Negative Numbers and Arithmetic's missing Chapter | Sociology and Pure Maths | N J Wildberger

[@MegaSquiff]

Very enlightening information! I’m an old guy who wishes he had teachers of your calibre way back when. Saved to ‘watch later’ and subscribed. A big thanks.

[@bobby-3x5x7mod8is1]

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.

[@PeterHarremoes]

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.

[@hywelgriffiths5747]

In fact Wallis attempted to represent complex numbers on a plane, but not satisfactorily. Stillwell talks about it in his book on math history

[@lucassiccardi8764]

What do you think of Florensky's representation of imaginary numbers?

[@jaydenwilson9522]

Counting begins at 1. Measurement at 0. TAKE 0 OFF THE NUMBER LINE! And get rid of negative numbers and imaginary and negative imaginary numbers with it! The Rule of Signs is WRONG! -a*-b ≠ ab as there is NO logical reason why they should flip like that! Especially when a*b DOESN'T FLIP BUT REMAINS AS = +ab sqrt-4 ≠ 2i either!

[@PeterHarremoes]

@@jaydenwilson9522 Multiplying by-1 means flip. Multiplying by -1 means flip twice which is the same as having multiplied by 1, which is the same as have done noting.

[@PeterHarremoes]

@@jaydenwilson9522 When we count a collection of objects the usual procedure is that we point at the first object and say on. Then we point at the next object and say two etc. until we reach the last object. This usual procedure does not involve zero, because there is nothing to point at for an empty set. One could introduce a different procedure where one initiates the procedure by saing zero and then point at the first object and say one and then continue as usual. With such a procedure word "zero" would mean 'now I start counting'. This procedure would give the result zero for an empty set. With such a procedure children would learn the meaning of zero intuitively so that they would not have to struggle with the concept in school.

[@toddymikey]

In simple vector systems, a negative is just a measurement in the opposite direction: eg 2cm + 6cm = 8cm is "go right 2cm and then go right 6cm" = you are now right 8cm of the origin. 2cm - 6cm = -4cm is "go right 2cm and then go left 6cm" = you are now left 4cm of the origin. When you reduce this to the basic idea, negation is the reversing of the direction. This is how multiplication of negatives makes sense: reversing the direction of the operation twice. -2 * 4 = reverse 2 * 4 = reverse 8 ... ie you are changing direction once. "Go reversed 2cm 4 times". -2 * -4 = reverse 2 * reverse 4 = 8 .... ie you are changing direction twice. "Go reversed 2cm 4 times ... in reverse". I remember learning this explanation when studying Turtle Graphics in high school. However, also as a child, also I remember studying an Egyptian translation that mentioned this justification on why the engineer had used negative numbers ... and then he immediately declared that negative numbers don't exist, but they were just useful for the arithmetic to work. I looked for this ... but can't find it. I'm not sure what version of Turtle Graphics ancient Egyptians used.

[@robharwood3538]

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!

[@robharwood3538]

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.

[@theoremus]

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?

[@ThePallidor]

​@theoremus Yes this is the problem. No one can say what multiplying two negatives together means.

[@robharwood3538]

@@theoremus I have a very ingenious explanation of integer multiplication in term of 'teams', but it is too long to fit into a KZbin comment... 😉😅 Seriously, though, I agree the teams metaphor only works intuitively for addition. But it's a good starting place, and then you can extend this new intuition with a *different* metaphor, such as, for example, involving number lines and 'mirror images' and 'reflections'. (The 'reflection' of the 'object' is a 'reflection'; but the 'reflection' of the 'reflection' gets you back to the 'object' again. 'Red' being 'reflections' and 'black' being 'objects'. Just one way of thinking about it.) As for connecting this metaphor to Box-arithmetic multiplication, I don't have a clear idea of that. Maybe this basic intuition might help Prof. W. or someone else come up with some other twist or idea that makes more intuitive sense? I'm just tossing out some brainstorming kinds of ideas.

