Why Geniuses were Confused about Negative Numbers ?

Mathematicians of the past used to struggle a lot with negative numbers. We discuss why in this video.
The idea was adopted from Martin Gardner's book "Penrose tiles to trapdoor ciphers", chapter 11.

(00:00) - minus 1 is weird
(01:16) - Euler is weird
(02:04) - A serious question
(03:25) - We should not ban negative numbers
(05:45) - Negative numbers in real life ?

#numbers

[AustinTheWeenieTickler]

Just realised how weird negative numbers are. Zero is the absence of anything. How could there be less than the absence of anything?

[HerbertLandei]

- * - = + made always sense to me: Take a coordinate system. x*y for two positive numbers can be represented as a rectangle x*y in the first quadrant. If you have a negative x or y, you "flip" this area over in the second or fourth quadrant, and this "flipping" makes it negative. However, to reach the third quadrant (bottom left), you need another "flip" (regardless whether you come from the second or fourth quadrant), and this makes the area positive again.

[MichaelRothwell]

Here is an algebraic approach to negative numbers.
Let's suppose we only know about the natural numbers (including zero) and we want to define negative numbers, the motivation being that we want to be able to subtract larger numbers from smaller numbers, e.g. calculate 3 - 5.

When we want to define new numbers or operations with numbers, we often assume that familiar rules will continue to apply in the new broader context, and frequently these rules are then sufficient to determine how these new numbers or operations behave.

One familiar example is extending positive integer exponents of numbers to zero, negative and fractional exponents using the rules of exponents.

Another familiar example is defining complex numbers by defining i such that i² = -1, then applying the rules of algebra for real numbers.

This method is valid, as long as, having defined our new numbers or operations, we demonstrate that the new system does in fact follow the rules that we wanted to hold true.

Let's apply this method to define the negative integers.

Let's say that the number -1 is defined by the property that when added to 1 it gives 0.

Since two numbers added in either order give the same result [commutativity of addition], we get:
1 + (-1) = 0 = (-1) + 1

Similarly, if n≥0 the number -n is defined by the property that when added to n it gives 0.

So n + (-n) = 0 = (-n) + n

One thing we notice is that adding -1 or -n in these two cases is the same as subtracting 1 or n.

But this is be true in general, at least as long as the number we subtract is no larger than the number we subtract it from.

For example, 5 + (-3) = 2 + 3 + (-3) = 2 + 0 = 2 [associativity of addition, 0 is the identity element for addition], the same as 5 - 3.

This gives us an idea of how to define the subtraction of a larger number from a smaller number: if I can work out 3 + (-5), for example, then I can define 3 - 5 to be this result.

Now 3 + (-5) = 5 - 2 + (-5) = 5 + (-2) + (-5) = 5 + (-5) + (-2) = 0 + (-2) = -2

So we can now define the subtraction of a larger number from a smaller number (as we wished to do) by saying that it is the same as adding the negative of the larger number to the smaller number.

In our example, 3 - 5 = 3 + (-5) = -2

The next thing we need to know is what is the negative of a negative number.

Notice that we want the (new) rule:
n + (-n) = 0 = (-n) + n
to apply even if n is negative.

For example, (-1) + (-(-1)) = 0.

But since (-1) + 1 = 0, we have (-(-1)) = 1.

Similarly, for any n ≥ 0, we have (-(-n)) = n.

We can now subtract negative numbers, using the (new) rule that subtracting number is the same as adding its negative.

For example, 3 - (-5) = 3 + (-(-5)) = 3 + 5.

Now let's think about multiplication with negative numbers.

If I take 3 of something and add 2 of that thing I get 5 of that thing [distributivity of multiplication over addition].

So if I take 3 of something and add -3 of that thing I would expect to get 0 of that thing.

So 3×4 + (-3)×4 = 0×4 = 0 [multiplication by 0 gives 0], and so (-3)×4 = -(3×4).

Similarly, 3×(-4) + (-3)×(-4) = 0×(-4) = 0, and so, since 3×(-4) = -(3×4)
(-3)×(-4) = -(3×(-4)) = -(-(3×4)) = 3×4.

We can now carry out addition, subtraction and multiplication with integers.

For a rigorous treatment, we would have to show that the system we have built does in fact follow all the algebraic rules that we wanted to hold and that we assumed in our definitions.

The consistency of the system is proof of its validity.

