Why Negative Times Negative is Positive - Definition of Ring | Ring Theory E1
ฝัง
- เผยแพร่เมื่อ 7 มิ.ย. 2024
- In this video, we introduce a structure in abstract algebra called rings, and prove why negative times negative equals positive in the framework of ring theory.
Chapters:
00:00 Intro
01:19 Short Answer
02:03 Definition of a Ring
06:02 Examples
08:53 Proof
12:31 Outro
🎵Music provided by Causmic
🎵Track : Soul Searching - • Soul Searching
I like the explanation that there are other possible rules for multiplying negative numbers, but that mathematicians found this rule to be more useful, so that's what we most often use.
The rules are whatever we make them.
yep, best answer that summarizes my intention as to why even introduce rings in the first place!! Since this video may have not been too explicit about it, you get a pin
Yes. I think this video explains why they are useful very well. If we want to define addition and multiplication as we do in a Ring, a negative times a negative is positive. He even goes through a series of proofs to show why this must be the case.
I guess EpsilonDelta could have spelled it out a bit more explicitly at the end, though.
@@EpsilonDeltaMain Hey thanks. I think it was clear in your video. I was just commenting my favorite bit.
Interesting take. Makes me think of what are necessary and sufficient conditions for an argument to be considered explanatory as opposed to illustrative. This dovetails into Godel’s Inconcompleteness theorem does it not?
I liked this introduction to rings, but this doesn't explain "the real reason why", it just shows that it must follow from several other properties we choose. This begs the question - what is the reason we choose these properties (distributivity, etc)? The answer is that it's properties we like as they describe things we want to describe, but the same reason justifies just choosing negative*negative=positive as an axiom on its own (with some other stuff) and then proving distributivity (or some other ring axiom we didn't include in this axiom set). Therefore the "real reason" is the car example at the start - because it describes natural things that we want to describe.
I think the only way to explain why a negative times a negative is positive is from some set of axioms. This answers "the reason why", but you say you want to know "the *real* reason why". That just results in trying to ask "why are the ring axioms the way they are", and you could try and find some even more fundamental axioms to prove the ring axioms from, but then you are still left with the question of "why are *those* axioms the way they are?" Your line of questioning is a never-ending regression of "what axioms can I use to prove the axioms I already have?". All you care about in math is the consequences of a set of axioms, not the reason they were chosen, although you may care to choose good axioms.
Okay that makes more sense. I was confused why this would be a better "why". That said, I always get suspicious when someone claims that there is a "why" in math - "why" requires an alternative, another way it could have been, some other axiom that could have been used, and a coherent reason to believe that the axiom really is the root and that replacing it should be assigned logical causality. the only "why" that makes sense to me is the evaluation order of the person or computer working through the problem - since math is made of equivalences and nonequivalences you can always go from anywhere to anywhere, afaict?
That's what he says in the beginning. It follows from the axioms we choose.
This is how math works. You creat a set of axioms that satisfy your needs. Math doesn't really tell reasons, it just proves statements.
@@tunafllsh well if you choose the wrong axioms…..?
Fantastic video! I can't wait to see what the rest of the series has in store!
Love this video!! Proving little theorems just like this one, ones that seem trivial to pretty much everyone, is one of my favorites! Understanding these basic and "trivial" things is crucial for understanding more advanced mathematical concepts imo. It is quite unbelievable that this is one of your first videos on this channel, it's awesome! Looking forward for more!!
completely agree, makes things in higher math so much more intuitive if you understand these
This brings back memories of first year linear algebra for me.
"You are just proving that 0 = 0 - wtf?"
Perhaps you are talking about the dimension of "Imaginary Time", discovered by Stephen Hawking.
I wonder if it is why ordinary time only runs in one direction, in our universe?
One of my students said, "It tickles my mind that a negative times a negative equals a positive."
Aww so cute!
You can also prove it using Peano's arithmetic.
The natural numbers are defined as a set that contains an initial element that we call 0. And there's a function S (for "successor"), where for every natural number *n* , *S(n)* is also natural; and *S(a) = S(b)* if, and only if, *a = b* ; also, no natural number satisfies *S(n) = 0* .
Given that, the natural numbers are {0, S(0), S(S(0)), S(S(S(0))), ...}, also known as {0, 1, 2, 3, ...}. Note that every natural number is either zero or a successor of another natural number, so we can use that to define the possible operations on this set.
Addition can be defined as:
a + 0 = a
a + S(b) = S(a + b)
And multiplication can be defined as:
a * 0 = 0
a * S(b) = a + (a * b)
Examples:
a + 1 = a + S(0) = S(a + 0) = S(a)
a * 1 = a * S(0) = a + a * 0 = a + 0 = a
By those definitions you can prove commutativity, associativity and distributivity, which will be needed for this proof. But it would be pretty verbose so I'm going to let it out of the comment (you can search it, though).
Also, you can define subtraction as simply as: *a - b = c* , if and only if *c + b = a*
However, Peano's arithmetic defines only natural numbers, if we want to extend it for negative integers, we can create an "imaginary" unit *w* (spoiler: we usually call it "-1") that by definition holds the property:
S(w) = 0
With that, we can simply apply the operations definitions:
a + S(w) = a + 0
S(a + w) = a
(a + w) + 1 = a
a + w = a - 1
a * S(w) = a * 0
a + (a * w) = 0
a * w = 0 - a
S(w) = 0
w + 1 = 0
w = 0 - 1
What would happen, though, when multiplying *w* by *w* ?
w * S(w) = w * 0
w + (w * w) = 0
w + (w * w) = S(w)
w + (w * w) = w + 1
w * w = (w + 1) - w
w * w = 1
Well, that's interesting, we just found another important property of *w* . Now, we're finally ready to prove that negatives cancel out on multiplication:
(0 - a) * (0 - b)
= (a * w) * (b * w)
= (a * b) * (w * w)
= (a * b) * 1
= a * b
Remember the spoiler I gave you earlier? So, we can use a simpler notation for "0 - n": we can simply write *-n* .
So, we can write *w* , or "0 - 1" as simply *-1* . And concluding my proof, we discovered that:
-a * -b = a * b
Thank you if you've read this far, if possible tell me what you think about this proof.
Ahhh yes, it is definitely cool system to consider, as there is a structured natural total ordering on N, and properties of the semirings are now theorems in the peano axioms.
Only problem I have with using peano axioms to prove negative times negative is positive is, well you need a video this long just to prove that 1+1=2 first in peano axiom, then extend the ordering to allow subtraction for all elements which is another video, and lastly, its use is exclusive to a totally ordered sets
I like your proof, but I think using Ring Theory is both simpler to explain and useful for a broader range of commonly-used mathematics.
For complete proof, you may check out the book Analysis 1 by Terence Tao
bruh
Nice 🤌🤌🤌
Why? Because (-) * (-) = (+) is most useful, so we defined fundamental axioms that lead to the properties we want once we use the logical inference of deduction on those axioms
yep, best answer that summarizes my intention as to why even introduce rings in the first place!! Since this video may have not been too explicit about it, you get a pin
Edit: oops, can only pin 1 comment at a time. the other person gave the answer first, so I would have to give it to them
@NikoR96 I am stopping reading after your "Six Apples" example to remind you that _Numbers themselves_ are also an invention of the human mind, and don't exist in reality, without someone to perceive the objects being counted....