[@theoremus]

@@robharwood3538 Rob, you must make a video on your red & black team, illustrating basic algebra. I look forward to it. 😁

[@Cihak200]

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

[@greenstar2108]

I don't share your philsoophy of mathematics more generally, but I never liked the ring theory 'justification' for negative x negative = positive. I would distinguish between numbers used for counting and numbers used for measuring (the distinction is blurred in undergraduate books on real analysis which forget to tell studnets that N < Z < Q < R is only correct upto isomorphism) . In the context of numbers for counting there is the chinese coloured rods model for negatives and positives, but no meaning can be assigned to multiplying negative numbers by negative numbers so the problem of justification doesn't arise. In the context of numbers for measurement, there seems to be a projective geometry type justification (Stillwell's 4 pillars of geometry allows the dedicated reader to derive a version of it, but the full version requires going through the tedium of Hilbert's system for Euclidean geometry it seems).

[@tapiomakinen]

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.

[@gandab]

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.

[@ThePallidor]

Negative numbers have a solid grounding additively but not multiplicatively.

[@tapiomakinen]

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.

[@sharonjuniorchess]

"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.

[@bat020]

One of the recent lessons from category theory is that commutative monoids are natural objects of study in their own right, and can be thought of as N-modules in the same way that abelian groups are Z-modules. And the multiplicative structure of N arises straightforwardly from the additive homomorphisms N -> N. Plus you can develop matrix arithmetic and much of linear algebra over commutative monoids. So I think commutative monoids are kinda the missing link here. And just like how monoids have an associated data structure, the free monoids, aka ordered lists, the free commutative monoids are unordered lists, ie boxes! So maybe that's why there is such a close connection between all these (as you say underdeveloped) ideas.

[@flatisland]

2:52 I got 0 apples and now I take away 4 apples. Now I got 4 apples but they are lying on the other side of the mirror and I can't reach them. Zero: the art of creating something out of nothing. Negative numbers: the art of creating things that don't exist. Both seems indeed illogical. Numbers are no objects. But once you apply them to objects things get a little strange... I have a question: ist there ANY example where you CANNOT avoid / circumvent negatives? E.g. if you have -273 °C you can simply use Kelvin instead and only have positive temperatures. Can't you always "move" things to avoid negative numbers?

[@whig01]

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.

[@СергейМакеев-ж2н]

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.)

[@rustedcrab]

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.

[@whig01]

@@СергейМакеев-ж2н Well yeah, LISP or Forth or other primitive languages are a good map to an arithmetical assembly language.

[@whig01]

@@rustedcrab Fractions are always this way, they can be simplified or not.

[@rustedcrab]

@@whig01 sure, but now it would be the same for every other kind of number? It feels overcomplicated or at least unergonomic...

[@krisdabrowski5420]

The analogy I use for negative numbers with people, is two sides pulling on a tug of war.

[@brendawilliams8062]

The genius of Dr. Wildberger is rising

[@ThePallidor]

OK but that is 1-dimensional so what would explain multiplying two negatives?

[@brendawilliams8062]

@@ThePallidor I just like pProfessor Penrose and you know I haven’t ever lived up to one of the Giants. Ask Professor Wildberger

[@christopherellis2663]

Credit and debt

[@bobby-3x5x7mod8is1]

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.

[@njwildberger]

Hi Thomas, it’s a great question and thanks for your interest. I do not have such a collection available. One reason is that it would be quite large. Do you know that you can download videos from KZbin?

[@bobby-3x5x7mod8is1]

@@njwildberger Yes, all my children are home schooled and about 10 years ago I downloaded your k-12 series to teach them. Even now I have old CD's with your work on them. Thank you kindly Sir..

[@therealbrewer]

So a semi-ring just doesn't have additive inverses? Is that the only difference?

[@MisterrLi]

What I appreciate most here is the non-negativity of critical thinking. Ok, so what's so natural about the natural numbers? By the way, is there only one set of them? What about natural numbers expressed in different bases, like binary, decimal, and why not unary (base 1), are these the same size and shape? They are clearly different when it comes to the numerals and their mappings to the natural numbers. Seen as sets, we can actually view one as a proper subset of another having a bigger base. Does that mean (disregarding cantorial theory for a second) that there are (infinitely) many different sizes of natural number sets? That would also totally impact the way we use these objects. Setting a transfinite number ω to the natural number size will necessarily be different depending on what base it is based upon. If you don't work in cantorian theory, where all countable sets have the same size (cardinality).

