c and b = d) and then using it almost as a definition for the rest of the series. And since if the impossible happens and you actually think about doing this, some possible cringy but kinda nice names are "from sets to real numbers" or "the journey to the real numbers". Another cool thing I just thought about is that you could later even make series about things like real analysis (or calculus of real numbers, whatever people call it), where you develop those theories based on the theory of real numbers you already have, and you could say that the first series is a "dependency" or a "prerequisite" for that series. One last thing: I know I am crazy for saying this, and that maybe I should kill myself or something, but if this series somehow happens, I think it will be nice to not make it full of memes... Crazy memes should be left for crazy clickbaity integrals. I think this series will benefit from not featuring a meme every few seconds. Fuck, I didn't want to write an essay when I started, I just wanted to make a small suggestion that will take like three lines...
@HanBurritoz
5 жыл бұрын
At that point you could just become a maths professor and get paid to give lectures.
@oraviram
5 жыл бұрын
@@HanBurritoz Good point. :P
@angelmendez-rivera351
5 жыл бұрын
Or Aviram Not only does HanBurritoz has a point there, but your entire comment was a complete waste of time, because 1. The series on abstract algebra for this channel already exists, and 2. THE WHOLE POINT OF THE CHANNEL IS TO BE MEMEY AND INFORMAL. Look, we’re not trying to take a college course every time we come watch a video. Some of us (correction: 90% of us) just come for the entertainment. If that is what YOU want, then that’s fine, you do you, there are a dozen channels in the Internet that do exactly what you suggested in the comment section. But there is a very obvious reason he has chosen the format he has chosen, and telling him to change it is like asking him to switch jobs. You may as well have asked him to switch to Vimeo instead.
@oraviram
5 жыл бұрын
@@angelmendez-rivera351 Wait, is there a series on abstract algebra? Then I may have missed it, so I can't really reply to that. But I feel like the second point is more important for me to reply to. First, I want to say that I haven't seen many people do what I wanted. Most courses have kind of a specific topic. For example, in real analysis you may construct the real numbers, while in number theory you construct...I assume the integers, but I haven't really studied it. But non of them construct all the basic building blocks from the ground up (at least not that I have seen, but I have very limited knowledge, so I would love it if you tell me I am wrong). Secondly, I didn't ask him to stop being memey and informal. I said that _if_ he decides to do the series, then that specific series should be a little less memey. And not that this doesn't mean being a boring lecturer talking for fourty minutes, it just means not having a meme attack every few seconds (like in the crazy integral videos). Like the 90% of us you mentioned, I don't come here for serious learning, and instead I come here purely for entertainment. The idea was definitely not to make this a serious channel, it was just to make a series that I think would be fun. Originally I didn't even plan for it to be this long. I was just a little annoyed at how he proved things without a very clear basis (not something I would usually comment about, though), and while thinking about it came up with an idea that seemed pretty cool. Then when writing about it I got a little carried away (and, as in this message, started rambling and repeating myself, because I am a terrible writer) and that huge message came out. It was even pretty clear to me that the idea will not happen, but I just wanted to share it, and had little hope that it will. I also just want to say that I wouldn't call it a waste of time. Maybe it was a waste for you to read it (although I am happy you read it, since now it seems like I wasn't very clear in the original message), but I enjoyed writing it. I probably wouldn't use my time in a much better way anyway, so it really doesn't matter.
@angelmendez-rivera351
5 жыл бұрын
Or Aviram He proved things from the axioms. That is as clear as it can get, and if you do not think that is “a clear basis”, then you have a bad understanding of how proofs work, in which case you should take a course on mathematical logic or watch videos on it.
@MathIguess
4 жыл бұрын
If I get 1 upvote, I will use Infinity_Boi as the name of a variable in my coding assignment.
