Negative Numbers and Arithmetic's missing Chapter | Sociology and Pure Maths | N J Wildberger
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- Опубликовано: 24 сен 2024
- The history of "negative numbers" is mostly a story of rejection and dismissal. These things only starting becoming a serious element of the mathematical landscape in the western world in the 18th century.
So in fact long before the well-known "number line" model of arithmetic took hold in students' minds, there was a simpler, more fundamental arena for arithmetic. It turns out that this largely missing chapter is now taking centre stage in the new development of Box Arithmetic.
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Yet another historic note: The idea of placing the negative numbers on the number line was introduced by John Wallis in 1673, and, like Descartes, he still only worked with one axis. He wrote "To go -3 steps forward is the same as going 3 steps back" (Processisse passibus -3 tantundem est ac 3 passus recessisse). The idea to represent complex numbers by points in two dimensions was published in 1799 by Casper Wessel.
In fact Wallis attempted to represent complex numbers on a plane, but not satisfactorily. Stillwell talks about it in his book on math history
What do you think of Florensky's representation of imaginary numbers?
Dr. Wildberger, I will need 13 of the Box Arithmetic Books! No one ever accused me of hero worship but you my Dear Brother, you are a conversation my life needed and will always need. Thank you for being driven to truth.
The analogy I use for negative numbers with people, is two sides pulling on a tug of war.
The genius of Dr. Wildberger is rising
OK but that is 1-dimensional so what would explain multiplying two negatives?
@@ThePallidor I just like pProfessor Penrose and you know I haven’t ever lived up to one of the Giants. Ask Professor Wildberger
Credit and debt
Hi Prof. Wildberger! I have an anecdote which might be helpful for you for communicating about positives and negatives, e.g. regarding box arithmetic.
When I was working as a tutor, I 'invented' a simple game to help students become more familiar and facile with adding/subtracting integers. Basically, I'd generate very simple arithmetical problems using a deck of cards, with the black cards representing positive integers and the red cards representing negatives. However, I would introduce the game as not being about 'positives' and 'negatives', but rather about 'teams' -- the 'black team' and the 'red team' -- with the idea that black and red are competing and will 'cancel' each other out. E.g. if aces count as 1, then a red ace will cancel with a black ace.
In the simplest version, I'd turn over two cards (e.g. a red 6 and a black 4), and ask the student two primary questions:
1) Which team wins in this case?
In the example, most students would very easily and intuitively answer that the 'red team' wins, since obviously 6 is greater than 4. Also, it was intuitively obvious that it didn't matter whether I had turned over the red card first or second in the pair (commutativity).
2) By how much does the winning team win?
In the example, most students could easily identify that the red team won by a total of 2, since it would take '2 more for black' for 'black 4' to become 'black 6'.
BTW, if both cards turned up the same colour, it would be equivalent to the other team 'not showing up', and the idea would then be to add the two numbers together. E.g. if the 4 was red instead of black, then the correct answers would be that 'red wins' by 10, since 'red 6' and 'red 4' combine to make 'red 10'.
Relevance to box arithmetic:
I don't recall exactly how you handle negatives in box arithmetic, but one possibility could be to think in terms of something like a 'vector' or 'array' or 'data structure' with one 'element' or 'slot' for the 'black part', and another element for the 'red part'.
So, if the 'data structure' looks like [black, red], then the example from the above card game would be [4, 6], and asking the student the two key questions would be to recognize that the 'red part' 'wins' over the 'black part' by 2 -- i.e. [4, 6] is 'equivalent' or 'equal to' [0, 2], or 'red 2'. In other words, -2.