[@njwildberger]

It is up to us, when we are creating an arithmetical system, to decide on which representations will be regarded as equal or equivalent.

[@darthbumblebee7310]

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.

[@lukiepoole9254]

Can you prove that the the circumference is indeed the greatest lower bound for all perimeters of the circumscribed polygons WITHOUT using biased functions and assumed-to-be-true value of "pi"? Can you find the value of pi if the opposite is true? Using three or four conjectures, there is only one unique solution. Is it algebraic? One can use distance formula to show that circumference can be algebraic by cube a chord length of C/4. Is it constructible and thus squares the circle? Is it self-similar which exhibits golden proportions? The first square which has the same perimeter length as the first circle, is tangent to the intersections of the second square and second circle which has the same perimeter length as each other, while the second square is also tangent to the first circle.

[@PeterHarremoes]

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.

[@mpcformation9646]

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.

[@PeterHarremoes]

@@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.

[@mpcformation9646]

@@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.

[@PeterHarremoes]

@@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."

[@mpcformation9646]

@@PeterHarremoes I suppose you ment « Descartes was not restricted to orthogonal projections », instead of a double negation… But again, I repeat my objection to your belief. The ancient are doing it the minimalist and most efficient way it should be done, yesterday, today and tomorrow. Since if you are on a virgin sheet of paper, it is silly and waist of time to draw an arbitrary second axis, parallel to your direction of projection, in order to express such arbitrary coordinates. So what you wisely do instead is to use your drawn segment on which you project the given point, project in the desired direction, and extract « coordinates » from this practical minimalist figure. It’s inconceivable that ancients, which minds were trained into the sharpest rigor and supreme elegant efficiency, would waist time making superfluous work and drawing when the Art of Geometry asks for minimalist sufficiency. So they are just doing exactly what a wise modern mathematician would do today, which is the universal and intemporel art of drawing : « Rien de trop », « why make things complicated when they can be done simpler ? », etc. Using a second materialized arbitrary axis is only usefull when there is a systematic use of this second axis. And the ancient had no problem building it if needed. But in pedagogical figures of their textbook, they isolated what is minimally usefull to be sufficient and the match with the elegance of drawing and the pedagogical efficiency. I therefore see no « conceptual opposition » at all, that you seem to claim, on this point at least, between ancients and moderns. And « historians » make often « technocrats shortcuts » that are dubious. Some slide into « attractive formulas » to « sell their work » and try to appear deep. They « force » historical facts and distort accuracy to « simplify » and round angles with mainstream inertia. It takes some courage to set records straight, especially when truth is inconvenient and unpopular.

[@ThePallidor]

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.

[@ThePallidor]

By the way, a VISUAL grounding would typically suffice for the same reason that a mechanistic one would. Box arithmetic may satisfy that.

[@njwildberger]

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...

[@hywelgriffiths5747]

​@@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

[@WK-5775]

​​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.

[@ThePallidor]

​@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.

[@postbodzapism]

Whats your alternative name for the semiring?

[@njwildberger]

That's a great question. I don't currently have a good answer.

[@jeremyjedynak]

​​@@njwildberger"rigs are rings without negative elements. (Akin to using rng to mean a ring without a multiplicative identity.)"

[@jeremyjedynak]

​@@njwildbergerAlso from Wikipedia: "The term dioid (for "double monoid") has been used to mean semirings"

[@jeremyjedynak]

Dioid for double monoid would definitely be a more constructive name than semiring.

[@russellsharpe288]

The reason that multiplication is extended to negative numbers in such a way that -1 x -1 = 1 and so on is that it would be very inconvenient to abandon the distributive law for addition and multiplication: we would like a x (b + c) to be (a x b) + (a x c) for all a, b and c, not just for positive ones. And then we immediately get 0 = (-1) x 0 = (-1) x ( 1 + (-1) ) = ((-1) x 1) + ((-1) x (-1) = (-1) + ((-1) x (-1)) which implies that (-1) x (-1) = 1, since 1 is the *unique* additive inverse of -1 (note that if -1 + x = 0 then by associativity x = 0 + x = (1+ -1) + x = 1 + (-1 + x) = 1 + 0 = 1).