@MathIguess
4 жыл бұрын
Say no more
@PapaFlammy69
4 жыл бұрын
ez
@hydraslair4723
5 жыл бұрын
Next: prove that in a field, 0 ≠ 1. ;)
@Euquila
5 жыл бұрын
My instinct tells me to do a proof by contradiction
@MrRyanroberson1
5 жыл бұрын
this and those are so close in pronunciation you might be able to accent your way out of it XD thøs field(s)
@MrRyanroberson1
5 жыл бұрын
though while i'm here, the stress timing of english is very much necessary
@pyroswolf8203
5 жыл бұрын
dafuq
@patricksalhany8787
5 жыл бұрын
Algebra is the hardest branch in maths, and that's why I like it.
@Guztav1337
4 жыл бұрын
No, topology is the hardest Bruh, you can't call something the most difficult, it is very subjective
@Alessar30
5 жыл бұрын
Best part is Infinity Boi on Papa
@nishatiwari9212
4 жыл бұрын
Wow
@paulthompson9668
5 жыл бұрын
What language was that from 7:05 to 7:07 ?
@青雲浮遊
5 жыл бұрын
This question is in my midterm :)
@ervinmacic9833
5 жыл бұрын
try proving a^0 = 1 or the infinite sqrt of n~1
@Reivivus
5 жыл бұрын
a / a = a^(1 - 1) = a^0 = 1
@Guztav1337
4 жыл бұрын
But why is that? What axioms did we use?
@tszhanglau5747
5 жыл бұрын
Papa piano would be proud Wait...
@joelsagflaatholmberg3922
5 жыл бұрын
That intro always catches me.
@forgalzz7
4 жыл бұрын
(1-1)(1-1)=0=1-1-1+(-1)(-1)=-1+(-1)(-1) => (-1)(-1)=1 That was 4 secs. Not watching the remaining 7:30
@benjaminbrady2385
4 жыл бұрын
>field trip Kek
@jonathanv.hoffmann3089
9 ай бұрын
Q.E.D. ... 🆒️........🔳
@alanjamey2777
2 жыл бұрын
I have one doubt how do use distribute law here for negative numbers ,you haven't proved distributive law for negative numbers.You are proving -x-=+ without even proving distributive law for negatives .The reason why why write A(B+C) = AB + AC is because LHS = RHS ,but can you prove it for negative numbers before hand????.Or is there some assumption or logical maths here.Or relatively correct mathematics like -X- is forced???Someone answer please.
@lukamitrovic7873
5 жыл бұрын
Never thought proofs can seem this fun xD
@Backen04
5 жыл бұрын
Beautiful. Also nice shirt
@xCorvus7x
5 жыл бұрын
Why don't you just demonstrate that any number is the additive/multiplicative inverse of its respective additive/multiplicative inverse? Then you could skip the commutation when factoring into (-x + x)*y.
@alxjones
2 жыл бұрын
This is a nice proof of the algebraic facts shown in the video, but it is *not* a proof that negative times negative is positive! Indeed, fields don't generally have negative and/or positive elements (see the complex numbers). One needs to consider a total order on the set, and show that compatibility with the field structure requires the order to respect some of the familiar statements, such as 1 > 0, if 0 > a then -a > 0, and if 0 > a and 0 > b then ab > 0.
@kalvin90210
5 жыл бұрын
Also, why don't the fields commute? I mean, after all, they are all arithmetic mathematical laws...??
@keyyyla
4 жыл бұрын
Oyaaaaaahahihia.
@davide8055
5 жыл бұрын
Can you explain why we change signs when multiplicating by a negative number?
@Demki
5 жыл бұрын
So we've started with the magma and added some associative and identities with natural numbers. We then bought a ring, and then we went and got our own integral domain. And now we own a field. What's next? do we module-ate some rocket outputs and explore the space?
@Demki
5 жыл бұрын
@@PapaFlammy69 I feel like that might have been too many math puns for one day. gotta *lie* down for a bit.
@623-x7b
4 жыл бұрын
So because the coefficients are the same even thought the result is zero you can still think of 0a but if the additive inverse cancels the coefficients it cancels to zero even though it already equals 0 that is trippy as fuck...
