Negative Numbers and Arithmetic's missing Chapter

3,793 views Sep 24, 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. *********************** My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at / njwildberger Your support would be much appreciated. Here are the Insights into Mathematics Playlists: • MathHistory: A course in the History ... • WildTrig: Intro to Rational Trigonometry • Math Foundations • Wild Linear Algebra • Famous Math Problems • Box Arithmetic • Elementary Mathematics (K-6) Explained • Year9Maths • Ancient Mathematics • Wild West Banking • Sociology and Pure Mathematics • Sociology and Pure Physics • Old Babylonian Mathematics • Probability and Statistics: an introd... • Boole's Logic and Circuit Analysis • Universal Hyperbolic Geometry • Differential Geometry • Algebraic Topology • MathSeminars • Playing Go • Diffusion Symmetry: A bridge between ...Here are the Wild Egg Maths Playlists (some available only to Members!) • Algebraic Calculus One • Classical to Quantum (Members only)[ • Algebraic Calculus Two](https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-) • Advice for mathematics exploration (M... • Solving Polynomial Equations (Members...[ • The Hexagrammum Mysticum (Members only)](https://www.youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_) • Algebraic Calculus and dCB curves (Me... • Dynamics on Graphs (Members only) • Playlist • Triangle Geometry • Explorations with q-series (Members o... • Six: An elementary course in Pure Mat... • Maxel Inverses and Orthogonal Polynom...************************

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Hello, today I want to talk about the history of negative numbers and talk about uh somewhat

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obscure missing chapter of arithmetic a very foundational missing chapter which I think we

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are just now starting to realize is absolutely uh critical for the foundations of the entire subject

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so we need to go back and think about the large scale structure and Landscape of um of arithmetic

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and this is uh pretty interesting because there's all kinds of sociological aspects because if you

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go back in time and consider the role of negative numbers over the Millennia so for most of History

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negative numbers were viewed as fictitious objects or just you know crazy things and things that were

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not really valid it's really only been in the last 300 years maybe even less that negative numbers

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have become you know reasonably well recognized and and these days are considered to be sort of

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On a par with ordinary positive numbers so we're going to talk about that because um the modern

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view that undergraduates uniformally have is that well the number system is just represented by a a

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line a number line and there's zero in the middle and on this side there's posit positive numbers

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and on the side there's negative numbers that's the modern view that we have that's historically

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not at all the way people used to think people used to think in terms of counting starting

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with one one was always the first number there was one and then two and then three and then

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four and five Etc and for thousands of years probably actually even tens of thousands of

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years that was arithmetic that was the domain of arithmetic okay those numbers of course we know

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that eventually um zero started playing some kind of role the Babylonians had a sense of zero they

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didn't have a symbol for it but they they uh took zero on board in leaving appropriate spaces in

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their sexagesimal representations of numbers so they they had a concept of zero they just didn't

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articulate it it was more clearly articulated by the Hindu mathematicians around maybe the

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fifth century but arguably also uh there's some evidence that U early Cambodian uh Scholars had

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a notion of zero which is roughly contemporary with that okay but then even after zero was

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accepted as you know some kind of number for a long time it was still regarded that zero was

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you know the lowest you could get you couldn't go lower than than zero and in fact uh blae

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Pascal the very famous French mathematician and philosopher famously said I know people

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who cannot understand that when you subtract four from zero what is left is

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zero that's blae Pascal that was the majority opinion for a long time so when people finally

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started trying to you know create models of numbers which included negative numbers they

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often had the negative numbers sort of in on in a different direction so there was like

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the positive numbers in this direction and the negative numbers in this direction and only you

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know slowly perhaps with the Advent of of studying complex numbers did this modern notion of a number

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line come into being okay but prior to that when mathematicians had to solve equations and they

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got negative solutions they would typically dismiss those those were just not valid and

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so uh studies of quadratic equations or cubic equations or or other kinds of equations were

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often um separated into various cases depending on whether you got negative numbers or not if you

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got a negative number you had to throw it out uh probably the first people who kind of got their

History

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heads around negative numbers were the Chinese so around 200 ad there's a famous book in which