[@flatisland]

I always wondered about (-1)x1 being -1. That implies that -1 is "stronger" than +1. Kind of a breach of "symmetry" because 1x(-1) is also -1.

[@russellsharpe288]

@@flatisland I'm not sure what you mean by a breach of symmetry. -1 is the additive inverse of 1 ie that number which when added to 1 gives zero (0). It's not a second multiplicative identity like 1, so you wouldn't expect it to multiply 1 to give 1. (If it did, we would have -1 = 1: see below) And multiplication is commutative, so 1 x (-1) = (-1) x 1. As for the result in both cases being -1, that follows from the definition of 1 as the multiplicative identity: the unique number which when multiplied by any other number leaves that number unchanged (just as 0 is by definition the additive identity: the unique number which when added to any number leaves that number unchanged). These identities are unique because if we had two of them (multiplicative say, let's call them a and b) we would immediately have both a x b = a (because b is such an identity) but also a x b = b (because a is also such an identity); and hence a = a x b = b. Similar argument for additive identities.

[@flatisland]

@@russellsharpe288 I understand all that and it's of course logically correct when you consider the definitions. However, I see no reason why -1 should not be granted the same properties as +1 (except for the sign) and why -1 * 1 couldn't just as well be 1. I only can explain that to myself by the assumption that negative numbers are flawed and not symmetrically definable, therefore inexistent. I know how how this sounds when the whole math community considers negative numbers totally viable but it's almost like the matter - antimatter asymmetry. It just don't make sense. Btw, I have a similar opinion about zero. Devision by zero and other peculiar properties it seems to have may hint a problem here as well.

[@russellsharpe288]

@@flatisland Is it that you think of the negative numbers as being just like the positive ones, merely (so to speak) 'reflected' in zero, so that -1 should play the same role for them as 1 does for their positive counterparts? That's true as far as addition and subtraction goes; but 1 ("positive one") is as fundamental *and unique* for multiplication as zero is for addition. So one shouldn't expect -1 ("negative one") to play the same role multiplicatively for the negative numbers as 1 does for the positives; at least as long as you want to extend the definitions in a way that preserves the usual properties of commutativity, associativity and distributivity. You don't have to do this of course: you could define eg both the sum and the product of any two numbers one or both of which is negative to be zero (or anything else) if you like. But the 'arithmetic' resulting would be arbitrary and awkward, and certainly wouldn't be very useful. Concerning zero: that one cannot divide by zero is a consequence of the fact that zero multiplied by any number is just zero again. The ratio a/b is defined to be that number which when multiplied by b gives a. But for b = 0 and a != 0, there is no such number; while for b = 0 and a = 0, all numbers satisfy the condition. BTW My own favourite bonkers idea is that none of 0,1, or 2 are numbers at all really, but rather just reifications of our basic notions of (respectively) nothing, something and difference. The first genuine number, coming along so to speak simultaneously with the idea of counting, is 3, and this retrospectively brings 0, 1 and 2 into existence, along with all the rest.

[@flatisland]

@@russellsharpe288 I guess that associativity, cummutativity and distributivity are preserved if you only use numbers on either the left or the right side of zero, right (with - * - = -) ? (haven't checked it yet but I'd think that's true) and the unique element on the left side would be -1. Problems only arise if you mix positive and negatives. I think there's more to it that negative numbers (and perhaps zero) are not really "real". I mean, as we have heard in the video, people up until 300 years ago thought so, including bright minds like Pascal and Descartes. Maybe "we" humans were not all that wrong. I could imagine to use a number system that makes use of the dimensions we have. Perhaps spherical coordinates could do it, I guess it's possible to assign only positive numbers there for all points in space.

