Points and Lines in the Affine Plane
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This is the first lecture in the Algebraic Calculus One course, which will present an exciting new approach to calculus, sticking with rational numbers and high school algebra. The course will be carefully framed on careful definitions, explicit examples and concrete computations. In this first rather introductory lecture, we introduce the basic geometric framework of the affine grid plane, or Cartesian coordinate geometry, and review what hopefully should be familiar concepts around points and lines. But we will also introduce a perhaps novel point of view coming from projective geometry, employing the important ideas of proportions. This first lecture has two parts. The second part is given by Anna Tomskova and is Projective Points and Lines. ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. 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 will be 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 https://www.patreon.com/njwildberger Your support would be much appreciated.
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[Music]
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hello everyone I'm Norman Wahlberg and
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welcome to the algebraic calculus so in
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this lecture we're going to be looking
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at the geometrical underpinnings of the
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subject which is basically sort of
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linear algebra or affine geometry and
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also the corresponding algebraic set up
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that captures those geometrical notions
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logically it's usually the algebraic
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setup which is primary for us but we
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like to have pictures in mind as well
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and notably the pictures are not those
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of Euclidean geometry so the usual
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pictures that you find in most calculus
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textbooks are not exactly in the
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direction where to thinking we're
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adopting rather a more I find geometry
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point of view and I want to explain that
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because it might not be that familiar to
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you so roughly what we're doing is we're
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thinking about geometry in the plane
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with a notion of parallelism first and
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foremost in our minds so the
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perpendicularity is not actually going
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to play a role ok so here is a picture
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of an XY plane maybe a little bit
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different from what you're used to
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because these axes don't appear exactly
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perpendicular to you and you can see
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that the spacing in the X directions and
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the spacings and the Y directions are
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not exactly the same
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although along the x direction we have
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equal spacing and along the Y direction
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we have equal spacing but this
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separation here in this separation here
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are not really cool corresponding ok so
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that's the the basic set up we might
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call that the affine plane and a natural
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question is you know how do you organize
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such a thing and you know it turns out
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there's a a Renaissance artist kind of
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point of view that allows one to create
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such a thing if you just have a
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draftsman stool that allows you to
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create parallel lines so suppose we have
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such a kind of a fine grid then we can
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talk about points and lines which are at
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least the
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first fundamental objects of study so a
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point for us will be something like this
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a equals 3 2 that represents the point
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which is 3 in the X Direction 2 in the y
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direction
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there it is there so what we do is we go
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to the x position here that's labeled by
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3 and then we draw a line through that
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which is parallel to the y axis and we
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simultaneously go up to the position on
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the y axis corresponding to 2 and draw
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through that a line parallel to the x
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axis okay and then it's the meet of
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these two lines that determines that
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point a the other kind of ingredient
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that we'll need is that of a line ok and
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a line for us is an equation like this X
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plus 2y equal to this equation describes
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this line maybe the name of this line is
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L ok it passes through the point 0 1 it
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also passes through the point 2 0 and of
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course through many other points so
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we've already become sort of familiar
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first with this basic setup where we do
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not assume perpendicular kind of
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standard Euclidean graph paper kind of
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framework ok this is somewhat more
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general all right so now let's make an
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official definition basically an
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algebraic definition we'll say that an
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affine point is a pair of numbers in
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square brackets for example this thing
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here
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there's square brackets and there's two
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numbers 1 and 5 over 2 and they're
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separated by a comma there's a left
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bracket on the side and a right bracket
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on this line these are square brackets
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ok so this is an affine point and in
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this case it's name is capital a so we
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say a equals the point 1 5 over 2 now in
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this course when we refer to numbers the
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default is usually rational numbers
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because this is a very algebraic
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approach so for us real numbers are
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highly problematic and we're going to
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avoid them almost entirely surprisingly
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but we are going to allow the
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possibility of extending our
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considerations to other fields also in
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particular later on to finite fields so
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ultimately the same kinds of definitions
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will also work if we replace our notion
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of number with the notion of a finite
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field number whatever that is you don't
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need to understand what that is at this
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stage okay but currently the default
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when I say number is it's a rational
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number like a fraction now this point is
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a purely expression oriented object but
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it can be interpreted geometrically in a
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wide variety of ways so here is one
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possible interpretation here is one kind
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of representation of an affine plain and
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then the point here a is one five over
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two here's another representation a
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different kind of affine plain one and
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five - there's the point a in this
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scenario so both of these are correct
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and illustrate this idea that it's not
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really the geometry that's primary it's
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the algebra that's primary and a given
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algebraic object will have many
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different geometrical manifestations or
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interpretations okay so that's that's
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our rough orientation to things okay
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we're gonna start with the algebraic
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thing we're gonna say precisely what we
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mean when we write something down