[gustavosilveirafrehse]

I like to think that multiplication by i is a rotation of 90 degrees, so multiplication by i^2 = -1 must be a rotation of 180 degrees. Why they are a rotation I am not sure but it works out hahah

[saraqael.]

I justify multiplication by negative numbers like this: -2 * 4 means going to the left by 2 a total of 4 times. 4 * (-2) means “backtracking“ from the right (= going left) by 4 a total of 2 times. -4 * (-2) means backtracking from the leftwards motion of 4 a total of 2 times => reverse the left motion => going right by 4 * 2.

[jamesexplainsmath]

Quick note: R\Q is the set of irrationals rather than just R. Nice vid btw! ;D

[victorpaesplinio]

I like the work to prove every single type of operation while defining the set of real numbers. But we need to have our foundation, a set of axioms to begin.

I will assume we already defined our axioms (the addition, multiplication, neutral elements, etc)

(i) -(-a)=a

One axiom tells us that for any number a there is a number (-a) such as a+(-a)=0. It is the opposite of a number.
The equation (-a)+x=0 has a solution x=-(-a), which means -(-a)=a. #

(Ii) (-a)(b)=-(ab) = a(-b)

The equation ab+x=0 had a solution x=(-a)b because
ab+(-a)b=((-a)+a)b=0b=0, by the distributive property axiom.
The same goes for a(-b). #
PS: we need to prove using our axioms that any number multiplied by 0 is 0.

Finally
(iii) (-a)(-b)=ab

By using (i) and (ii):

(-a)(-b) = -(a(-b))=-(-(ab))=ab #

[angeldude]

Negative numbers are greater than infinity? This is a completely normal thing in some branches of mathematics, specifically ones that close the number line into a loop, where the largest conceivable number, infinity, is exactly identical to the smallest conceivable number, negative infinity.
Additively, negative numbers are just the additive inverse of positive numbers. Multiplicatively, negative numbers are a reflection about the origin. (If you want a little extra, try thinking of what kind of operation you could do twice to get the same thing as a reflection about the origin. ;) )

[MasterHigure]

3:30 I am pretty certain that humanity went from naturals to positive rationals to all rationals, not from naturals to integers to rationals.

[senerzen]

The video starts with the wrong assumption.
Negative numbers aren't really smaller than their positive counterparts. They have the same magnitude (size, greatness or smallness) but in the opposite direction. Otherwise |-1| = |1| wouldn't be true. Instead of thinking minus numbers as getting smaller, which is wrong, one needs to think them as getting bigger but in the opposite direction.
"<" or ">" is comparing whether a number is on the left or right side on the number line. These signs are about directions and not really magnitudes. So, the terms "smaller/bigger sign" is actually a misnomer. A more accurate name would be " < to the left sign" and "> to the right sign".
2 < 3 actually means 2 is to the left of 3 in the number line, which also happens to correspond with saying 2 is smaller than 3 ( |2| < |3| ), but that is rather a side effect and not the real meaning of the sign. -1 < 1 doesn't mean -1 is smaller than +1. -1 is as big as +1 ( |-1| = |1| ) but it is facing the opposite direction, hence on the left "<" of +1.
This resolves 1-/1 = 1/-1 issue, because minus is not about magnitudes but direction, hence, no paradox.

[kanhaiyalalrajput]

3:44 to 3:50 this is one of the great insights.❤

[zapazap]

The number 1 is lesser than the number -2 if we are considering the magnitude of these numbers, ie the distance from the origin. The numbers 1 and -1 have the same magnitude.

Don't conflate magnitude ordering with linear ordering.

Cheers! :)

[livedandletdie]

to move a negative step, is to make a positive step, but in the opposite direction.
So -2×-4=8 because whether you start at -2 or -4 if you flip the numberline about the origin, you'll flip the signs and you can do the math as normal.
There's a better understanding one could get if they saw, it all as move n steps right for positives and n steps left for negatives. And as flipping the signs for -2*4 and 2*-4 doesn't change anything, just flipping the signs for 2 negative terms shouldn't do anything, because if you flip the sign once for each negative sign, the answer will be correct.

[kiiometric]

It can be proved that -x-=+ with vectors: suppose we have two unit vectors -i, if we take the dot product it'll be (-i).(-i)=1.1.cos(0)=1.
Next, we'll do the cross product, (-i)x(-i)=1.1.sin(0)=0.
Finally the geometric product which is the sum of the two calculated products is (-i)(-i) = (-i)x(-i) + (-i).(-i) = 1 + 0 = 1, so in conclusion, (-1)(-1) = 1

[gorkem]

But, lesser and greater are relative. You can mean -1 debt and 1 debt with these numbers.