In your earliest Math Foundations videos, I recall that you started out using a single stroke | as representing the most basic number. In the card game, I used aces as 1s, but when dealing a red ace and a black ace, the answer would be 'it's a tie', neither team wins, so 'by how much?' is answered as 'zero'. In 'vector' or 'data structure' form, [1, 1] is interpreted as the integer 0, rather than 'presuming' 0, and writing it as [0, 0]. (In such an interpretation, 'red 2' might have to be written as [1, 3] instead of [0, 2], perhaps. Well, unless you introduce 'whole numbers' or N0 as being an extension of 'natural numbers' or N+, in which case you could build integers from N0 instead of N+.)
Anyway, I think there's something very intuitive about thinking of 'negative' numbers or 'integers' in general as actually consisting of two 'teams'. There's something very 'natural' in thinking about numbers represented by 'team members', so like 'black 4' means 'four members of the black team' for example. I think people naturally relate to 'quantities' when they refer to people or other living things or animals, or even to portions of foods like pieces of fruit. With the 'teams' interpretation, it's very easy to intuit how numbers will behave if you then decide that there are two teams in question, and the teams are competing against each other.
Or in the case of box arithmetic as two 'boxes' perhaps, a 'red team'/'red box' and a 'black team'/'black box'. Operations would then include 'reducing' or 'simplifying' intermediate results -- of adding the contents of X's black box with Y's black box, and simultaneously adding X's red box with Y's red box -- by 'cancelling' reds and blacks until one (or both) of the resulting Z's boxes are empty.
Perhaps something like: Each 'integer' is a 'data structure' (with appropriate box-ish name like a '2-box'?) of the form [ [ blacks ] [ reds ] ]. So [ [ 4 ] [ 6 ] ] is 'simplified' to [ [ 0 ] [ 2 ] ]. Or maybe [ [ ] [ 2 ] ]. Or something like that.
Anyway, I just found this card game with the 'team' interpretation as a *very* effective way to communicate the meaning of 'negative' numbers to the students I was tutoring. Even students in later years of high school would often benefit from this interpretation after years of struggling with more abstract or esoteric concepts (such as, perhaps, even the particle/anti-particle interpretation). Hopefully, it might also be useful for your endeavours.
Cheers!
In terms of 'semi-rings', I guess you could say that 'rings' can be constructed from two semi-rings, a 'red' semi-ring and a 'black' semi-ring. Thus, 'semi-rings' are more fundamental, and rings are derived. Following historical development.
That is pretty clever, Rob. I like the teams approach. This might work well for addition but how do you use it to illustrate multiplication? Why is red card times red card a black card but black card times black card is black card?
@theoremus Yes this is the problem. No one can say what multiplying two negatives together means.
The most intuitive way I have seen to justify multiplication two negative numbers is by thinking of multiplication as the method by which straight lines are plotted in the cartesian x-y plane.
If a negative multiplied by a negative were negative, or if a negative multiplied by a positive were positive, then we would get a V shaped graphs when we try to plot out something like f(t)=2*t or f(t)=-2*t.
If multiplication is to work in such a way that these functions produce straight lines when plotted in the cartesian plane, then it follows that a negative multiplied by a negative is positive is satisfied.
Interesting to think about this stuff. When I'm thinking of this stuff there's also the debit/credit system of accountancy too. We need the additive inverse to keep track of the balance of a account, and that's the only way to have it set up it seems.
Negative numbers have a solid grounding additively but not multiplicatively.
I have thought of that, too. More than I care to admit. Subtraction seems legit to me, as long as there is something to subtract from. You kind of cannot credit if there is not enough debit somewhere in the system. Your negative money is always somebody's positive claim. Just my two (positive) cents.
"that's the only way to have it set up it seems". Not so. I remember devising term which I called a 'Trebit' which allowed for a greater analysis of financial information. Not sure of its relevance in pure mathematics yet though but it opened up another dimension to looking at financial reports in a more meaningful way.