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typically as an expression involving
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certain symbols and then we're going to
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interpret that in perhaps a different
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number of ways geometrically alright and
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our second definition is that of an
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affine line okay so a line for us is an
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equation in the variables x and y of the
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following form ax + b y equals C where a
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B and C are numbers with the crucial
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condition that a and B these two numbers
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here are not both 0 one of them can be 0
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but not both of them that's what a line
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will be for us so for us a line is not a
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set of points
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is not a set of points even were a set
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of points that we would have to worry
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about you know how to encode this set of
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points onto our hard drive which is
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basically an impossibility unless you're
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working over a finite field then you can
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do it but over the rationals you can't
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describe all the all the points on this
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and that doesn't even have any meaning
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to me because there's all these
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difficulties associated with dealing
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with numbers which are so big that they
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don't fit into the universe okay so I
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don't want to go there so it's much
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better if we just think about a line not
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as a set of points but actually it's
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just an equation now however we do have
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to say that there's some there's some
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ambiguity here because a given equation
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can be expressed in alternate ways so we
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could perform algebraic manipulations
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for example the line X plus 3 y equals 2
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okay we can multiply that equation by 2
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to get the line 2x plus 6 y equals 4 and
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we would want these two to be the same
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line or maybe we could bring them two to
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the other side here get minus 2 plus X
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plus 3 y equals 0 or maybe we could
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solve for y bring the X to the other
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side and divide by 3 to get y equals
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minus 1/3 X plus 2/3 so these are all
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the same line for us now maybe that's a
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little bit unsatisfactory and I hope you
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do feel that a little bit okay that it's
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a little bit ambiguous because now how
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is our computer going to tell whether
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this line is equal to this line and that
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in fact will motivate the short well
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talking about an alternate form of
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describing lines involving projective
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ideas okay so we're going to get to that
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but at least at this stage here with
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this connect with standard high school
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terminology so we now have points and we
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have lines as linear equations in X and
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y so we know what a point is we know
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what a line is now we have to describe
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the relationship between them we say
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that a point a which is say x0 y0 lies
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on the line L given by the equation ax
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plus B y equals C precisely when a x0
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plus b y0 equals c in other words
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the equation is satisfied if you plug in
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the values of x0 y0 let define a into
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X&Y there's alternate notation or
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terminology for this we equivalently say
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that the line L passes through a or
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sometimes we say that the point a is
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incident with the line L great ok
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another important definition is this
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basic idea of parallelism which I said
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was at the heart of affine geometry so
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we should say when two lines are
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parallel so the line l1 say with
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equation a 1 X plus B 1 y equal C 1 and
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the line l2 which is a 2 X plus B 2 y
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equals C 2 they are parallel precisely
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when the following relation is satisfied
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when a 1 times B 2 minus a 2 times B 1
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equals 0 now in this situation we will
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sometimes write that l1 is parallel to
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l2 by using two vertical bars to
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represent parallelism this is a very
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important kind of expression that we're
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gonna be seeing quite a lot in this
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course alright so this is our first
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example of it it's describing
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parallelism of two lines and I've
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arranged the indices here so instead of
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having a B and C and D E and F it's
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maybe a little bit easier to remember
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this if we go to the additional work of
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using indices so ones for this first
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line and twos for the second line it
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somehow makes things consistent and
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reduces the number of variables
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altogether even at the cost of a slight
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amount of extra work with the indices so
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that's something that would have to play
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off against each other quite a lot in
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the course you know
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so anyway I just point that out that
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sometimes it's nice to use indices to
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sort of organize or book keep the
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objects that we're interested in okay so
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there is the definition of parallelism
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for lines a few more hopefully familiar
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definitions will say that points a B and
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C are collinear precisely when they all
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lie on a given line L and we'll say that
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lines K L and M say are concurrent
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precisely when they all pass through a
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given point a so for example here's a
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picture of three collinear points they
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all happen to lie on this line so the
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point a point B Point C it's a point on
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a here is the y intercept of this line
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it's the point 0 2 at the point B here
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is the x intercept of this line it's the
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point 3 comma 0 and here's another point
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on it say C which has x-coordinate 1 and
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then to find the y coordinate we would
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sort of plug x equals 1 here and solve
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for y and get 4 over 3 here is an
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example of a point with 3 lines through
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it so these are three concurrent lines
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the point is to 1 and lines are well
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here's the line x equals 2 yeah that's a
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valid line here's the line y equals 1
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which is a valid line and here's some
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other line it happens to be X plus y
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equals 3 and we can check that that
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actually does pass through a because 2
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plus 1 does equal 3
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okay so we have collinear points and
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concurrent lines now we want to augment