Food for thought 🤔
@NikoR96 Also, I suggest you use another even number besides two for your example because two plus two equals four, but coincidentally two _times_ two equals four, as well.
@@nikor9640 You are introducing superfluous stuff that isn't needed. I honestly hate when people respond like this because it just shows you have no idea what you are talking about. You are literally saying 1 rock times 3 bugs doesn't make sense, therefore 3 * 1 doesn't actually equal 3 since you cannot multiple rocks and bugs.
My original comment is entirely correct, and you won't find a single mathematician that would disagree, because I just spoke straight facts. You had to introduce extra concepts, such as 'debt,' then you pretended that that is equivalent to the negative sign. You are basically introducing units, which is dimensional analysis.
You need to remove this from your thoughts and start at a lower axiomatic level. If you want to build upon rings and introduce a logical concept of debt, you could do that, but I honestly don't even understand why you even responded when it isn't relevant to the discussion whatsoever. It is incredibly ironic that you think your superfluous additions are 'more fundamental' than a literal first axiomatic approach
I was thinking about this the other day. Cool video
great job on this one! I will say although I've personally been studying them for years, another example of an unusual ring (all i can think of offhand is quotienting a polynomial ring by an ideal but obviously there are easier examples) would have helped and inspired more curiosity if I was watching this years ago. All in all, great video, and I'm thrilled that it's a series and more is coming up!
This is an awesome video cannot wait till you become popular.
I just love the way how you gave us a sneak peak of the upcoming video very much
Also, the video itself is very interesting, nice 👍
Really high quality stuff! Keep making videos
I'm partial to an approach that to some degree combines the two examples you mentioned with the car and the complex numbers rotiation -- one where we imagine treating arithmetic as placing or removing arrows on the number line, sort of as simple roadmap instructions on how to arrive at the answer by simply counting our way there (starting at 0, a rightwards arrow with size 2 plus a leftwards arrow with size 3, brings you to -1). Multiplication is handled by treating it as statements of how many sets do we have of some arrow.
A crucial point to make it work is to also state that we can remove an arrow even if it isn't explicitly said to be there - we'll just assume that it was added previously. That is to say that there's no functional difference between "go 3 steps to the right" and "assume that you had earlier been told to go 3 steps left, now undo that" - if you were at 0 when you got either instruction, both of them would bring you to 3.
How does it work for multiplication?
(+2)(-3): Start at 0, assume that the vector in question has size 3 and faces leftwards - place down such a vector two times. We end up at -6.
(-2)(-3): Start at 0, assume that the vector in question has size 3 and faces leftwards - remove such a vector two times. We end up at 6.
In other words, we can treat one operator as whether we are adding or removing an arrow, and the other operator as the distinction for which way the arrow points.
This is no formal proof of course, but I've explained this way of thinking about it to kids at the junior high/middle school level who struggled with grasping the intuition behind multiplication with negatives, and almost without fail it's like a disco ball gets electrified behind their eyes.
I imagine it's possible to express this rudimentary vector-arrow-simplification idea in terms of the Peano formalisms, though I don't know how rigorously (S is equivalent to a rightwards arrow, the inverse of S is equivalent to a leftwards arrow, both have the size of the unitary, etc). But whether it's workable from my starting point or not, isn't it possible to construct the same proof as you've done in this video using Peano, and if so, wouldn't that on some level possibly be "even more" mathematical?
Brilliant!
I didn’t understand the (-2)(-3) example where you said to “remove the vector two times.” What do you mean by “remove”? I followed everything you said up to that point. I like this idea since I am also a school teacher, and visuals are best for young learners.
Fantastic video, you are amazing at explaining math concepts :) thanks
Awesome video dude, I never thought of it that way lol. Love to see new channels making videos with manim.
I've got a video on this on my channel. But this is like the greatest of all epic explanations I've seen so far. Great job!
This simple explanation works for me:
2 x 2 = 4
2 x 1 = 2
2 x 0 = 0
2 x -1 = -2
2 x -2 = -4
1 x -2 = -2
0 x -2 = 0
-1 x -2 = 2
-2 x -2 = 4
This is actually very eye-opening... many people think that math is naturally inscribed into reality, until they find out about Gödel's incompleteness theorems and other things, like the fact that the result of multiplying two negative numbers is actually agreed upon 🙂 Seems like another brilliant channel has been born just now 🙂
@@caoinismyname So you say if I multiply 2 apples by -3 I should get -6 apples? ;-) These nice "explanations" you posted are actually just analogies or metaphors, but many of such "math phenomena" have nothing to do with direct experience in reality as we know it. What's even more mindblowing is that through such abstract thinking you can actually end up with something real, something that happens in this world ;-)
@@caoinismyname Wow, dude, what an ego! Look, the truth is nobody cares who you are and what prizedid you receive in your petty math olympiad. There are no authorities in science. On top of your enormous ego, you've clearly misunderstood the video. The author indeed referred to the difficulties of even imagining negative numbers and not mentioning manipulating them in real life. Get off your high horse, kid.
@@caoinismyname perhaps we should stick to a more civil tone that knows nothing of trying to score points by ad hominem attacks?
@@caoinismyname So, for internal logic, it is useful to caution one against (1) being too full of himself (as you did) or (2) acting like a genius (as you did), while (3) posting one's awards won (as you did)?
@@caoinismyname not some math's ,most maths you can't give this debt concept in multiplication of two negatives
Excellent video, and I love the way the proofs were done systematically to lead us to the whole point of the video. I wish I had this channel back in college to help me with my math major!
Wonderful video that I have been waiting for years.. Thank you..
I mean, this rule is the most intuitive.
-5*3=-15 can be read as "I borrowed $5 three times, therefore I owe $15", it then stands to reason that a negative multiplier must have the inverse effect: -5*-3=15 "I lent $5 three times, therefore I'm owed $15".
Other rules can be defined, but this one maps fairly nicely to real life and the constraints thereof.
why would the difference of lending or borrowing lead to a sign change on the 3 instead of the 5?
@@davidbarroso1960 yeah, you're right there is a flaw in the argument. You can't lend or borrow for a negative number of times as they HAVE to be Whole numbers.
@@davidbarroso1960 Yes his logic is flawed but I have a refined version of it... Logically in words, without mathematical notations, (-2)×(-3) means.. For example You have a certain number of candies and You get a temptation of eating 2 candies, it happens 3 times but You control Yourself and Don't (1st negative) eat (eating = subtracting 2 candies each time which is the 2nd Negative) 2 candies for all the 3 times that You were tempted to.. So, when You're asked how many candies did You save, Your answer will be 6 candies
Or that you're -$15 down
I asked this question to my maths teacher in 5th grade and she probably couldn't understand that I was trying to think abstract and she thought I was dumb
Grade school teachers despise math so much, I'm surprised we have anyone that can use math in spite of it.
Sir you just won a new subscriber ! Loved this video .
I really like that you're starting at the axioms. nice video !
Your content is very good, so I would appreciate if you keep uploading.
Subscribed
I would, too. 👍
But, I will not subscribe, because while I would like _some_ of your videos, not all of this kind of material "Will fit easily into my brain", so to speak.