[angelmendez-rivera]

0:30 - 0:44 It is not nonsense. Consider the function f : {–1, 1} —> Z such that f(x) = –x^2. Then you will find that f(–1) > f(1). Nonsense! Right? No, of course not. There is nothing nonsensical about it. All it demonstrates is that f is not an isotonic function: it does not follow from x < y that f(x) < f(y). There is nothing wrong with that. I watched ahead in the video, and this does not have much to do with what the video is about. The video is about abstract algebra and introducing groups and rings to the audience, so commentary about "lesser" and "greater" elements does not belong, as it is bound to confuse people by mixing up concepts that should not be combined without proper treatment of the subject.

1:20 - 2:04 Well... yeah, if you start with the wrong assumptions, you generally get the wrong results. But you are equivocating the ideas of "negative numbers" with the ideas of "being less than something." Negative numbers are all about adding numbers and subtracting numbers in a number line, but while being "less than" is related to number lines, it has nothing to do with arithmetic.

2:38 - 2:46 For the integers, yes, but not in any other context.

2:50 - 3:17 I think this severely misunderstands what multiplication does. And this is why "multiplication is just repeated addition" should really not be taught once you encounter new number systems like the integers. It should only ever be taught in the specific context of the natural numbers. Addition of integers is just analogous to sliding a ruler a certain amount of units from a starting point. Multiplication of integers is just analogous to taking a rubber band with labeled units and stretching it by a given factor. Multiplying by a negative number gives you negative stretching. What does that even mean? Well, it is simple: if I multiply –2 by 4, I stretch it by a factor of 2, arriving at 8, but then I flip the rubber band around, about the center, to get –8. Negative stretching is just stretching and flipping. So if I start at –4 and I stretch it by a factor of –2, then that means that I stretch it by a factor of 2 to end up at –8, and then I flip the rubber band around, to end up at 8. It really is simple. There is nothing unintuitive about it, it just gets explained poorly. Something being unintuitive is completely different from it being poorly explained by school systems. But really, there is a bigger issue with the video, and the issue with it is that the video takes multiplication of a negative and a positive for granted, claiming it makes intuitive sense, without actually providing an explanation. I think this is fatal to the point being communicated, because the thing is if, whatever explanation one has for Negative*Positive = Negative also immediately explains why Negative*Negative = Positive. If someone does not understand why the latter is true, then that means they also do not understand why the former is true, regardless of how strongly they mistakenly believe they actually do understand it. So if you want your audience to question themselves about Negative*Negative = Positive, then you should also tell them to question Negative*Positive = Negative. Otherwise, they really are not going to develop the natural understanding that they need. What I am saying is that the video is right to try to use this topic to introduce the ideas, but the execution is not effective, in my opinion. And also, this has nothing to do with how dividing by bigger numbers positive numbers reproduces smaller ratios. This is what I meant when I said the video is confusing "less than" concepts with arithmetic, even in contexts where they are not related.

[degrees]

Well, multiplication by a positive number is a scaling of the number line (we can take ℤ, we can take ℚ or ℝ if we like, or ℂ… any ring considered as a module over itself, though usually it won’t be visualizable as a “line”). In this vein a negative number makes a scaling with a reflection added. Zero is the degenerate case which connects both (if we have some kind of topology we can make sense of that saying). This is a good way to go if we’ve already discovered that addition of a constant is a translation.

We can omit any intuitions about what is a “scaling” just positing we talk about functions f that preserve addition: f(x + y) = f(x) + f(y). Then, as far as we’re not yet came out of ℚ, we don’t need continuity—there’d be only a single function f which maps 1 to any given number. Now we can define multiplication x ⋅ y = f(y) where f is such an additive function that f(1) = x.

We can also forget about our prior numbers, taking these functions to be our new shiny numbers (the ring of morphisms of an abelian group). We can define addition argument-wise: f + g = h such that h(x) = f(x) + g(x); this h will be additive too. And multiplication as function composition: f ⋅ g = h such that h(x) = f(g(x)). We definitely have a zero function which sends everything to zero, we also have an identity function that returns its argument as it was. This is the explicit form of what we’re doing implicitly at some level when we do geometry (or linear algebra) with numbers, be they ℝ or ℂ or say quaternions.

[abt]

🤔😀 please Make a new video on decimal digit exist in univer or not .