It is so comforting to learn that Pascal and Descartes struggled with the notion of negative numbers, too. Not a bad company to keep. I am also a bit zero-skeptic, especially when doing binary math; it seems like a mere place holder. If it were a "real" number, surely you should be able to divide by it, too, but it seems that nobody knows what happens if you do. I have kind of accepted that anything raised to the zeroth power is 1; anything but zero. Or is it 1, too? If it is, you would need plenty of time and patience explaining me that. Cool video, by the way, and thanks for making me feel passionate about numbers.
You can model integers in the range [-n,+n] using natural numbers and modular arithmetic.
Consider two sets: M = {0,1,...,2n} and set Z = {0,1,...,n-1,n, -n, -n+1,..., -1}.
Define maps between those sets:
MtoZ(x) = x x>=0 ? x : x+(2n+1)
So if you encounter something like (-2) * (-3), what you do is MtoZ( ZtoM(-2)*ZtoM(-3) mod 2n+1 ).
You can use similar approach to model the origin-centered square-shaped subset of Gaussian integers as well - you know, when the natural number corresponding to "-1" is actually a square
Dr. Wildberger, do you offer a collection of all your "Insights" videos for sale anywhere? I want to purchase the whole playlist from you. I am terrified that one day I will look for your work online and it will be gone.
Whilst the 'pure mathematicians' in the west rejected the notion of negative numbers for a long time. Their use was readily accepted in matters of commerce & the copying of the idea of Debits & Credits imported into Italy from the Middle East in the 14th century. There is no doubt that the Chinese were familiar with positive & negative numbers which they represented as black & red chequers on their counting boards that they had been using for a considerable amount of time before the west. Even today the convention is that Debits are black (+) & Credits are red (-).
Yes but what about multiplication and division?
I like the idea of operator-tuples, a negative number can be described by (-, 1, 2). We can infix that as 1-2, and simplify to -1. Zero can be defined as (-, 1, 1) in this system, without having to presuppose it.
This has flexibility for using other operators, so fractions then are included, (/, 1, 2) for one half. And of course nested, (-, 1, (/, 3, 2)) is -1/2 in simplified form. And it allows us to restrict our number definition to counting numbers only, while extending them to represent all kinds of multidimensional forms as well. The operator can of course be more complex.
From a philosophical grounding, there is always at least a unit of something defined, so there is no question of zero or negative of what. And to make it pronounceable, you can use text operators like (sub, 1, 2) instead of using the operator itself here, which maps to a sort of mathematical assembly language then.
At this point you've reinvented Lisp.
Might as well introduce lambda expressions into this notation.
(But then the lambda expressions are eventually going to need _types,_ and that's a whole other can of worms.)
So like generalizing the numerator and denominator? Then we would have infinite ways of representing the same numbers, like 0... Interesting but not convinced of the tradeoffs.
@@СергейМакеев-ж2н Well yeah, LISP or Forth or other primitive languages are a good map to an arithmetical assembly language.
@@rustedcrab Fractions are always this way, they can be simplified or not.
@@whig01 sure, but now it would be the same for every other kind of number? It feels overcomplicated or at least unergonomic...
Hopefully box arithmetic won't have to go without negative numbers for thousands of years.
I prefer to see numbers in pairs that cancel to zero. Thus, 20-20=0-0=0
Zero is the pivot between positive and negative numbers.
Of course, 20-3=17-0 & 3-20=0-17. The so-called number line is not arithmetic. Negative quantity reflects positive quantity
Hi Christopher. I have a new channel dedicated to woodworking.
A historic note: Descartes introduced analytic geometry, and he represented positive numbers along a line, which we may call the x-axis of a coordinate system. He did not use a second axis. Instead he plottet a point (x,y) by drawing a segment of length y from the x-axis.
To my knowledge, this is not an accurate representation of Descartes Geometry in his appendix of his « Discours de la Méthode ». Indeed, the great geometer even starts for instance by giving geometrical constructions to compute inverse of a « positive number » (magnitude), based on a triangular construction where Thales theorem rules.
