Box Arithmetic with Polynumbers
Box Arithmetic with Polynumbers | Math Foundations 238 | N J Wildberger
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We introduce fundamental terminology and notation for Box Arithmetic, putting the duality between mathematical objects and their anti objects at the centre of the development. We also move to a more geometric, visual approach with actual boxes over inline representations.
Is this the arithmetic of the world?
Big thanks to my Patreon supporters and Members of the Wild Egg maths channel.
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I'm Norman Wildberger today we're carrying on with this new approach box arithmetic where we
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are trying to put the Duality between objects and anti-objects sort of foremost in our arithmetic
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so this is a very novel approach to to arithmetic because it's really building on a 20th century
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physics understanding discovery that of the fundamental Duality between particles and
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anti-particles in our sort of subatomic World which is really a remarkable development in
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20th century physics and really ought to rank up there with relativity and quantum mechanics in
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terms of just the significance of it so what we're going to do today is to proceed by more or less
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concentrating on notation nomenclature I'm going to introduce variants to the things that I've
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already been talking about sort of now with this view of trying to put this Duality Center Stage
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for us okay so roughly what we're going to do is if we have some object okay then the anti-object
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will be denoted by a hyphen object that's our general orientation okay that will be used quite
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frequently and correspondingly when we want to talk about both these objects whatever they are
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and the anti-objects together I propose to include them in a a bigger sort of category
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or classification by using a capital so in this case capital O object okay so a capital O object
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will include both these little o objects and there are anti-objects so that's just a very
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general kind of thing so let's see how it works for these um these boxes and the arithmetic of
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them so the basic idea is that we're going to build up arithmetic using boxes and now
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centrally crucially we are going to suppose that we have two different kinds of boxes black boxes
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and red boxes okay not just one we have two different kinds and these are somehow opposite
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or antes of each other so we'll agree that the black ones are to be denoted by boxes these are
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the boxes and the anti-objects or a boxes will be denoted in Red so if we have some box with
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something inside it let's say it's the Box called B and then we consider a very identical box which
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has the same things inside it but now has a red boundary instead of a black boundary then we'll
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refer to this as b super a or the anti of B okay now both these little B boxes and the a little
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B boxes are included in the broader classification of capital B boxes okay so when we write capital
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B then we're referring to both objects in black as well as their anti-objects in red
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okay so to start the story we look at the simplest kind of boxes which are the zero boxes but now
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there is two of them not one okay so our world starts with two things the empty Black Box just an
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ordinary box and it's anti-object you know in red it's a red box so I'll refer to this first one as
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zero and we'll refer to the second one the red one as anti-zero or a0 consistent with our notation
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and when we write it numerically we'll write just an ordinary zero for this black one and a zero with an a superscript for the anti-0 or a zero and both of these together will be
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included in this type called Capital zero okay so when we talk about a capital zero we could
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be either talking about this zero here this black one or this anti-zero here this red one
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okay what can we do with just zeros well we can add them now when we were just talking about msets
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before we just had empty M sets and then it was obvious that the empty M set plus the empty M set
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was the empty M set but now things are a little more complicated because we have these two kinds
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of fundamental zero objects so we have to specify the addition for all possible pairs so a black
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empty box plus a black empty box is another black empty box and similarly a red anti-0
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plus a red anti-zero is a black box or a zero and otherwise black plus red equals red and red
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plus black equals red so we could say this also in the language of zeros and anti-zeros so zero plus
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zero is zero zero plus anti-zero is anti-zero anti-zero plus zero is anti-zero and anti-zero
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plus anti-zero is zero so this is a pretty fundamental kind of thing and we Justified this
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because we want the the arithmetic that we are going to create to be somewhat consistent okay and
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um and so this is a good way to proceed okay now we can extend this consistently to sums of
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more boxes or anti-boxes in this case just zero boxes so for example if we have uh an anti-0 and
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a zero and a zero and a zero and an anti-zero and an anti-zero and we add them all up okay so
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then we can do it in in a number of ways but we can just think about really what what's crucial is the number of anti-zero so that's going to determine the the type of the resulting thing
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in this case there's an odd number of red ones or an odd number of anti-zeros so the result is
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going to be another anti-zero zero anti so the sum depends only on the parity of the number of
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anti-boxes if there are an even number of anti-boxes then the result is zero well if there are an odd number of antibboxes then the result is anti-zero okay so now we're going to
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define recursively the um what a box is what a capital B box is
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and we're going to start the initial objects are the two zero boxes the zero box and the
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anti-zero Box okay and we declare that these are both capital B boxes okay and then there's only