@deeptochatterjee532
5 жыл бұрын
In the first part didn't you assume that the additive inverse is equal to negative times the number?
@joshuabrown1033
5 жыл бұрын
VERY NICE MY FELLOW DUDE
@magnuswootton6181
2 жыл бұрын
why do we just do axioms, when we can actually use them to solve something!!!!
@oni8337
3 жыл бұрын
3:31 "This is nothing but 0" *Hmm yes the floor here is made of floor*
@42networks
5 жыл бұрын
Elements don't have to commute to cancel with their inverses
@088muhammadsiddiqwiraawald6
4 жыл бұрын
Please more abstract algebra video papa 😊😊😊😊
@immersionmusic
4 жыл бұрын
What happens when a German marries a Singaporean? Singrish
@Henrix1998
5 жыл бұрын
0:44 why there is "1≠0"?
@ДмитроПрищепа-д3я
5 жыл бұрын
Because we're talking about a field, which is a nontrivial ring.
@eladnic
5 жыл бұрын
Thank you for these two proofs!!!!!!
@sofianemohammed8048
5 жыл бұрын
flammable maths what's a mathematical field ? 😊 please
@willnewman9783
5 жыл бұрын
It should be noted that you dont need a field here, all of this stuff works in rings. And you dont need the ring to be commutative nor have a multiplicity identity. I also didn't see asosiativity of multiplication being used.
@SpatialGuy77
5 жыл бұрын
will newman - associativity...FYI 🤓 ...is one of the four properties of addition, not multiplication. Just sayin...
@willnewman9783
5 жыл бұрын
@@SpatialGuy77 I don't know exactly what you mean. In any ring, including fields, assosiativity of multiplication is a very important axiom. It is included as the 5th axiom when he lists the field axioms at 0:37.
@SpatialGuy77
5 жыл бұрын
will newman I was correcting you spelling and making a lame joke - please relax...
@angelmendez-rivera351
5 жыл бұрын
Well you are either intentionally dense or you paid little attention to the video. When we talk about multiplication of negative numbers to give positive numbers, the context is the real numbers, hence field axioms.
@willnewman9783
5 жыл бұрын
@@angelmendez-rivera351 I did watch the video. So I guess you are calling me stupid. Instead of getting into am arguement with you, I will just say that nothing in my statement is wrong, and that mathematics is all about generalizations, and this result generalizes to any ring, as I stated.
@Lemurai
4 жыл бұрын
I can honestly say as a chemist, I have never used any of this math to do my job. It’s quite sad really, this is novel stuff a hobbyist would be into, but it won’t help you find a job.
@isaacsilbert8976
5 жыл бұрын
It's field-tastic :P
@baminasha9718
3 жыл бұрын
How ya doing head mathematician✊
@nicolasmarin7289
5 жыл бұрын
Could you have proven that zero x something is zero by breaking the zero on the left into -1 and 1 and then distributing? There you could just cancel them which is in accordance to theorems.
@askyle
5 жыл бұрын
Sure, if you first prove that (-1)*x = -x
@angelmendez-rivera351
5 жыл бұрын
Ariel J. Birnbaum The fact that adding negative numbers is equivalent to subtracting positive numbers is trivial. In fact, that is the definition of negative numbers.
@askyle
5 жыл бұрын
@@angelmendez-rivera351 We're working with only the field axioms; you can't take these "trivial facts" as granted. And in fact what I said had nothing to do with subtraction (which you can indeed define as a - b = a + (-b)) or "negative" numbers (which may or may not be a thing, depending on the field under discussion). What I said was that Nicolas would have to prove that multiplying by the inverse of the multiplicative unity of the field is the same as taking the inverse of the multiplicand, regardless of the field you're in, based only on the field axioms.