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our number system a little bit think
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about numbers in not just an absolute
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way but in a kind of a relative way so
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for example if I have two objects and I
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want to sort of compare them let's say I
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want to compare their weights one way is
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to sort of weigh this one and get a
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number and then weigh this one and get a
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number and then compare those two
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numbers but another way is to not worry
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about the actual absolute measurements
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of those weights of the two things but
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just the relative weights
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so maybe we can establish that this one
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weighs twice as much as this one without
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actually individually weighing them both
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so we get a proportion the weights are
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in the proportion two to one so this
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kind of relative notion of numbers was
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very important to the Greeks who thought
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geometrically okay rather than
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arithmetic li primarily but so they had
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maybe one segment like this and maybe
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another segment
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you know half the size and they would
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say well this the proportion this
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segment to this segment is two to one
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that kind of thing so we want to allow
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ourselves some flexibility in terms of
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being able to use this kind of notion of
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number sometimes okay so we're gonna
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introduce the idea of a proportion here
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okay so what is a proportion well there
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are different kinds of proportions the
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simplest ones are the two proportion so
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a two proportion is an expression A to B
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where there's a colon here okay so a
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colon B where a and B are numbers in
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other words typically rational numbers
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not both zero one of them could be zero
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a could be zero but then B cannot be 0
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or B could be 0 in which case a could
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not be 0 with the following convention
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that we agree that a certain proportion
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is equal to another proportion in a
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certain situation we say that a to B
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equals C to D precisely when ad minus BC
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equals zero hopefully that looks a
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little bit familiar ok so this is the
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fundamental notion of equality between
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proportions
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a three proportion is something similar
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but we are comparing three things not
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two things
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so it's an expression a 1 to a 2 to a 3
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where now a 1 a 2 and a 3 are numbers
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not all 0 with the corresponding
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convention that 1/3 proportion say a 1 a
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2 a 3 equals a 2nd 3 proportion B 1 B 2
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2 B 3
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precisely when a1 b2 minus a2 b1 equals
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0 and a2 b3 minus a3 b2 equals 0 and a1
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b3 minus a3 b1 equals 0 so the same
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kinds of relations occurring in a2
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proportion are occurring here but three
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different ways corresponding to pairing
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the ones and the twos the twos and the
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threes and the ones and the threes no
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proportions are kind of close to my
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heart because I wrote a book called
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divine proportions k this is my book on
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rational trigonometry which I hope
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you've all heard of and I hope that
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maybe some of you have read or if you
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haven't that you will read it at some
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point because then you'll learn the very
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important new geometrical understanding
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of rational trigonometry and the the
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notion of proportion encoded here plays
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a kind of a central role in setting up
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this entire new theory of trigonometry
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ok so it's a quite a useful thing even
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though it goes back to antiquity we'll
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see that it's very useful for us so the
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proportions satisfy some nice little
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propositions the first one captures the
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idea of an equivalence relation okay
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that that proportions really provide us
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with an example of the equivalence
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relation that amounts to three sort of
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simple properties that proportions
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satisfy the first one is very simple
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that a to b equals a to b now you may
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think well why do we have to check that
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house that a proposition is that obvious
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well kind of but maybe not entirely
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because remember we defined equality in
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a very precise way we say that this
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proportion is equal to this proportion
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when ad minus BC equals zero so there's
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something to check before we can state
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that two proportions are equal in this
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case you have to check that this number
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times this number minus this number of
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times this number is equal to zero okay
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and obviously it is
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but that's still something that you have
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to check if you had some more convoluted
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notion of equality then maybe this
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property wouldn't actually be satisfied
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and you might wonder why you're actually
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considering that so but anyway that's an
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important problem here's another kind of
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obvious property that a to B equals C to
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D is really the same as C to D equals a
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to B so in other words if these two
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proportions are equal then the Easter
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proportions are also equal and the last
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one transitivity that is a to b equals C
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to D and C to D equals E to F then we
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can conclude that a to B equals E to F
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you might like to prove this for
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yourself
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okay the corresponding facts also hold
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for three proportions so there's a
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corresponding set of statements for
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three proportions and here's another
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useful way of thinking about this
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equality of proportions that a to B
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equals C to D precisely when C is lambda
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a and D is lambda B for some nonzero
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number lambda so in other words this
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proportions obtained from this one by
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scaling up both of the numbers by the
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same nonzero amount because it's another
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way of thinking about what equality of
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proportions means that it's invariant
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under nonzero scaling so that's great so
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we've learned a little bit about basic a
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prior geometry and proportions
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[Music]