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one more thing which is a kind of an inductive definition so we agree that if we have X Y
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up to say Z these are already defined boxes capital B boxes then we'll agree that so are
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these two things the Black Box containing X and Y and so on up to Z call it a and the
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anti of that object which is a red box containing exactly the same elements containing X Y up to Z
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okay so this thing here's a box because it's black this thing is an anti-box because it's red and these are antes of each other because the contents are otherwise exactly the same the only difference
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between these two is the color of the bounding box okay and just a little bit of additional notation
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we'll say that in this case X Y up to Z these are the elements of the capital B boxes a or anti-a
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so this is really the the story of the objects that we're going to be considering we're
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just going to create recursively more and more complicated boxes capital B Boxes by recursively
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creating Box by nesting them and once we have nested boxes then we can use those as elements
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of further nested boxes and that's that's what our arithmetic is going to consist of and that's uh
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going to be a much wider and bigger arithmetic than we currently have okay much wider and bigger
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to distinguish these two types of capital B boxes so we'll say that a box is a capital
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box of positive type and will denote the positive type by this particular sample like a plus sign
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with a circle around it okay not to be confused with addition so this is just a symbol denoting
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this positive type which is associated to a box that is something that's got black around it
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similarly an anti-box or a box is a capital box of negative type it
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will be denoted by a minus sign this is not a Theta it's a minus sign enclosed in a circle
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okay so these are the two possible types of a capital box you know Capital box you can ask is
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it either a positive type or a negative type and to tell you just look at the color of the outside
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box if it's black then it's positive if it's red it's negative independent of what's inside
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okay so then we can express this summation of zeros this aspect of counting the number of
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anti-boxes by saying that the sum of zeros is a zero again whose type is the product
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of the types of the constituents okay well so we have to Define what we mean by a product of type
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so that's sort of the obvious thing so we'll agree that positive times positive is positive
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negative times negative is positive positive times negative is negative and negative times positive is positive so that's just the same as it is in ordinary life okay so that's how we can say
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um how to determine that the type of uh sum of of zeros let's say but in fact of more General boxes
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so for example here uh here we've got three black boxes and three red boxes these are
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the corresponding types underneath the Plus or positive type positive type negative type positive
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type negative type negative type to determine the type of the sum okay which is going to be another
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zero but what kind of zero what type of zero we just form the product of all these types okay and
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well that's going to be negative time because there is an odd number of negatives
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so you get a negative and that means that the resulting sum is an anti-box or anti-0 in red
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okay The crucial Annihilation creation sort of property that we we insist that
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objects and anti objects have is that if we have a box capital B and it's anti-box
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ba b super a or anti B if they both occur in a box m a capital B box M okay then they
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mutually annihilate each other this means that they both disappear they both disappear
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that's the crucial relation that we have between boxes and their antes okay so let's look at some
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examples so here is a box it has two elements in it and these elements are antes of each other
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because the contents are exactly the same there's two red boxes inside them but one's black and
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one's red so these are antes of each other that means they mutually annihilate and the result is at empty box or a0 here is a red box that has three elements inside two are these red boxes
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and here's a black one the contents of all three are the same so this first red box for example
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this is an anti-box it's the anti of this this box here because the contents are the same there's two
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black boxes inside them so these are opposites or aunties they're going to mutually annihilate each
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other they're going to disappear and we're just going to end up with this remaining fellow so we end up with this red box with a red box inside and two black boxes inside it over here we've got a
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black box with four elements inside and um there's some cancellation because inside this first box
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there's a zero and an anti-zero and they're gonna then mutually annihilate each other so we're just
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going to get a an empty box there for that one there and over here this one here and this one
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here are anti's of each other because the contents are the same there's two red boxes in each of them
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but one's black on the outside the other one's red on the outside so these are aunties they're going to mutually annihilate each other poof they're gone so that just leaves us with this
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red one down here in the corner additionally but now we have a black box and empty box and here's
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a red version of that so this is zero this is anti-zero they mutually annihilate each other
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there's nothing there left and we just end up with a single empty box of the resulting thing is zero
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okay so now we can introduce natural numbers and then integers so a natural number first of all is
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a box of zeros okay now notice there's that's a it's a small B box that means we're talking