@angelmendez-rivera351
5 жыл бұрын
Ariel J. Birnbaum (-1)x = -x is true by definition. The notation “-x” is quite literally an abbreviation of “(-1)x” by construction. Calling -a the additive inverse of a is a convention precisely because (-1)a is the additive inverse of a. Now, yes, this must be proven, but you presented your question incorrectly. Also, it is not unrelated to subtraction at all, contrary to your claims. In fact, you require subtraction to prove the claim, using the theorem that for all x, x - x = 0, which is a corollary of the field axiom x + 0 = x.
@askyle
5 жыл бұрын
@@angelmendez-rivera351 it's actually the other way around. The field axioms demand (among other things) that there exists an additive identity (that we denote as 0), and furthermore every field element has an additive inverse (i.e., for every a there is b such that a + b = 0; it can be proven from the group axioms that there is a unique b with that property, so it is well defined to denote it as -a). To reiterate: -a, in the context of Abelian groups and the different algebraic structures built on them (like rings and fields and vector spaces), is defined as the additive inverse of a, and *not* as the product -1 * a. These do turn out to be the same as a consequence of the distributivity property (which is also required by the axioms), but that is something that must be proven from the axioms.
@matron9936
4 жыл бұрын
In non ablelian groups -a•-b=b•a
@MCJonCRLKLLR
4 жыл бұрын
If something exists negative times
@namannarang4208
5 жыл бұрын
But again why?
@abedelbasetabusheikhah2598
2 жыл бұрын
You're amazing💖💖💖💖
@kalvin90210
5 жыл бұрын
What is this 'field' papa keeps referring to? The set of arithmetic rules? Please tell me because I want to start learning abelian groups.
@kalvin90210
5 жыл бұрын
Got it, thank you!🍄
@chokza0238
4 жыл бұрын
1 is not 0 thank you wikipedia
@alexandersanchez9138
5 жыл бұрын
This is a direct approach, but I prefer to introduce the lemma that (-a)=(-1)a early on. This is easily provable once you have established that 0a=0=a0. Finally, the result in the video follows from the simple lemma: -(-1)=1. Also, I was convinced that you were going to use the scene from “Stand and Deliver” at the beginning of this video. If you haven’t seen it, I think you’d get a kick out of it.
@StefanKoran
5 жыл бұрын
@4:00 you write down for the left side -ab. But shouldn't it be 0-ab. The identity 0+a=a is strictly seen not the same as 0-a=-a as 0+(-a)=-a would be the corret step. or can we simple assume, that this is given?
@helton8549
5 жыл бұрын
Mas não respondeu a minha questão! O + vezes - dá - por convenção ou é fundamental.
@kikosilva96
5 жыл бұрын
Por exemplo no caso dos inteiros, convenciona-se uma data de coisas que 'fazem sentido'/'dão jeito' tipo comutatividade da soma, existência de simétricos, 0 é elemento nulo etc e também algumas para o produto misturado com adição como por exemplo a propriedade distributiva, e isto tem como consequência por exemplo que (-a)*b=-ab e (-a)*(-b)=ab. Portanto na verdade vem indiretamente de outras convenções bastante 'normais' de se fazer.
@YitzharVered
5 жыл бұрын
Rigorous proof that a ÷ a = 1?
@oraviram
5 жыл бұрын
But that's, like, the definition of "÷ a". It's like asking for a proof that a + 0 = a. :P
@SpatialGuy77
5 жыл бұрын
If a / a = 1, then logically A / a = 2
@tracyh5751
5 жыл бұрын
a ÷ a is shorthand for a*a^(-1) which is 1 by definition of a^(-1). It's axiomatic.
@SpatialGuy77
5 жыл бұрын
Tracy H 🙄
@uchihamadara6024
5 жыл бұрын
0:04 Good english mein fuhrer - I mean papa flammy. I have a question though. How can we prove the following statement, using only the fact that = is an equivalence relation and the field axioms? I ask this cause you do use this fact in the video. And as a matter of fact, this lemma was also taken for granted in my own algebra course. If a = b, Then a + c = b + c for any c in the field F
@nate4511
4 жыл бұрын
slgihtly subscribed for the diffeq-ult pun......