[@jaydenwilson9522]

Counting begins at 1. Measurement at 0. Take 0 off the number line!!!!!!! If you're counting apples and have 7 apples, there’s no reason to start at 0, because 0 represents having no apples, not some apples. Yes, when you measure, you're essentially counting a length, but there's a key conceptual difference between counting and measuring: Counting deals with discrete objects. You count individual apples, people, or cars-things that come in whole, countable units. When counting, you're concerned with "how many" of something exists. In this context, zero means absence-no apples, no objects. It represents a state where there’s nothing to count, which is why zero feels unnatural in the counting process. Measurement, on the other hand, deals with continuous quantities. For instance, when measuring length, you're determining how much space something takes up, and you can measure in fractions, like 1.5 meters or 0.25 inches. In this case, zero makes sense as a starting point because measurement is often about determining a distance or amount from a defined origin. Here, zero means "starting position" or "no distance yet." In short, putting 0 on the number line confuses magnitude with direction. These two concepts are fundamentally different: Magnitude is a measure of the size or quantity of something. Direction is a conceptual orientation in space, which tells us where this quantity is directed. Confusing these two conceptually different ideas is indefensible logic, as it muddies the distinct purposes that counting and measurement serve.

[@jonorgames6596]

What would you say to students who struggle with (-3)*(-4)=+12?

[@jeremyjedynak]

Two wrongs makes it right.

[@santerisatama5409]

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!

[@SOBIESKI_freedom]

@@santerisatama5409 ***Debt (pronounced "det")

[@ThePallidor]

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.

[@hywelgriffiths5747]

Multiplication by a negative number flips the number about 0, then multiplying a negative by a negative flips it round to the positive side

[@joshuadelacour1106]

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.

[@njwildberger]

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.

[@ThePallidor]

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.

[@WK-5775]

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.

[@christopherellis2663]

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

[@AlenaWoodworking]

Hi Christopher. I have a new channel dedicated to woodworking.

[@jeremyjedynak]

Hopefully box arithmetic won't have to go without negative numbers for thousands of years.

[@sharonjuniorchess]

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 (-).

[@ThePallidor]

Yes but what about multiplication and division?

[@sharonjuniorchess]

@@ThePallidor I have found that teaching kids about positive & negative numbers using different coloured counters makes understanding the laws of integer arithmetic that much easier. Once they have mastered addition & subtraction then it is not such a big step to recognize that multiplicatrion is repeated addition & division is repeated subtraction. However Norman's work on the laws of fractions can also be applied to positive & negative numbers. I can't point you to the actual video but it is worth watching if you can find it (probably in his foundation series).

[@Kraflyn]

link to negative boxes pls thnx

[@KarmaPeny]