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about a black thing okay and it's a small Z zero that means we're talking about an empty Black Box
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so it's a box consisting of a lot of empty black boxes that's a natural number
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so aside from zero the fundamental new natural number is this one here it's a black box with
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a single black box inside it that's the number one so in art world that is the number one but we have
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other things which are also natural numbers for example this thing here is the number four what we usually call the number four and this thing here is also a natural number because it's also
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a box of zeros it's a box that has no zeros in it but nevertheless it's still a box of zeros and so
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that's zero so zero also is counted as a natural number now a big departure from usual arithmetic
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and a natural number okay just consistent with our notation to get the anti of something we
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just stick an a with a hyphen in front of it so an a natural number is an anti-box of zeros
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so for example here for ante there's four black boxes inside a red box so this is the
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anti of the four above here and the anti of this thing is zero anti so these are both uh
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anti-natural numbers okay so we got these two types little Nat little n not end a little n not
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and now a natural number with a capital N so a capital natural number is either a natural
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number or an a natural number so we're going to enlarge the view of what a natural number is at
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least when it has a capital in front so the use of the capital is yes indeed rather crucial here to
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keep the distinctions that we want okay so the the capital N Nat is now a type
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that includes both natural numbers and their antes
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these anti-natural numbers these are objects which are not found in traditional mathematics
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we've been doing mathematics for thousands and thousands of years and I think this is really the first time where we have anti-natural numbers and I'm
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claiming that they are actually playing a very important you crucial role in our arithmetic
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okay and to bring that to the fore we introduce the idea of an integer
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an integer without a capital is a box of capital zeros it's a box of capital zero so that means
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it's a it's a box it means it's black but the elements in it can be either zeros or anti-zeros
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an a integer is an anti-box or an a box of capital zeros so it's again a box but now a
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red box consisting of a whole bunch of zeros and the zeros are capital zero so they could be either
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empty black boxes or empty red boxes so we have these two type little i int and a
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int and now consistent with before we'll define a capital I integer Capital integer is either an
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integer or an anti-integer and we'll introduce Int now to refer to either integers or anti-integers
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so for example this thing here is a natural number but it's also an integer it's a box of zeros
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this is actually a little I integer
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this thing here is an a integer it's a red box consisting of zeros
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this thing here is a black box also consisting of of capital zeros so this is another integer
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this is what we would normally call minus three is what we call three this thing here
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we don't have a traditional name for it so we have to make up a new name this is three ante
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the anti of this thing we also don't have a traditional name of here's what it looks like
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so this would be -3 anti or three bar anti these are all examples of capital I integers
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right okay so additional boxes this fundamental operation okay which at the multi-set level
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we've seen is absolutely fundamentalists what distinguishes the data structure of multi-sets from all the other ones is that multi-sets support this most Elementary of operations the additional
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you just dump stuff out into another box so we're doing very much the same thing here but
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we have to just be a little bit more careful a little bit more subtle because we have to keep
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in mind the colors of the boxes either black or red so to add Capital boxes what are we going to
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do we're going to form a new capital box whose elements are all the elements of the original
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Capital boxes and whose type is the product of the types of the original capital boxes
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so we're going to do just as we did before we take the contents of all these Capital boxes that means they could be boxes or anti-boxes they're the outsides could be black or red I'm going to dump
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them all out okay the the elements all get put into another box but the color of that new box now
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has to be determined it's determined by the color of all the various original boxes that we are
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taking the stuff dumping out of okay so we have to take the product of the types
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of the original boxes to determine the type uh or the color of the final box
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so for example here we're adding these three boxes all right so first what we do is we just
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take the elements and dump them out into a box so these two guys here get copy there these two
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guys here are copied down here and these guys here are copied here okay and then we put this in a box
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but we have to determine what's the color of this bounding box so we go up to here and we say okay
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these are type well this is this is a negative type this is positive type this is positive type
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so the product of those types is negative that's why we have a red bounding box
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but there is some simplification possible here because this thing here and this one here are
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antes of each other so they're going to cancel okay and also this zero and this anti-zero are
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also going to cancel so we're just going to be left with this object here and this one here
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so we get an anti-box overall since negative times positive times positive is negative