🤔➡️😲➡️💥
This was great! The background music was a little off-putting though; felt like I was watching Trash Taste!
When I learned what multiplying by i really means, negative times a negative made perfect sense. This is a nice explanation but above my level lol
Really great explanation, excellent presentation. Subscribed!
I think of negative numbers as opposites. If you multiply 2 by 2, you get 4, but if you multiply 2 by -2 you get the opposite of that which is -4, if you multiply -2 by -2 you get the opposite of *that* which is 4 again.
Exactly. It simply follows from definitions of positive and negative numbers and definition of multiplication. Why is negative times positive a negative? Because 'negative times' means changing the sign (opposite). Changing the sign (opposite) of negative is positive.
I strongly believe people first realized negative times negative is positive before defining rings that were probably inspired by that fact.
It's like saying "why are wheels round?" and then presenting how cars drive on round wheels and showing that it's much better than square wheels.
This presents the mathematical demonstration not the historical reasons lmao.
The people didnt "realise" negative times negative gives negative, but they prooved it using maths that can be, and where later seen as a part of set theory.
@@butwhoasked1821 Rings can be modelled in set theory, but do not require set theory. (One needs the naive notion of a collection on which to define (say) closure, but this need not entail the theoretical framework of ZF, and can even be done absent talk of collections altogether by introducing types.)
@@butwhoasked1821 All knowledge builds up on previous knowledge, so historical progression matters. If there was no ground to "realize" negative times negative is positive, they'd never carry it all the way to the rings and set theories.
I hope I was clear enough
@@butwhoasked1821 Brahmagupta is the earliest known person to give rules for multiplication involving negative integers. Brahmagupta viewed positive numbers as fortunes/gains and negative numbers as debt/losses. Using these meanings, it is entirely reasonable to say Brahmagupta "realized" a negative times a negative is a positive, based on what he was using his numbers to model. In the end, he did define the rules for multiplication involving negative numbers, but this definition was based on what would accurately represent the scenario he cared about.
From our modern perspective on mathematics, we can delve deeper into abstract reasoning to show why Brahmagupta's original definition is so robust, but doing so does not negate the history.
Used to really confuse me when I was a kid why positive x positive was positive but negative x negative was not a negative. Very cool video.
I always had a passing interest in math but never really entertained that interest, and always wondered about quirks like these. I liked I was able to understand the explanation too despite my lack of knowledge in the subject.
I think that one can explain this at a simpler and maybe more fundamental level, considering the very meaning of multiplication as a series of repeated addictions of a number.
Let’s start with (+2)·(+3) = +6: this means that we need to sum up the number +2 three times
(+2) + (+2) + (+2) = +6
But these are two addictions. The third addiction, that is missing, is the one with the number 0:
0 + (+2) + (+2) + (+2) = +6
We sum up the number +2 three times starting from 0.
What does it means then (+2) · (-3) ? The only logical extension of the meaning of this operation is that -3 stands for subtracting 3 times the number +2 from 0:
0 - (+2) - (+2) - (+2) = -6
So one can conclude that (-2) · (-3) means that we need to subtract 3 times the number -2 from 0
0 - (-2) - (-2) - (-2) = +6
Nobody usually talks about the number 0 but this is, I think, the starting point of every repeated addiction (or subtraction) between Integers (i.e. multiplication). After all they are called Relative Numbers because their positions on the numeric line are relative to 0.
If we want to give a fully convincing explanation anyway, we need to explain why the subtraction of a negative number work as it does, meaning why 0 - (-2) = +2?
This can easily be viewed thinking about a subtraction as an operation that gives the offset between the starting point and the point of arrive on the numbers line, that can be positive or negative (positive if we move in the positive direction of the numbers line; negative otherwise).
So 0 - (-2) = +2 because -2 is two units far from 0 and moving from -2 (the starting point) to 0 we go in the positive direction. The positive direction of the numbers line is the one in which the numbers grow bigger.
Your intuition is correct
- and it represents the distributive law. Using it, would make your argument ten times shorter.
I guess you are not well-versed with distributivity, so I praise you for your intuition 👍
How do we know the 2d directional thing applies to real life tho? Or is there any real life application that depends on the fact that two negatives multiplied is a positive?
@@difforno There certainly is! Look up one of Stephen Hawking's greatest discoveries, which he called "Imaginary Time"!
It is a very real dimension in our universe, even thought (arguably) we do not experience it directly.
In fact, I wonder if it is what keeps time running in only one direction in our Universe, instead of either way;
If you have a number you obtained by squaring a negative number, and you undo that operation by square rooting it, you cannot get back to having a negative number, your answer will always be positive.
Maybe, maybe not.
Anyway, look up imaginary time. 👍
@Rocco Don't listen to Kimchi, your explanation is great, better than this video! 👏👏👏👏
Have you considered sending a resumé to the producers of Epsilon Delta videos?
@@TheNoiseySpectator Wow, nice, thanks!
The complex number explanation is particularly good because there is evidence that brains actually understand negation using wetware for rotation. That is, the wetware that controls and understands turning around and walking in the opposite direction.
Dope video. 100% showing this to all my classes.
Superb!
It is intuitively much simpler. Negative numbers are VECTORS. They have both magnitude AND direction. Direction is defined relative to an origin POINT and a reference direction (usually the positive axis). The multiplication OPERATOR for vectors MULTIPLIES MAGNITUDES and ADDS DIRECTIONS.
This sounds like a better explanation, but is there a visualization for this?
Little elaboration would be appreciated.
Basically. Also the ring theorem he explained is essentially a vectorial space
You said "negative numbers are vectors". Are positive numbers vectors too? Thanks
@@pjay3028 It's possible to consider any number a vector; a scalar (standard number on its own) can be considered a one-dimensional vector. If you think of standard multiplication as the same thing as a vector dot product (probably the closest equivalent), one way to represent it is |v1| * |v2| * cos(angle between). Two positive or two negative numbers go the same direction, so angle between is 0, and cos(0) is 1. Otherwise the angle between them is 180, and cos(180) is -1. It's one way to think about it, but probably not very useful.
Your videos are so relaxing. I miss school math 🥲
I really appreciate this explanation because I didn't know about rings and should learn more about them. I thought I had an intuitive geometric explanation for -a*-b > 0, but I never really though to question it too deeply and thinking about it just now I realized it was, no pun intended, circular reasoning.
Abstract algebra is the pinnacle of math fun.
I imagine the base rule for this is simply (-1) * (-1) = 1. And I imagine it's because in any ring, the rule is that if you multiply something by -1, you're taking 0 (the additive identity) and subtracting the number. So it boils down to explaining that when you subtract a negative, you're adding a positive.
Thank you u just taught me something that baffles me for half of century!
It's appreciable how you have put this in front of us... Great effort...
But this phenomenon can be easily understood with mirroring effect ...
The classical definition of multiplication that Descartes gives in the Geometry is based on simple proportion theory. It can be shown, using that definition, that a negative times a negative equals a positive. It is really fascinating that you dealt with the question using ring theory! Cheers!