In my opinion we should attack the arguments behind real numbers and infinite objects on the basis of their ridiculousness in the real world. Any so-called 'abstract' objects sound 'other worldly' and should be clearly identified as our enemy for their obvious absurdness. Anything with supposedly no basis in reality is inherently nonsensical. But you have set about creating your own alternative fundamental theories that are also heavily abstract in nature instead of having a basis that is entirely founded in physical reality. Given that you appear to not insist on a real-world foundation, I presume you must disagree with part or all of the following: If our human brains are merely finite biological computing devices, then as with traditional computing devices, quantities would only exist as data items within the device. Quantities would then be created as-and-when required. There would be no reason to delude ourselves that all numbers must already exist somehow in some mysterious metaphysical way. It would then be trivially obvious that only a finite amount of 'numbers' could possibly exist at any possible time. The problem of implied completed infinities started around 2 and a half thousand years ago in Ancient Greece when they discovered that the diagonal of a unit square could not be expressed as a multiple of any known length. They could have interpreted this as evidence that '√2' cannot evaluate to a constant and thus it must be wrong to believe that perfect shapes can exist (even imaginary ones). Instead they refused to contemplate that perfect shapes might not be possible; after all, their Gods must have perfect forms. So they decided that irrational lengths must exist, and that this was a secret previously only known by the Gods. A fixed/static length is by definition an unchanging value. An unending addition of non-zero values is, by definition, a changing value. These two cannot be the same thing since a changing value cannot equal an unchanging value. And so whereas '√2' can be interpreted as an algorithm (or computer program) it cannot be said to be a constant. But this trivial contradiction was either not acknowledged or it was ignored (and still is today). A vast amount of flawed logic has followed in an attempt to prop up this ancient mistake. This includes the axiomatic approach in which axioms and rules of logic need not have any basis in physical reality, as well as the belief that we can work with the concept of completed infinities. This is cloaked by claims from authority figures that success is all that matters, and that nothing more is required to justify the mathematical approach to reasoning. An alien from another world might conclude that most of Earth's mathematics is pure garbage. It might say humans are deluding themselves since they blindly accept the status quo and refuse to accept the blatantly obvious fact that it has foundational principles based on complete nonsense! It might point out that if we apply the expression "1 + 1 = 2" to the question of how many standard-sized violins will fit inside a standard-sized violin case, it forms an invalid statement. If mathematical expressions must relate to specific physical scenarios, their validity can be scrutinised and tested against empirical evidence. However, if we perceive mathematical expressions like "1 + 1 = 2" as existing independently of reality, their validity relies solely on subjective interpretations. One person might construct an imaginary scenario where the statement is false, while another argues it's true. Does it truly make sense to allow multiple interpretations of the 'mathematics' game, where the same statement can be valid in one version but invalid in another? This means that we can never be certain about which version/framework is the best version. The choice of which version should be adopted by the mainstream becomes a popularity contest rather than a rigorous evaluation. There is no way to ensure that the prevailing system is the best system and none of its assertions can be considered to hold any claim to being superior to the negation of that same assertion. The visiting alien might add that it is bizarre that humans don't prefer the narrative of mathematics to mirror the symbol manipulations they're actually performing. Weirdly they seem to prefer for it to weave a mysterious fairytale that is supposedly only loosely tied to reality! For instance, after seeing how we humans use negative and positive numbers, an alien might think that it should be obvious to humans that when they use the symbols '+' and '-', they are assigning particular meanings to the signs depending on context, such as forwards and backwards, credit and debt and so on. It should be obvious to humans that there is nothing in the real world that inherently corresponds to a negative sign or a positive sign. These are just arbitrary symbols that we attribute real world meanings to. For example, if a person takes some steps backwards then it makes no difference if we call this the positive direction or the negative direction. This example shows that the signs themselves have no meaning until we assign meanings to them. But the alien might despair when it discovers that humans appear to believe in the existence of non-physical mathematical concepts including the concepts of negative and positive. Humans don't appear to even realise what they are actually doing when they claim to be multiplying two signed numbers. What they are actually doing is multiplying two unsigned numbers and then obtaining the resulting sign from a lookup table. The lookup table was constructed by humans to produce the desired outcomes. They aren't actually multiplying the signs. Worse still, the multiplication of so-called complex numbers and even quaternions operate in the same way. If they operate in the same way as signs then they are signs. Complex numbers consist of 4 signs (as indicated by the dimensions of the lookup table) and quaternions consist of 8 signs. But due to the weird approach taken by humans they came across these concepts in haphazard ways. They don't even realise these are just signs and instead they believe that mathematics has mysteriously revealed these strange concepts to them. They reinforce these weird beliefs by dismissing the fact the lookup tables were devised by humans to give desirable results. Then all that remains is the apparent magical power of abstract mathematics that must (they conclude) form the underlying fabric of the universe. These beliefs are held by people considered to be amongst the cleverest people on the planet. The alien can only conclude that all humans are crazy creatures that are incapable of letting go of their primitive supernatural belief systems!

[@MisterrLi]

"If our human brains are merely finite biological computing devices..." I find it hard to believe that the brain should MERELY be a computing device. Now that we have access to real such machines, the AI systems, we don't have to try to imagine such devices. The finitistic attack against finite systems that claim they include infinity is interesting though. You could set up a theoretical cognitive machine, replacing a human, and see what is possible within that kind of advanced but finite thinking machine. That way you don't have to include consciousness in the thinking process, only calculations using abstract symbols. So, can such a thinking machine have a theory and a concept of infinity? How about representations of natural numbers? The concept that there will always be a bigger natural number, doesn't that lead to the concept of infinity? Because, if there can not be a last natural number, the number of natural numbers must logically be without finiteness and be a number that doesn't have that property. A finite thinking machine should be able to make those simple conclusions and then create infinite sets, not based on a one-to-one construction but on properties of objects representing numbers and sets of numbers.

[@ayscix]

I love this, but I can't help but think, "what is wrong with ridiculous notions in a ridiculous reality?"

[@WildEggmathematicscourses]

@@ayscix our world is just the opposite of ridiculous

[@brendawilliams8062]

Thankyou

[@michielkarskens2284]

“A proper mathematical theory in a geometric sense.” 😂 Circular argument

[@Steve-ro7fj]

Please explain...