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and we note that the sum of natural numbers is also a natural number and the sum of capital integers is also a capital integer we get however an a integer
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if there's an odd number of a integers in the sum and otherwise we get an integer
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okay and now we have a result that we can convince ourselves of that in in this great generality of
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addition of arbitrary Capital boxes that it's actually commutative and it's associative
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now I can talk about the multiplication of capital boxes so to form the product a times B of capital
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boxes A and B we take the capital box of all possible sums a plus b or little a plus little
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B where little a belongs to a in other words is an element of a and little B is an element of B and
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of course we have to take each element with the corresponding multiplicity that is we're going to
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take each element no matter how many times the similar kind of thing appears where the type
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of a times B so the resulting type of this this big box is the product of the types of A and B
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so it's the same rule as we used for addition of boxes also for multiplication
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so in terms of a special case the special case is just the products of zeros of capital zeros
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and that multiplication table is just like the addition multiplication table so 0 times 0 is 0 0
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times anti-zero is anti-zero anti-zero times zero is anti-zero and anti-zero times anti-zero is zero
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and I remind you that we had justification we had good reason for defining things this way
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but curiously it is a table which is identical to the addition table for Capital zeros
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and of course we notice as we have before that there are some Curiosities about this maybe even some disconcerting aspects in particular this
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property that's zero times anti-0 is anti-zero that's disconcerting because it's different from
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the familiar rule that zero times anything is zero right that's a rule that we're used
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to for any a but this does not hold here it holds much of the time but it does not hold
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when capital A is an anti-zero in that case the right hand side is not zero it is rather anti-zero
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okay and here's an example so here's a box a and here's an anti-box B okay The Roof Is
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Missing there there it is there okay and so to take their product what do we do well we have to
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take all possible sums so we start with this one here maybe this this box here plus this one here
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and then this one plus this one and this one plus this one so this first row we're adding this empty
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box to each one of these three and so now for the second row the second element here first of all
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I'll add it to this thing and then add it to this one and then add it to this one we get these guys here and so now we have to evaluate these by our prior motion over Edition so we do that so adding
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zero to something doesn't change it so we get uh just this thing here this thing here this thing
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here and then when we do the sum here okay here we have summation so we're going to get three black
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boxes inside but the outside is going to be black because negative times negative is positive
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over here we have negative times positive so we're going to get an anti-box with three things inside
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and uh is that right with three things inside
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no not with three things inside that should actually be two things inside shouldn't it um because we're just adding this thing here to that yes so pardon me um this thing here uh should
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be eliminated it's just a red box with two black boxes inside it and lastly this thing here have
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I done that one right um we've got four black boxes yes and the type of the bounding box is
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a negative so it's really red okay so it's almost correct a calculation of a times B
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and then we have a theorem that multiplication of boxes is also commutative and associative
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now that we have both addition and multiplication of General boxes we can ask about how they relate to what extent does the distributive law which is this
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thing here hold that a plus b times C should be a times C plus b times C
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Okay so we've already observed that the distributive law doesn't always hold and
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that's evident even just at the level of the zeros the capital zeros because if we have
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a zero plus an anti-zero and we multiply it by anti-zero and zero plus anti-zero is anti-zero so
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we're getting into zero times n to zero which is zero on the other hand the distributive law asks us to calculate zero times anti-zero plus anti-0 times anti-zero this thing here is anti-zero this
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thing here is zero and their sum is anti-zero so the distributive law does not hold generally
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that's a little bit disconcerting but it's actually not that bad when we realize that
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we have the following result that the distributive law actually holds for a b and c boxes
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that is their black things okay so I'm going to just show you why that is so in
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considerable generality suppose that there are three boxes are the a b and c and a has say
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elements X Y up to Z B has elements k l up to M and C has elements p q up to R okay then when we
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calculate a plus b times C so there's a plus b just the the sum of the two uh M sets and it's
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going to be black okay times C here C which roof is up here um so what we're going to get is we're
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going to get the M set of all possible sums and since it's Black times black we're going to get
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black okay so it's positive times positive is going to be positive so we're going to get elements X Plus P X Plus Q up to X plus r and then y plus p y plus Q up to Y plus r and then uh
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Z Plus P all the way up to Z Plus p z plus Q Z plus r is from that top row and then down here
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K plus P K plus Q up to K plus r l plus p l plus Q up to L plus r all the way up to M plus p m plus Q