I was inclined to agree with you. I like the video's explanation, but like you said (if I'm interpreting right) it's just about thinking about it in terms of area being a positive manifestation (I guess I'm at a loss for an official term, but with magnitude of size being only able to be described in absolute terms, I just took it as it being multiplying coordinate values in a point and accepting that area ought to be positive. Sorry if this comes off as sounding like some pedantic rephrasing.
Absolute values are one way of thinking about it. If you would be interested, I can email you a short explanation of the definition, and how it applies to negatives.
@@drakesmith471 I'll write up a doc, and leave a publicly available link for people to read at their leisure 👍
@@thomasaldredge653 sorry about that effort you’re going through. That said, thank you.
@@drakesmith471 nah you're good. It's not a big deal, I'm working on this stuff for a thesis anyhow
My assumption before watching the video:
using distributivity and commutativity,
4=(-1+3)(4-2)=-(4-2)+3(4-2)=-4+(-1)^2*2+3*4-3*2=2+2*(-1)^2, if (-1)^2=1 then this is correct.
Sorry for the bad example though.
Ah, I see you're letting it handle itself by defining that single given instance, and assuming one could do that, I like that. You could use the cyclicality of i to a power to justify that, at least, so I'd think. Given that i^5 is i again, it implies the negative canceled to produce a 1*i.
While teaching high school math, I explained it with a vector example (without using the word vector .) Forward is positive, reverse the gear and it's negative, reverse the reverse and you're back to positive.
I remember my teacher doing the same.
Your great at explaining
They have shown us all those proof at first year of my College. I regret it was shown so late, because it's fascinating to understand why math works like it works.
I understand why it's not shown in primary school, but I think it should be taught in middle school.
I think the main problem with math education is that it focuses too much on the applied side too much. so students are taught basic algebra just to be able to learn calculus so they can be useful in engineering and finance and such. to be honest, I believe that is probably the best solution for an average person though, and we who are actually curious about how things work are few
@@EpsilonDeltaMain You are right. Even tho I'm not super into math, my college had a lot of it (Computer Science). And finding out for the first time, why basic math works like that was really interesting.
@@EpsilonDeltaMain I think your perception is greatly skewed. The vast majority of humans are stupid and cannot even learn algebra, let alone calculus (even though the intuition of calculus is super easy). I was always in the higher math classes, and even then, a bunch of people didn't get anything
They don't even teach negative numbers until 7th grade now; math education is in the dumps; same with science education. It is as if they want a dumber population
The people that are actually interested in this stuff learn on their own; I haven't really learned anything from school and just taught myself everything. Why wait to college to start 'real calculus,' when I can teach myself calculus at age 13, then real analysis at 14? Then I got a differential geometry textbook at 15 and was already solving problems in math and physics at what would be considered real college level and not those stupid AP classes
I think the entire bachelor's degree in college is actually just high school level, if they didn't dumb everything down so much. And for those that don't go into higher learning, they should do trade skills and stuff that is actually useful to society. not everyone needs to know how stuff actually works
So high school should end at age 13 now, then what is considered college should be 14-17, or those that do trades can learn trade skills, then be capable of working at 15 (two years of trades would be more than sufficient). Then, by the time we go to university, we would essentially be pursing PhDs
I even thought college was way too slow and a waste of time and I finally felt that undergraduate stuff should personally be at the level of what graduate work is now
@@EpsilonDeltaMain I agree. The argument often goes that math like this is too abstract for students if they already struggle with the concrete stuff. I think it is the other way around. I think some students struggle with math because the rules and methods that you learn seem arbitrary and there is little understanding of _why_ it works.
As an example I was tutoring my cousin and she was struggling with derivatives and at first I did not understand why this seemed so hard for her... at some point I figured out it was because she had no conceptual understanding of what a function is. Once I've explained that and why the rules are what they are she had no trouble with even the more difficult exercises.
@@Kaepsele337, yeah you're right. If maths were taught better with the concepts clearly explained I'm certain a lot of people would see that it's not that hard.
I am not sure what I was hoping for from this video, in the end it helped me to clear out what I already knew: Mathematics are a set of rules chosen because they fulfil certain requirements better than others, the explanation lies on an axiom that can be seen either as true or false but that in itself is void of reason other than its suitability in a set of operations. I am not a mathematician and I've learned axioms and their consequent statements almost by heart as they are useful if not vital on my everyday life and work as an architect. Through my experience in life maths has been a language to express and eventually solve problems more than anything else and as such (a language) its basis are defined to better suit and define the expression of problems involving the physical world that surrounds me. In a way I speak fluently English, Italian and Spanish whilst I only dabble in Mathematics.
as an architect you are not confronted with math but merely with calculus as any engineering profession. Math is the study of "essential relationships" stripped from all else what can disguise that. But this is my personal description for it. And it is not meant to cover all aspects of Math
The only reason arithmetic and algebras such as ring theory are widely known (albeit in the math community) is that it has a practical engineering and scientific use.
Enjoyed vid. Explained very well. Will be subscribing
I was watching the Rings of Power videos, and this came up as suggested. The Ring theyory, indeed, is the one Ring to rule them all.
'Group' of Hobbits have other plans.
It's as simple as it was described to me in grade school:
If I give you a negative I am saying you gave me something.
If I want to do something negative times I must undo the thing I did.
So if I gave you a negative -3 apples -1 times then I have given you 3 apples by reversing you giving me 3 apples.
I feel like this is MUCH more clear in my head than when I try to write it out.
Alternatively if I take -3 apples I would be giving you 3 apples, so if I take negative 3 apples negative 3 times then you would be giving me 9 applies.
I feel like reversing the context here made it slightly clearer.
To me, the most intuitive and simple explanation is to "factor" the multiplication out. For example:
(-2) * (-3) can be rewritten as - (-2 + -2 + -2), which is -(-6), which is, finally, 6.
So why is (-1)(-6)=6 then?
But with rings, multiplicative inverse (factoring / division) is not assumed, so you can't factor.
Negation of negation is positive under observable events.
@@sq7507 That would be -(-6), which is 6? I don't see your point
@Basil I see. Sorry about the confusion.
This channel is awesome
I always kinda viewed it as the reverse function of “multiplication is adding a number to itself X amount of times”. So (2)(3) would just be 2+2+2. If these numbers were negative, it would just be (-2)(-3). Because its not adding but subtracting instead, it would look like 0 - -2 - -2 - -2. The reason theres a zero is because there is an infinite amount of zeros in any equation. 2+3 is equal to 2+3-0. This is how i saw it at least
When I taught K-4 I used the debt analogy.
If you have a debt you "have" a negative quantity of currency. Canceling debt is negation of a negative.
If one clerk cancels your debt and a second clerk cancels it too that is cancellation 2 times.
I owe $5 so I "have" -$5. The first cancellation gives me +$5 so I pay my debt and I'm back to zero. -$5 + {-1 x (-$5)} = $0. Cancelling (negating) debt two times is multiplication by -2. So -2 x -$5 = +$10. My balance went up by $10 from -$5 to +$5. (-$5) + {-2 x (-$5)} = +$5.
No need to get into group, ring, field algebra.
I'm sorry. I did start thinking long time ago like you did. I first owed $5 then I got bogged down because with the first cancellation, I saw that I did not owe $5. I failed to see why I have $5.