And as a consequence of such configuration, two of the three lines of the triangular structure are the working « axis of coordinate », their intersection being the « origine ». Which shows furthermore that, contrary to what is usually thought, Descartes frames are not necessarily orthogonal.
More widely, without limiting himself to « cartesian coordinates» based on right « angle » projection , Descartes uses as well « parallel projections », which are in fact the « covariant » ones, in complement of the more usual « contravariant » ones. The last holds on Pythagoras, while the first hold on Thalès and are more general.
@@mpcformation9646 The modern idea of a coordinate system is something like this: The first coordinate of a point P is found by projecting P onto the first axis, and the second coordinate is found by projecting P onto the second axis. In the early days of coordinate systems the second coordinate was often considered as the distance from P to the projection of P on the first axis rather than a projection on a second axis.
@@PeterHarremoes I perfectly understood what you had previously said. As well as what you are now claiming. But both of your caracterisations are in my knowledge, « forced » and bias. Because you present it as a sort of conceptual « opposition » between ancients and moderns. Which is obviously a simple smoke screen creating a « historical » illusion.
Because the actual difference between modern and ancient time, is that we are raised since kindergarten with preprint grid on papers, whereas the ancients had rare and expensive virgin « papers » (clay tablets, send plates, parchemin, animal skins, etc). So the behavior was undoubtedly more minimalist and straight forward in ancient times.
And as such they used the minimum but sufficient effort to project a point on a given line, given furthermore a directive line to project along. But that is exactly what we still do today on a white virgin sheet of paper. Or what would do a carpenter on a flat piece of wood. People simply adapt the Principe of projection, to the situation, choosing the most convenient and minimalist way to perform it.
So I see nothing here that can be claimed as a « conceptual opposition » between ancients and moderns. That is to answer your last claim.
But there is more toward your first claim on Descartes. What you are believing is obviously simply not true. And more widely, even the mainstream claim that he invented « Cartesian coordinates » is also biased. Because as I said, Descartes didn’t restrict himself to right angle « axis ». He simply did what all the geometers did for millennia, used parallel and intersecting lines to construct geometric figures, mechanical patterns and geometric machines, as the one he opens his « Géométrie » in his 1630 « Discours de la Méthode », to compute the inverse of a given « number ».
He uses parallel and right angle projections to attain his goal, with « axis » at any usefull relative « angle ». And what Descartes brought as a « coordinate revolution » was in fact his systematic correspondance between algebra and geometry, using one approach to solve the second, thus leading to formulate geometry problems in algebraic form, thus equations. That was the turning point and the corner stone of his revolution.
@@mpcformation9646 Yes, Descartes was not restricted to non-orthogonal projections. That is why I did NOT claim that he used orthogonal projections. I DO claim that Descartes only used one axis. For this I rely on the figures (or replicas of figures) I have seen. These figures were made by Descartes, Fermat, Pascal, Barrow and others. I have definitely not seen all their figures, but I also rely on Katz where he in his history of mathematics on p. 484 writes about Descartes vs. Fermat: "Both used as their basic tool a single axis along which one of the unknowns was measured rather than the twoaxesused today, and neither insisted that the lines measuring the second unknown intersect the single axis at right angles."
There's a physical/mechanistic grounding for negative numbers additively, but I've been unable to find one multiplicatively, at least not one that allows for consistent grounding of all aspects of arithmetic.
By the way, a VISUAL grounding would typically suffice for the same reason that a mechanistic one would. Box arithmetic may satisfy that.
One possible approach commonly used is to consider dilations and their compositions. If we dilate the plane from a fixed point by a factor of 2 and then a factor of 3 then the result is a dilation by a factor of 6. In this context a dilation by a factor of -2 makes sense (including a reflection in the fixed point) and then one can convince oneself that the composition of dilations by -2 and -3 is a dilation by 6. But then the natural question is: OK, but what about addition? How do we incorporate that? Its possible of course, but perhaps a notion of vector arithmetic is required...