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they're solving systems of of equations systems of linear equations and they're definitely working

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with um with negative numbers um I think they also had a system of rods maybe black and red Rod or

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something representing positive and uh um negative numbers so uh the Chinese and probably also the

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Indians um you know had some kind of notion of of negative numbers before that idea became popular

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in the west and that really only happened you know in the 18th century D Hart who was uh the father

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of the idea that geometry should be connected to arithmetic and whose name is now associated

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to The Familiar cartisian coordinates that we all draw pictures of in high school and undergraduate

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days they cart would not have approved of this because he only believed in positive numbers

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so he would be happy with a quadrant you start with zero and there's the x- axis you can start

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with that zero and there's the y- axis but going these other directions no those aren't negative

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numbers those don't really exist okay so we have to appreciate that there was a long struggle uh

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in the ma mathematical world you know to slowly accept negative numbers and to get one's head

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around the even the meaning of arithmetic with negative numbers the famous French playwright

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stendall who is also quite a good mathematics student um famously wrote about uh his struggles

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with this theorem that says that minus * minus equals plus he couldn't understand the logic of

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that like why why why is that and he would ask his teachers and they wouldn't be able to come

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up with a proper explanation and indeed even nowadays it's a an obstacle for uh teachers

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who are teaching arithmetic to Primary School students when you get to negative numbers you

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have to explain that you know -2 * -3 is + 6 and if you try to find some uh some environment that

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explains that it's not so easy to do right you can you can talk about you know -2 means like

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you know instead of having two cows I owe two cows to somebody okay that means I have NE -2

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cows yeah okay maybe um but then what is -2 * -3 how do you explain that that's supposed to be six

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in fact the very idea that um you know that that numbers um are closely connected with measurements

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and typically like segments and um measuring like lengths or or areas or or volumes um that

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that idea which goes back to Greek times was an obstacle for people to accept negative numbers

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because what does it mean to have a negative area what does it mean to have a negative volume right

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sort of Ironically in the algebraic calculus okay we find out that actually the negative values are

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crucial if you want a consistent theory of area or volume volumes or anything like that

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you absolutely pretty well absolutely need to take on board uh integer arithmetic at least okay you

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need negative numbers because the orientation of a triangle determines the sign of of its area Okay

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so so so ironically that um that obstacle that was around for thousands of years actually um

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you know it has to be surmounted before you really understand have have a proper mathematical theory

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of of measurement in in geometrical sense so why are these things important well first of

Why

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all obviously there is a educational aspect to this right this has been a a source of difficulty

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for educators for a long time but it's also interesting from a pure research point of view

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because I claim that if you adopt a historical point of view okay the historical point of view

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is that the most important numbers are positive numbers and that's where we do arithmetic okay

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that is definitely not the modern point of view if you go to a university class in your first

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week of undergraduate course you'll you'll find out that the number systems that we use are all

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sort of modeled on a line there's well there's no natural numbers that's quickly introduced

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but then we quickly move to the integers you know and then the rational numbers and then if

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you're in an analysis class to the real numbers or perhaps complex numbers but this idea that

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that we should actually think clearly about a chapter of arithmetic which is solely concerned

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with the historical dominant object which is the natural numbers we should have an arithmetic which

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is separate from these larger arithmetics and has almost a primary kind of place right in the center

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of our arithmetic well that's what turns out to happen with this new box arithmetic that I've

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been developing in my math foundations series okay so box arithmetic is a dramatic new large

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scale enlargement of of arithmetic which it makes it extend in lots of other directions but gives

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it a consistency and a uniformity and I think sort of an aesthetic of appeal that currently we don't

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have and when we're developing this box arithmetic we find out that sort initially what we do is we

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adopt a A Primitive point of view um the basic objects are natural numbers which are just boxes

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of empty boxes that's what a natural number is and it takes some sophistication as I hope some of you

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have have seen the videos where I I explain how you introduce negative numbers really relies on a

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kind of a corresponding mathematical notion to the the physical development of particles and anti

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particles coming from the Dirac equation and Paul D's crucial insights in the 1920s 1930s and so um

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so there's an aspect of sort of modern physics which ends up motivating some of the structures

