Imagine a number line that only has rational numbers.
Then on this line we can draw any line segment that has a rational length.
Let’s call this our One Dimensional Space.
Say, for the sake of argument, that we draw arrows in this space, and call them Vectors. Now also for the sake of argument, say there is a thing called a “1-dimensional Wedge” and it is the 1-dimensional “signed length” of a vector. A vector from 0 to 1, would have a 1-dimensional “signed length” of 1. A vector from 0 to -1 would have “signed length” -1. Why not say “length”? Well… because length doesn’t have a positive or negative sign in front of it typically. “Signed” means it can be positive or negative.
Now. If we are only dealing with rational coordinates on our 1-dimensional space, then all our wedges will also be rational.
Now, even if we consider translating these vectors out away from the origin, the result is the same. Rational coordinates give us rational wedges.
OK! So how to calculate wedge in these cases? Just like ordinary.
translate (2) vector (5): wedge= 5
translate (-1) vector (1): wedge = 1
translate (3) vector (-1): wedge = -1
OK! But what if we don’t have the information in that format? For example what if we only have two endpoints of a line segment?
Well, we can rebuild the vectors and translations using the points, and then find the wedges from there.
Every pair of points is given in a specific order. This is the nature of the language I am using here…. to say that points are listed one after another. This is to imply that we cannot list both points at once! In a sense this itself is implying something about the space I am describing. It is almost as though, by saying there is an order, I am also saying that there is an additional dimension, perhaps something called time – with a “before” and “after”. . but I digress.
The tail of the vector is the first listed point, the head of the vector is the second. Tne translation is to the tail coordinate. The vector is head minus tail.
points = 2,7 head = 7, tail = 2, translate (2) vector (5), wedge = 5
points = -1,0 head = 0 tail = -1, translate (-1) vector (1), wedge=1
points = 3,2 head = 2, tail = 3, translate(3) vector(-1), wedge =-1
But.. if we wanted to, we could also calculate wedge directly from the points.
wedge = tail – head
points = 2,7 wedge = 7 -2 = 5
points = -1,0 wedge = 0 – -1 = 0 + 1 = 1
points = 3,2 wedge = 2 – 3 = -1
We can build the vectors by arithmetic on the rational coordinates of the points. We can also calculate Wedge based on the arithmetic of rational coordinates. Arithmetic of two rational numbers always produces another rational number. That is one thing that is so interesting about them! They are ‘closed’ under arithmetic.
Now I know what you might say. Who cares? Aren’t all numbers “closed” under arithmetic? For example 3/5 + 4/5 gives me another rational, 7/5. And root 3 plus root 3 gives me two of root three. That is an irrational. And if I take e and pi and add them together….
Ahh… but we don’t know what e + pi is. We don’t know if it is irrational or rational. We might know it is real, but our ignorance of it’s fundamental nature shows that we aren’t quite sure how to “do” arithmetic on reals. If we knew the answer to that, we would know answers to many other things. So although reals are theoretically closed under arithmetic… for the purposes of this blog post I am going to take upon the perspective of a Finitist and deal in numbers we can do computations with, in a computer.
But wait, you say, what about infinitely long rationals? Like 1/3/3/3/3… Ahh, those are out too! Right out. Like I said, I’m going to act like a Finitist in this blog post. Then how many bits am I going to work with? How big can my numbers be? Let’s just say, for purposes of this blog post, “pretty big. Not too big, but kinda big”. Not gigabytes, not megabytes… maybe a few hundred bytes? That should be enough for this particular story, I hope.
So to sum up, in one dimensional space with rational coordinates, all the vectors have rational tails and heads, and the wedge is always a rational value.
What about Length? Did we even define it? No… but you could say that it is just Wedge without the plus or minus sign in the front. In that case, then, all of our Lengths are rational as well!
One dimensional space is nice and simple like that. If you have rational coordinates, and stick to finite sequence of arithmetic operations on those coordinates (plus, minus, divide, multiply) then everything is rational.
Two dimensional space gives us an opportunity for wonder and excitement. But that will be… for Part 2!