@David-km2ie
5 жыл бұрын
This is so satisfying
@fatima8709
4 жыл бұрын
i always thought of it as an invisible -1 thats being cancelled from both side of the equation lol
@mackinleymenezes8286
5 жыл бұрын
Really cool concept
@SpatialGuy77
5 жыл бұрын
‘Cause two wrongs don’t make a right and if you ain’t got sum, you ain’t got none!?
@executorarktanis2323
4 жыл бұрын
How old are u dad?
@MCJonCRLKLLR
4 жыл бұрын
Haha funny accent
@rome8726
5 жыл бұрын
Rigourous proof.
@FarisSkt
5 жыл бұрын
Papa Flammy with dem METH KNOWLEDGE
@yahavitah2791
4 жыл бұрын
Forgot closer
@rjmorpheus
4 жыл бұрын
MIND BLOWN!!
@salahahmed1902
5 жыл бұрын
Hi can you help me to solve laplace for sin(lnt)?!
@herr0_056
4 жыл бұрын
Kadse
@bulldawg4498
5 жыл бұрын
Yet another solid math video for the benefit of us math mice :)
@Ferolii
5 жыл бұрын
Can you please do a video about integrity domain?
@xXPROxCAMPERXx
5 жыл бұрын
at 5:56 can’t we just say that, since (-a•b) is the inverse element of (a•b), by definition (-a•b) + (a•b) = 0 ?
@SpatialGuy77
5 жыл бұрын
YukoAsuka - the inverse of a•b is (a•b)/1
@angelmendez-rivera351
5 жыл бұрын
SpatialGuy Stop the jokes.
@SpatialGuy77
5 жыл бұрын
Yes - got a bit excited. Advice taken.
@TheBlueToad
5 жыл бұрын
Why? Because Jaime Escalante told me it was.
@julianedinak6107
5 жыл бұрын
But what means 1+2+3+4+..=-1/12 in real life(
@angelmendez-rivera351
5 жыл бұрын
Юлиан Единак You prove this by smoothing sums or by using either Borel convergence or some other rule.
@julianedinak6107
5 жыл бұрын
@@angelmendez-rivera351 Ramanujan summation
@hexa3389
4 жыл бұрын
at first, I thought this is a joke video about Spivak's Calculus.
@seismicdna
4 жыл бұрын
What is that
@hexa3389
4 жыл бұрын
@@seismicdna the best calculus 1 textbook ever. Its rigorous as heck.
@seismicdna
4 жыл бұрын
@@hexa3389 cool, ill check it out
@Shafixy
4 жыл бұрын
Визуелно, због пресека двеју цртица - x - = +
@thedoublehelix5661
4 жыл бұрын
How is it a group if its not closed. There's no multiplicative inverse of zero
@ahlpym
4 жыл бұрын
The non-zero elements of a field from a multiplicative group, so 0 isn't in that group. How 0 interacts with multiplication follows from the axioms of a field (specifically distributivity).
@thedoublehelix5661
4 жыл бұрын
@@ahlpym thanks so the field technically isn't a group under multiplication
@ahlpym
4 жыл бұрын
@@thedoublehelix5661 The entire set? No. For the reason you stated.
@David-jj7dy
5 жыл бұрын
your still going at it with your high art self.
@AstroTibs
5 жыл бұрын
If you turn your car around 180° in the road and then shift into Reverse, you _are_ going in the correct direction...
@angelmendez-rivera351
5 жыл бұрын
SpatialGuy What? No.
@MCJonCRLKLLR
4 жыл бұрын
Fies
@henloitsdiego
5 жыл бұрын
just rediscovered you. absolutely my favourite math KZitemr now. you beat Grant somehow, congrats
@gogl0l386
5 жыл бұрын
I honestly don't want to be mean but Grant is definitely the king of math videos on KZitem. That being said, papa flammy beats Grant regarding meme game and spicy rigorous proofs.
@xCorvus7x
5 жыл бұрын
@@gogl0l386 Grant the king of maths on YT? Do you know the Mathologer?
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