@@jeffleung2594 The story is that two clerks saw the cancel-debt order and both "cancelled" the same debt without the other's knowledge. To cancel debt is to add. Negating a debt is subtracting a negative. This was done twice. -2 x -$5 = +$10.
The original balance was -5. 10 (2 cancellations) was added. -5 + 10 = +5.
Clearer?
@@George4943 yeah, thank you😄
This makes much more sense than this complicated video.
@@dubio77 Yah, advanced math theory not needed.
Long story short it's popularly agreed arbitrary decision.
That if you stick to the rules along with other arbitrary rules, you can predict the future with physics.
Love this. Do you have one for this exact doubt but with division?
I Recognized the music from Trash Taste podcast
This was a very informative video , great explantion
Follows from the distributive law and 0x = x•0 = 0 (which also follows from the distributive law):
Assume A + a = b + B = 0. Then
AB
= AB + aB - aB
= (A + a)B - aB
= 0 - aB
= a(b + B) - aB
= ab + aB - aB
= ab
Note that x - y is short for x + (-y) here.
Lemma:
0x
= 0x + 0x - 0x
= (0 + 0)x - 0x
= 0x - 0x
= 0.
I like this a lot, though it may need some explanation.
What cmilkau is noting is that one can proove that A * B = (-A) * (-B) using the distributive law (Where -X is the additive inverse of X). They need 0x to equal 0 for the proof, which they also proove in the Lemma using the distributive law.
Engineering lecturer here: this a good introduction to rings but does not attempt to explain why we would want integers to follow the rules of rings. I find the most satisfying explanation to relate to a personal bank account, where someone can make deposits but others can take debt tokens to the bank as well. This way negative numbers are introduced. Then, the bank can offset the negative tokens with positive deposits, reducing the total number of tokens in the account. Lastly, we allow for interest, so that debts or deposits can grow through multiplication. In setting up this sort of system we can motivate for the axioms of rings and then later give them the appropriate name.
What rules of rings would you consider superfluous for our naive use of numbers that did not involve multiplying negatives?
Computer Science lecturer here: Agreed. The way I see it is that this framework is suitable to a wide range of mathematics and other fields' purposes (which usually is as good as we can get for really fundamental stuff), but saying: because this is leads to a good framework for X, Y and Z would make for a really short video.
Though honestly, what really bothers me is the clickbaity language.
For context, this is a very general overview of ring theory. We went through the axioms and basic proofs regarding them in the first lecture of my first year abstract algebra course.
There's so much more and it's a lovely field :) (hehe pun).
you make great videos
the way i learned it was that in multiplication and division, that you cancel the negatives out with each other. so if it's a positive times a negative, then there are no other negatives to cancel it out and therefore the result will be negative.
but if there are 2 negatives, then they cancel out and the result is positive.
but if there are 3 negatives , then 2 cancel out, leaving 1 remaining negative, so the result is negative.
basically, if there is an even number of negatives , the result is positive, and if there is an odd number of negatives, the result is negative.
.
this works in language as well.
if you take a positive word, like "good", and add a negative to it, like "not" then it becomes a negative word as in "not good" = "bad".
if you add two negatives to that, then it becomes positive again, as in "not not good" = "good" or "i'm not doing nothing" = "i am doing something".
Bill and Ted understand this on a primal level. when something bad happens and they say it was "heinous", but then sometimes, when something good happens, they say "that was non heinous", but when something really bad happens, they say "that was non, non non, non non, non heinous", they deliberately say it with that cadence, they say it once, then pause , then say two more "nons" again and again in pairs, to convey the desired level of intensity, then close it with one more "non" so that way, all the nons cancel out to make whatever happened "heinous" again
Thus, our universe must have an even number of dimensions, not an odd number.
Because the inverse operation of squaring something is finding the square root, and such an answer is always positive. Not just Squaring but working with any even numbered exponent (^2, ^4, ^6, ^8 etc).
That is not true for the _cube_ rooting of a number obtained by cubing its factor, or "^ing" by any other odd number. Including eleven.
You want proof?
Otherwise, time could run both forward and backward and, we know from our own experience in this universe that it only runs forward.
@@TheNoiseySpectator but then we have "i = square root of negative 1" which should not be possible, since any number squared always results in a positive and yet somehow, somewhere, there is a number that can be squared to result in negative 1. . .
amd this number shouldn't exist. .hence, it is "imaginary", and yet, plugging it into equations make them work.
and i'm not educated enough in maths to understand it. i never made it to calculus
A very simple real-world example is the repeated removal of debits. Debits are negative, removing them is negative, repeating this is the multiplication. And the net effect is positive value added to your account.
That was really a nice video! Thanks^^ what's the music in the background?
I already knew that thought process that deduce the conclusion but what I would like to know is an example that take it to something more concrete as we can see with other mathematical operations or is that that example doesn't exists?
I have a Mathematics Degree with a Major in Statistics - so I shouldn't be commenting on this subject. That said, I would offer that as soon as we accept that a number like 2 exists, and it represents some distance from 0, it should follow that there are other directions away from 0 that must also exist. Humans have always had an innate understanding of rotations - we look left & right and we look behind us to see what might be trying to eat us - so the idea of rotating around our personal 0 has always existed - even if we didn't think mathematically this way, or could put it into formulas. Humans didn't fight the notion of mathematics that described rotations like we fought accepting that negative 2 bananas exist - no one accepted this for a very long time. If 2 represents 2 steps forward, then it is very easy to accept that we can rotate the +2 180 degrees to represent -2 steps behind us - but that of course opens the mind that we could have first rotated +2 by 90 degrees and then rotated this halfway stop another 90 degrees to get to the thing we are calling -2. Thus I am therefore going to DEFINE -2 to be a 180-degree rotation of +2 (which also happens to equal two separate 90 degree rotations). Let's pick a symbol (i) to DEFINE a 90-degree rotation - thus Mathematically -2 = 2 * i * i (2 rotated 90 degrees twice). I would similarly Define -3 = 3 * i * i. And now we are ready to ask the question, what is (-2) * (-3)? The Answer is 2 * i * i * 3 * i * i = 2 * 3 * I^4 = 6 * i^4 - and since i^4 is a 360 degree rotation back onto itself, (-2) * (-3) = 6 because we have rotated entirely back towards the direction that we have DEFINED 2 & 3 to exist in. I've used this explanation to explain these concepts to young children and they easily 'get it' - whereas if I ask educated adults why (-2)*(-3) = 6 - they really don't know why.
This is not actual proof because there is no proof how this system correlates with the algebraic and other systems we use .In this logic - x - is plus , but what is difficult to prove is the correspondence to algebraic system with your rotation system.
The thing this is missing is the justification that "negative" and "additive inverse" are the same or related. The concept of "negative" has to do with ordering, which isn't mentioned at all. If you have that the additive inverse of a positive number is negative, then you've got a complete proof, but otherwise you've just proven a fact about inverses and not negatives.
Ordering is more specific than inverse. The ring of integers modulo 5 has inverses but no order structure .
I'm pretty sure the term negative for integers is defined by the additive inverse property, so the proof is implicit in the definition.
@@turtledruid464 No, the term negative for integers is defined by the order. A negative integer is an integer which is less than zero. The fact that the additive inverse of a positive integer (one which is greater than zero) is a negative integer is a theorem, not a definition.