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up to M plus r on the other hand when we look at the other side of the distributive law a times C
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plus b times C so here's here they are again okay so we have to look at a times C plus b times C
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then what do we get well here's a times C so X Plus P X Plus Q all the way up to X plus r then
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y plus p y plus Q up to Y plus r all the way up to Z Plus p z plus Q up to Z plus r and something
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similar for B times C and then when you look at these things and compare their sum with uh this
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you see that you're getting exactly the same elements okay so yes indeed we do get a plus
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b times C so things work uh because everything is in black and so the black times black is just
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is black okay so we see that the the problem with the distributive lie is not really a deep problem it's just a sort of a surface problem having to do with the color of that outside box
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okay so now let's go to poly numbers okay which is um actually sort of where most algebra takes place
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okay but now we have this sort of enlarged point of view so first of all a definition a poly number
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or a poly is a box of capital integers box so it's black okay that's a small poly that's a type an
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anti-poly number or a poly is an a box of capital integers that's type a poly there okay so then a
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capital poly number or a capital poly is either a poly or an anti-poly and we have the type Capital
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poly so this is a little bit different because formerly we call these things integral polynumbers
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so roughly speaking I want to elevate integer arithmetic that's sort of the the nicest
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arithmetic in some sense okay and so our polys are going to be associated with integer arithmetic
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that's sort of really nice algebra so for example the fundamental new poly that we get okay this
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Alpha thing is where we have a a box inside a box inside a box and what's inside here is the number
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one so this is a box containing one Alpha squared will then be the the square of this thing so we
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have to multiply this thing with itself that means we have to make a box whose element is the sum of
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this thing in this thing so when we add these two things here we're getting one plus one which is two so we're getting the the the box which has a single two in it and so on Alpha cubed will be the
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Box containing a single three Etc so for example we can combine these things so this thing here it's a box it's got a zero there a zero there a three there so this thing is 2 plus Alpha cubed
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but in the world of poly there are now more possibilities because these examples are all
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just black things but we can have black things and red things so in particular this thing here
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is of Interest so so we have a black box which contains a red box which contains a black box
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so this is minus alpha or our preferred notation Alpha bar why is that well because if you take
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Alpha which is this and you add it to this thing when you add these you're going to get a black box with this thing inside it and this thing inside it but these are edges of each other so
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they cancel and you're just going to get zero all right another possibility is this thing
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here which is a black box containing a black box containing a red box that's Alpha to the minus one
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or another way of looking at it it's the Box containing minus one or one bar and why is
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that well because if we multiply Alpha here's Alpha with this this object okay then what we
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have to do is we have to form the Box because we're taking the product to two boxes of this
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box here and this box yes now when we add these two things okay when we add these two things
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well the contents will annihilate each other and we're just going to get an empty box which is one
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so that's why this thing deserves to be called Alpha to the minus one because when we multiply
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it by Alpha we get one or Alpha the zero and you can check that another possibility is to take a
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black box containing a red box containing a red box that's minus Alpha to the minus one
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so um that's just sort of the start of polynomial arithmetic but it's as we've seen before
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um in some sense a bigger arithmetic than usual because of the existence of these negative powers
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of alpha playing essentially just as important role as positive Powers so this is a bit of a
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departure from the calculus point of view towards polynomials it's a bigger picture
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and very briefly then of course the next stage up there's a whole hierarchy of of stages or
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maybe we should say the next stage down are those multi-numbers okay and similarly we can define a
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multi-number or a multi that should be a multi is a box of capital police well an a multi number or
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an a multi is an a box of capital polys and then a capital multi-number or a capital multi is either
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a multi or an anti-multi and we have this new type and this this world here is something of
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a new world we haven't explored it very much okay but there will be some interesting twists in this
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in this multi-number world as we'll see but we can see this you know entire Chasm of arithmetic
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below us so we're just at the top rungs with the usual algebra polynomial algebra and you
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know there's this vast Chasm underneath us with all kinds of complexity because these things can be can be uh you know layered and we could get blacks and reds uh layering in all kinds
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of interesting ways and we should expect that the behavior of these objects depends sort of on the
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the relative colorings of these various nested layers uh so in principle this is a a huge and
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and maybe sort of almost mind-blowing arithmetic that we have in front of us but lots to explore
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and of course a crucial question is you know is this the is this the arithmetic of the world
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I'm Norman Wildberger thanks for listening