@@njwildbergerHi Professor Norm, couldn't you just have dilations of the line instead of a plane, with multiplication by a negative number including a reflection about 0? And then addition is just translation along the number line, adding a negative number going the opposite way to a positive one
As I wrote in reply to another post in this thread, there are lots of examples in physics:
1. The torque exerted on a lever by a force. The distance of the point where the force acts to the pivotal point has a sign, and the direction of the force is represented by a sign too. The opposite force on the oppsite side of the lever causes the same movement.
2. Attraction and repulsion of positive and negative electric charges. Same charges => repulsion, oppsite charges => attraction. Changing the sign of both charges does not change the direction of the force.
@njwildberger I thought of a possible way using the two classic examples of debt and competitive scoring: imagine a game like Monopoly except you can take out unlimited loans of $200 each, except you cannot take out more than 2 more loans than any other play has. You can then be down by 2 loans compared to someone else, which translates to having a $400 cushion or potential to tap into relative to them, thus:
down by 2 loans × down by $200 = up by $400
-2 × $-200 = $400
However, as with the example of dilation and the example of force on a lever the above commenter gave, it's unclear what it'd mean to ADD "down by 2 loans" to "down by $200." How do you add loans or dollar allotments to dollars themselves? Again there's vector arithmetic, but that's expanding the scope.
I'm fine with the concepts of positive and negative but I loathe the concept of positive or negative "numbers". Now you can call me pedantic, however +- "numbers" are vectors, not numbers, more explicitly, numbers adjoined with a group structure. Digging into this canonical construction and considering alternatives allows for a lot more opportunity for exploration, even the non-group constructions I find to be quite enlightening.
Hi Joshua, I think this is an interesting and indeed valuable point of view. We ought to ponder the various assumptions we make about basic concepts.
I like this. By adding negative "numbers" to arithmetic, mathematicians had already altered the concept of a number to that of a vector. Rigor was missing from the beginning.
(Some will object that with the red vs. black examples, it's not really a vector but two separate counts set up against each other, but then "2 red times 2 black" is undefined.)
There are only counts and ratios of counts.
In the usual setup, you need to know first what are numbers before you can define what are vectors. More precisely, vectors are the elements of a vector space - so the central definition is that of a vector space. That definition needs a given set of numbers, called scalars in this context, and these scalars need to be organized in what's called a field. So one starts with a field F, and defines what's a "vector space over F" as a second step. The most prominent fields are those of the rational, the real or the complex numbers.
In a field, you have addition and multiplication acting together in a nice way, and the field F itself is the simplest non-trivial, namely the 1-dimensonal, vector space over itself. In this sense, every number is a vector, but you have to agree upon what's a number (better called a scalar) in the first place.
Whether thre is a notion of positive and negative numbers depends on the chosen field. In the rational numbers, this makes sense, but in the complex rationals or in finite fields it doesn't. In short, some fields are ordered fields, but a field F need not be ordered to provide the scalars for vector spaces.
Whats your alternative name for the semiring?
That's a great question. I don't currently have a good answer.
@@njwildberger"rigs are rings without negative elements. (Akin to using rng to mean a ring without a multiplicative identity.)"
@@njwildbergerAlso from Wikipedia: "The term dioid (for "double monoid") has been used to mean semirings"
Dioid for double monoid would definitely be a more constructive name than semiring.
What would you say to students who struggle with (-3)*(-4)=+12?
Two wrongs makes it right.
When you are feeling down and overwhelmed with negative emotion, go to bank and take a loan. Multiply your negative money with your negative emotions and tada, you are not in dept anymore!
@@santerisatama5409 ***Debt (pronounced "det")
That part of arithmetic isn't grounded on anything solid, so there's no way to justify it. There's no physical correspondent to a negative times a negative.
Multiplication by a negative number flips the number about 0, then multiplying a negative by a negative flips it round to the positive side
link to negative boxes pls thnx
Thankyou