@@zapazap Ordering is completely independent from any algebraic properties. You can have either one without the other, both together, or neither one at all.
It's hard to explain, but it just makes sense.
Very Well explained
I guess we need rational numbers for division.
Exactly, rationals is the smallest ring extension of the integers that allows division for all numbers except 0
@@EpsilonDeltaMain eeey gottem
@@EpsilonDeltaMain it's probably makes division difficult for computers
When I was a kid we were told to think of multiplying as " lots of".... so 3 lots of 2 = 6. So if we say, 3 lots of -2 , that will = -6 because it is -2 + -2 + -2. So -3 lots of -2 must = 6 because it is 3 negative lots of -2. So the double negative being a positive analogy is very good.
But what is a ‘negative lot’?
Oh helllllll yes! That felt awesome, thanks! Subbed!
Excited for next part. and I know jack shit about complex numbers, but I think you either mean modular system or natural numbers in foreshadow.
Your introduction as to why reminds me of a vector space, not so much a ring.
But to be fair, I don't recall enough of my studies on Groups, Rings, & Partially Ordered Sets.
they are similar after all since they are both sets equipped with two operations.
ring is a set with internal addition and internal multiplication
and vector space is a set with additional set called scalar field (which indeed is a very special kind of ring) with internal addition which we call vector addition and a multiplication between a scalar and a vector called the scalar multiplication
Mathematicians always make simple things look difficult. The explanations is so simple - any child would understand it: if the number of people to each of whom you owe 3 apples decreases by 2 people - it is like you gained 6 apples. THAT IS IT!!!
I like to call this a "Tom and Gerry" approach, because it reminds of when Tom (the cat) decided to learn to play the piano. He opened the first page of his book and found just one note. He turned to page two and found two notes. Similarly on page were three notes. But when he turned to page four it was black with notes and, miraculously, Tom was able to play them all fluently. It has been my life experience that every single maths teacher, maths text book and maths video is like Tom's piano book. Mathematicians, like musicians, seem not to comprehend that 'mere mortals' cannot progress at their pace - should I apologise for being dim?
Well, no. But when do you think you got lost here?
@@hugofontes5708 I have always felt that the learning curve in mathematics lessons steepens much too quickly - I have certainly found it to be so. For example, you introduced complex numbers and set theory immediately after an example of a car going backwards and forwards. My understanding is that educational research has demonstrated the need for very carful consolidation of the basic concepts before more abstraction ideas are introduced.
@@geoffhurrell8478 Yes but this video says "ring theory" in the title. It's from the field of "abstract algebra", and it is not geared towards beginners. If you want to understand this video I'd recommend 3b1b's linear algebra playlist first - visual algebra with examples, and only then come back here for abstract algebra
@@AssemblyWizard That's not the title - where does it say that? I viewed the video because it asked: What is negative two times negative three? Do you not get my point? Those with an aptitude for mathematics and who were fortunate enough to receive good mathematics teaching seem unable to comprehend that most people neither had a maths brain nor good teaching. I'm a musician, but if you came to me for piano lessons I wouldn't show you a simple five finger exercise in lesson one and expect you to play Liszt in lesson two.
@@geoffhurrell8478 The title is "The Real Reason Why Negative Times Negative is Positive, Intro to Rings | Ring Theory Part 1". It says "ring theory". Of course I understand what you mean, but you clicked on this advanced maths video and then commented that it's too advanced. If I clicked on a Liszt tutorial video I wouldn't then complain that it's too advanced, because I intentionally clicked on an advanced video. I agree that the level could have been communicated more clearly (and the clickbaity question is misleading), but just because you don't understand advanced math (yet) doesn't mean that others should not be able to watch videos on it.
Btw I disagree with the "maths brain" thing, this is bad mentality towards learning, math is a subject like many others and requires effort, contrary to popular belief
This video makes something so simple extremely complicated.
It is about double logical negation. A negative number represents the opposite direction. Multiplying two negative numbers means the opposite from the the opposite. Like: I'm not going left = I'm going right (assuming I am going somewhere). Or: I don't have credit at the bank = I have saving account at the bank (assuming there is a financial obligation between the bank and me).
My way of explaining to my kids is that to remove something is a negative. And a debt is also a negative. To negative times a negative is I’m removing a debt. Which is positive !
Wow! How old are your kids?! 😲
I learnt this through jujutsu kaisen
I cannot hear the background music without thinkinv of trash taste damnit
You must've had an amazing precalculus class! (Non-US, I presume...?)
I didn't understand the underlying geometric intuition behind complex numbers until well into grad school if I remember correctly -- and I had a Math minor and very math-intensive major (CS) as well!
Really?
They explained it to me in my high school pre-calc class (Yes, in the USA)
Might have helped that I was into programming a bit.
@@kamikeserpentail3778 this illustrates both of the main problems with American public education (in general, but ESPECIALLY for math):
➡️ wildly inconsistent curricula and standards from state to state -- or even between districts with differing SES within the same state (which leaves many students not only sorely unprepared for college level courses, but also unaware of their unpreparedness, causing them to struggle more than they should to try to catch up with their peers, which will inevitably discourage some of them enough to quit that program)
➡️ the part of a lesson (again, generally, but particularly in the case of math education) that is both the most important for motivating attention and capturing it long-term, and the one we most frequently disservice as a poorly thrown-together afterthought (if at all!) is the *motivation* -- the "why should I care about what the squiggly symbols on the overhead mean?"
I went to a US high school, and my school didn't teach it, and maybe since I took an accelerated path where we take calculus by 10th grade, so maybe they skipped "unimportant" parts that wasn't related to functions. but I have had many students from multiple different high schools who learned it during their high school precalculus, so I thought it was fairly common
Clear as mud. Thanks.
nashe
ya to.... "trash taste" music.......
Really cool
What branch of Mathematics do I need to study in order to understand this? I know that your channel has something to do with real analysis since epsilon delta was something I somewhat learned in Calc 1
This is intro to abstract algebra
Slightly distracted by the music from trash taste lol
Lol "These examples look outside in without fundamentally justifying why it's the case" *proceeds to list unjustified properties of an unrelated mathematical construct called ring and explaining through the lens of its properties*
Justification implies some sort of goal/reason, I didn't catch any except "I like rings and their properties and want to show them to you", but that is your reason not "the real reason" as the title says and it's just as good or bad as saying it's a complex rotation.
If it's a real fundamental reason, I want to see it emerge as you build numbers and multiplication and explore the design space to show numbers can't exist without rings(which is of course false, there's many systems out there) so the question shifts to why we make the choices and how do they result in the concept called ring.
Whereas this video is backwards, boring textbook style, here's a thing and its properties let's see how it solves our problem, the properties are even numbered before shown, completely disregarding that someone had to choose and formulate those properties.
But does anything above matter? No, as you said (-2)*(-3) can be anything you want and just like your video. Just know that by saying "real reason", "fundamental", "define any way you want" instead of "rigorously applying a mathematical framework I like" you attract my demographic which finds the video comically bad, dislikes, unsubs and leaves.