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that we need to have to have a consistent uh theory of negative numbers in this box arithmetic

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world but it's also legitimate to say okay that's fine but what about if we just restrict ourselves

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to the sort of the Primitive boxes where we don't have anti- particles or anti objects that's a

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consistent world of arithmetic so if you just have natural numbers and we'll include zero okay so 0 1

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2 3 4 5 Etc so the two operations are addition and multiplication okay and they're they're Pleasant

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um they uh they're both commutative they're both associative and we have a distributive law we have

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an identity element for addition which is the zero and we have an identity element for the

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multiplication which is the one okay so what kind of a an object is that well most undergraduates

What

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would struggle to say what that actually is so they learn about Rings okay so when you extend

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this to the integers well then you get what's called a ring and the ring has the property

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that um you also have negatives okay so there's like a minus one so you have have negatives and a

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lot of algebra is built on ring Theory what about this more restrictive thing that's represented by

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the natural numbers that's called a semi- ring okay and it's much less studied than rings and

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even the terminology suggests that well a semi- ring is you know something weaker than the main

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object of interest is obviously a ring and then if you weaken the conditions in some

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suitable way then you get a semi- ring not not quite so important that's very different from

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the historical the historical tradition though where arithmetic with natural numbers was the

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main game for thousands of years okay long before anybody even dreamt of negative numbers so um in

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my box arithmetic um series I'm going to be emphasizing this that that there is um a Val

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kind of chapter of algebra having to do with semi- rings uh but also um sort of the the

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semi- ring equivalent or analog of a so-called integral domain so the natural numbers have

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the property that if you multiply any two of them and you get zero it means that one of them has to

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be zero that's what's usually called an integral domain property but it's usually in the context of

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rings so in this more restricted context of a of a semi- ring it's a very unpleasant terminology um

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we might say this is like a I know a semi- domain or something like that but uh the reason why this

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is all going to be important because in the Box arithmetic course I'm going to be explaining how

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to understand fractions which are the sort of semi- ring equivalents to rational numbers so

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there certain operation where you take quotients in a certain way and then you you extend your ring

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if it's an integral domain to a field okay so there's a a simpler version of that where we

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just work with natural numbers and we do the same thing okay we get what in my math foundations

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course I call fractions just things of the form 3 over5 no negative numbers at all okay but you have

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a valid arithmetic of such fractions including addition and multiplication and crucially division

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assuming of course that you're you're not not having zeros in denominators but aside from

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that you have actually a nice framework and undergraduate courses don't even have a good

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language to describe what's going on here but that's going to be really crucial in the Box

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arithmetic world it's going to turn out that that that construction works very generally for for all

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of the boxes not just for the the natural numbers of the polinomial so it's it's it's it's it's it's

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quite an exciting development so I want to alert you to this idea of trying to separate in your

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mind okay these various arithmetical systems that you have in your mind most of them based on fields

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I want you to separate out the natural numbers okay the natural numbers are have a good claim to

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being the most fundamental arithmetical system that's by far the oldest one that we have and

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so we should be giving it real proper names not names which are subordinate to more complicated

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things that only came in the last few hundred years right so there's a a disconnect between

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the mathematical reality and our current modern view of it it's a very interesting sociological

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question how do we how do we get ourselves in these binds well of course one explanation is

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because well it's not convenient for us to think too carefully and precisely about arithmetical

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issues because if we did well then we would have to acknowledge the obvious sort of silliness of

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so-called real number arithmetic which is you know basically completely Bonkers I mean people

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can't even give any non-trivial examples of of this arithmetic seriously you ask an analyst

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give me some some solid examples of real number arithmetic the best they can come up with is

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stuff like uh Pi + Pi is 2 pi really or you know cosine 7 + S of 5 is uh cosine of 7 plus s of 5

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I make fun of this all the time but it really is quite sad but you see if we looked carefully at

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arithmetic then the obvious um you know and and ridiculous weakness of real number arithmetic

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would become so blindingly obvious that we'd all you know have to be completely embarrassed and

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so it's best not to look too carefully at these things but there are great benefits in looking

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carefully at fundamental arithmetic as you'll see in the Box arithmetic series of lectures

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