Why Negative Times Negative is Positive - Definition of Ring | Ring Theory E1

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.

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.

Fantastic video! I can't wait to see what the rest of the series has in store!

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.

One of my students said, "It tickles my mind that a negative times a negative equals a positive."

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!!

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

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

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?

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!

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.

This is an awesome video cannot wait till you become popular.

This was great! The background music was a little off-putting though; felt like I was watching Trash Taste!

I was thinking about this the other day. Cool video

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 👍

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 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 🙂

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.

Awesome video dude, I never thought of it that way lol. Love to see new channels making videos with manim.

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.