The real world and mathematics are two distinct things. The reasoning here is that multiplying negatives is not easily justifiable with real world examples, however we can, using mathematics, identify the main properties of integer summation and multiplication, which people have no problem justifying as it is quite intuitive in the real world, and reason what multiplying two negatives would to make sense in order to keep those properties. It is very common in mathematics to try and identify the main structure of the concepts we are studying and focus on the structure itself rather than the original concept.
You are right, however, in that multiplying negatives could amount to anything we define, if we so wish. Unfortunately most of the mathematical concepts we take for granted since we learn them as school children, like rational numbers, continuous functions, and especially real numbers, are entirely unnatural and have a lot of mathematical heavy baggage behind them. Even the way we represent any number in base 10 has some heavy baggage, like number theory, infinite series and convergence. Mathematics is entirely unrelated with the real world, and these "simple" concepts we take for granted need the "boring textbook style" to be explained properly and for us to have some insight on them. Believing that you can understand mathematics fully without actually doing mathematics, just with analogies to the real world and simplifications, is akin to believing in magic.
@@trewajg I agree and I think you misunderstood my comment.
The boring textbook style I refer to is using outside knowledge in the form of already defined properies(rules) which magically fit whatever problem in the intro.
I want to see the design space of those properties explored, how choices were made and coming up with formulations for the properties(whose number you don't even know).
If you don't go through this process of discovery and exploration and instead start with the framework you know it works, whatever is described is confirmation bias, not fundamental or the real reason.
What bugs me about it is that he tries to build this complex proof to show how the negatives can be pulled out and canceled using these ring properties, yet the very first theorem he shows looks as if it presupposes the property: (-a) + -(-a) = 0.
@@xxportalxx. that one comes from the fact that every element in the ring has additive inverse. The inverse property of addition. Every element has its respective negative. No presupposition there.
@@HaramGuys yes it derives from that, however the additive inverse itself is just a + (-a) = 0, the idea that the inverse of an inverse is the original seems to me to be the entire core point being discussed here, so imo he basically jumped the most important bit. Proving the rearrangement is trivial imo, the core idea is that -(-a) is a. This is the whole reason the 180° rotation explanation works, the product being the multiplicand scaled by the multiplier is all well and good, it's how the negatives cancel that was being discussed.
It's good to know that this fact follows from the axioms of ring theory.
I don't know complicated stuff but if you are down sad crying listening to a sad song that you can relate to makes you a lil better
Isn't it simply that multiplication is multiple additions?... could that thinking work to prove this? (-1 x -1= 1) is where we are adding negative number negative times? Following the pattern: -1x2=-2, -1x1=-1, -1x0= 0, -1x-1=1, -1x-2=2. There is clearly symmetry here. I think it's just a slight step forward from accepting negative numbers "exist" in the first place (i.e. the symmetry to positive numbers), right?
I’m currently to stupid to understand, but I tried... i give myself a+b=c for effort
You aren't dumb, people just never get taught how logical proofs and axioms work. Axioms are assumed, and then you go from there using logical rules. So why is -6 * -6 = +6? Because if we arbitrarily decided so, and cooked up some axioms and logic for self-validation.
However, using this cooked up math in physics, science, and engineering works extremely well for predicting the future, so it gets accepted by people and taught in school.
Honestly at this point I would rather just think it in terms of complex numbers.
So a negative number in the Argand Plane would be a complex number with an argument of 180°(namely |z|•e^iπ). When multiplying with an another negative number, the exponents add and you get |z1||z2|•e^i(2π)
Since 2π is equivalent to a full rotation from +ve x axis, we end up back on a positive number :)
I had a middle school student ask me this. He really wanted to know and could not let it go (future mathematician?). So, first I did a proof that (-1)*(-1) = 1 borrowed from real analysis, but then I had to do something simpler because he (and the other students) did not get it. So, I went back to the number line idea of and the idea the multiplication is actually like a short-cut for addition. In the end, I had to use the idea of transitivity with some equivalent expressions to show that it worked for one particular case. I have often thought that perhaps before learning about counting or arithmetic, students could be taught about mathematical structures and their axioms so that we can use these ideas to explain things in terms of fields, rings or groups.
It depends a lot on why we've decided to teach them math. When we teach them math "to be logical", then it's pretty important that they learn why the way we've decided math is the way we do things. When we teach them math so they can calculate tips at restaurants, then it doesn't matter and they just need to do it the way they're told.
Teaching your students the multiplication is repeated addition is why it's so hard to grasp the concept of neg x neg = + Students should learn the truth from the beginning. Multiplication is repeated addition or repeated subtraction.
@@Duiker36 it is probably best to teach to think for themselves and then they teach themselves how to calculate tips. But, I would need to do a study to verify.
@@simpleman283 do you have any research to confirm your claims or are you simply speculating? Also, it seems that you contradict yourself, so I am not completely certain what you are saying. Actually, if you want students to know the "truth" we would never define subtraction, since it is actually just addition where a unary operator (negative) has been applied to a number. Or, if you like the inverse of some positive integer.
@@JDMathematicsAndDataScience
Take a few minutes of your time & read.
Yes, I have done some research. I stumbled upon a video & had a go w/producer. This is the result from 2 years ago. I replied to a comment & now I've copied & pasted the following here for you.
Ramy Melhem posted the original video.
Kim Robert Kanel
Kim Robert Kanel
1 year ago
I'm an English teacher in Japan, and have had my students try to explain these equations, and I have to say that this is the best explanation of neg. X neg. = pos. that I have come across. I still have a couple issues with it because I just see it as somehow ultimately reduced to zero. But then again, my friends call me the ultimate reductionist. haha. Anyway, you are an excellent teacher and your students are very lucky.
1
Ramy Melhem
Ramy Melhem
·
Ramy Melhem
Ramy Melhem
1 year ago
Thank you very much. 🙏🙏means a lot. This explanation is the actual reasoning behind the inventor's methodologies.
Simple Man
Simple Man
1 year ago
The best it could be is 0. In his last example - 2 x -3 = + 6 he then reveals the 6 that was in the bank, if you are just seeing the expression ( what he shows is not an equation) how do you know what was in the bank. I want to see it on a number line.
Ramy Melhem
Ramy Melhem
1 year ago (edited)
@Simple Man -2 x -3 = 6 is an Equation and not an expression. I'm not sure that I understand your main question, but it sounds like you are doubting how there could be $6 in his bank account at the end. Consider it like this, if you started with $6 in your account, then racked up $6 credit card debt, then your bank account is worth zero. If your dad said he will pay off your cc bill then you are back to $6 again.
Simple Man
Simple Man
1 year ago (edited)
@Ramy Melhem -2 x -3 = 6 is an equation, agreed. I was talking about
- 2 x -3. To me it is just a rule we follow to all stay on the same page, like PEMDAS. They could have made the rule from left to right no matter the operation. Anyone can see -3 + -3 + -3 + -3 + -3 = -3 x 5 = -15 or -5 + -5 + -5 = -5 x 3 = -15. In your explanation you told us to ignore the $6 that was in the bank and look at the expression -2 x -3 then pull it down after we got to 0. In order to evaluate an expression we need all the information, there could have been $6 in the bank or $7. To me subtracting groups is not multiplication. If that is truly subtracting groups then we need a new math symbol such as (G). -(2)G(-3). I know you put a lot of thought and effort into your presentation and I am not trying beat you up in any way but it still is not very clear.
Ramy Melhem
Ramy Melhem
1 year ago
@Simple Man I'm honestly not sure if you are trolling with your arguments but I will assume that you mean well. First off, I didn't invent this explanation. I am re-explaining it the way the founder of this entire methodology explained it in his book 1500 years ago. Now allow me to attempt to counter argue your statements.
1. -3 + -3 +-3 + -3 + -3 is not -3 x 5 but rather 5 x -3. It won't impact the answer, but it does make a difference in terms of understanding the literal meaning of the expression. We have 5 groups of -3 in 5 x -3.
2. I suggested to ignore the red tiles (+2 x +3) to draw attention to the blue tiles which represented the credit card statement so that we could derive an expression for it.
3. It's absolutely and utterly irrelevant as to whether we started with $6, $7, or $82.32 in the bank account. What matters is that removing a debt increases your net-worth. That is basic financial literacy.
4. It's not up to you or me to decide whether or not "removing groups" is or isn't considered to be multiplication. Multiplication in its essence refers to deriving "repetitive groups" of any asset, debt, abstract, or non-abstract entity. X=groups plain and simple. You can either 'add' groups or you can 'remove' groups and both would be considered 'multiplication'. There is no need to over-complicate it. Why would you need to start a new math symbol 'G' to represent what an 'x' already does perfectly. How would that change Anything?
I feel like the gap in your understanding arises entirely from a mathematical literacy perspective. You understand the difference between an Equation and an expression perfectly fine, but you aren't totally clear as to how multiplication expressions and symbols truly function.... and then you aren't completely sure as to what the positive and negative symbols in front of each number represents. I go through that in the first 3-min of the video. And all that is totally ok because we are all here to learn.
Let me know if you have any other questions. I'm happy to help.
Let me know if you have any other questions.
Simple Man
Simple Man
1 year ago
@Ramy Melhem Thank you Mr. Melhem for taking the extra time to explain. I would still be lost about this without your last reply. I rewatched the video with a more open mind. I've never thought of the # 3 as a group but indeed it is. My lack of mathematical literacy is due to dropping out of High school in the 80's. I read all of the comment and most of the replies so I can tell that you are the teacher that cares. If I had teachers like you when I was in school I never would have dropped out. Someone asked about 10^0= 1 and you took time for them. Right now I am watching Professor Leonard's TH-cam Prealgebra Lecture's. He showed an example for - x + but not one for - x -. So I searched for this answer but no other video even comes close as you already know. I found in the comments about Brahmagupta, after looking him up I discovered why you used red as earnings. Before this turns into a book I'll just say thank you again you are the teacher and I am the student.
Ramy Melhem
Ramy Melhem
1 year ago
@Simple Man ok so you weren't trolling lol. Sometimes it's hard to tell someone's intentions just by reading a written reply. Hey, There's nothing wrong with needing to improve math literacy. Thays why I make these videos for people. We are all here, myself included, to build bigger brains haha. I'm happy I was able to open your mind to a new idea, and anytime you've got something else to debate, then I will be ready. I'm always happy to help.
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Kim Robert Kanel
Kim Robert Kanel
1 year ago
@Simple Man Yeah, I still think of it that way...in a way. Because, the equation is premised on the idea that you are already in debt. But the more I have asked about this, apparently there are higher math functions that you, Simple Man, and I, Kim, are not familiar with, or have ingrained intuitive doubts about. And that is why we have guys like Ramy. Honestly, I still don't completely buy it, but at least I can explain the way others see it. Maybe it's like Quantum Mechanics, if you think you completely understand it....you don't. Sorry for the ramble...
1
Simple Man
Simple Man
1 year ago
@Kim Robert Kanel Hello. No need to be sorry for wanting to understand that is why we came here to this video. I wanted to see a number line example of how this works. He threw me off with the money left in the bank and talking about groups. Like I replied to him I've never thought about the number 3 as a group. But after thinking about it I realized it must be. In order to get to 3 you must have 1+1+1. So 3 is a group of 1s.
I do not fully understand how -2 means subtract in multiplication but I see it a lot more clearly now that I rewatched the video. I was thinking - 2 was a fixed number. Now that I can see all numbers as groups it makes all the difference in the world to me. If you look at -6 as your net worth but then that debt is gone your net worth has become 0, but it went positively in your favor by 6. Even though your net worth is 0 you did go in the +6 direction because the -6 is gone. If -2 means subtract and x means groups we will be subtracting 2 groups. I can see (-3)as (-1)+(-1)+(-1) being one group. We are told to subtract 2 groups so we need another group of (-1)+(-1)+(-1).
Simple Man
Simple Man
1 year ago
I wrote to much and got cut off but here is part of the rest of what I wrote.
After subtracting all 6 (-1) we will have moved 6 places in the positive direction. Rewatch the video and read his replies to me. Don't give up. Let me know what you think.
Ramy Melhem
Ramy Melhem
1 year ago (edited)
@Simple Man You, sir, have learned so much over the past few days. If anything, it's because u had to struggle with the concept with it in your mind first. And this explanation and reasoning you just put forth is 🔥🔥🔥. Way to go Simple Man.
1
Ramy Melhem
Ramy Melhem
1 year ago (edited)
@Kim Robert Kanel Kim, yes the question is premised on the idea that you are in debt, but it doesn't have to be that way. Multiplying positives and negatives was invented to account for money 💰...thats the origin of it. The answer you get when u multiply integers always reflects the "change" in your net valuation and net worth. So any positive answer Always means ur net valuation has increased, whereas any negative answer Always indicates that your net worth has gone down. You current financial situation could be in surplus or debt...that part is irrelevant. All that matters is, did ur net worth go up...Or did it decrease.
1
Simple Man
Simple Man
1 year ago
@Ramy Melhem Yes I feel like I have taken a big step in the positive direction of understanding this concept thanks to you Sir, for turning a light on to something I had never seen before. Your right I did struggle with this and now it feels good. Understanding is so much better than just knowing and following a rule. Students are everywhere but good teachers are hard to find. Thank you Mr. Melhem you are one of thoses very good teachers and I'm glad I found this video.
2
Because that way has been accepted at peer reviews as convenient for general use across mathematics using common multiplication.
Multiplication can be defined other ways as we know but for most people it is good to think about it and ponder why learned people decided on that option and why it has existed for so long.
If that rule failed it would be a major revision across the whole of mathematics?
When you assume that distributive property holds for negative number you are kinda doing a cyclic proof -(-a) should be = a for the distributive to hold true
The "prime the intuition" example I always use is with money, specifically debt.
If you're one dollar in debt, then you have a net of -1 dollars.
To "remove" that debt - getting back to net 0 dollars - is to subtract that debt.
Thus, in dollars: -1 - (-1) = 0.
But another way to remove a one dollar debt is to gain a dollar, so -1 + 1 = 0.
Thus -(-1) = +1.
You can either add one dollar to remove a one dollar debt, or equivalently, you can subtract the one debt to remove it.
Subtracting a one dollar debt is the same as adding one dollar to a one dollar debt.
I kind of expected a defining line why it is the way it is, but I guess thisworks too. I’ll stick minus times minus as ”Minus reducing itself” as a go to explanation.