0
$\begingroup$

My text book of Coordinate Geometry defines straight line as follows:-

Equation of a line:- Equation of a straight line is an equation in $x$ and $y$ which is satisfied by the coordinates of all points on the line and is not satisfied by the coordinates of any points which does not lie on the line. Thus, in order to find the equation of a line and establish a relation in $x$ and $y$ which contains only $x$ and $y$ and known quantities.

I don't see the significance of this definition of line, at least at my present level of mastery of the subject(which you can say is infinitesimal).

Is this just a formal way to define a line, or does this help in proving some very non-intuitive results in geometry. Some illustrations to show the significance of this definition would be appreciated.

Also, I think that defining the line in terms of points holds some algebraic relevance like relating the solution set or something like that. I am not very clear what I am trying to conclude by trying to come up with an algebraic connection but I think that the book implies to think of the solution set of the equation. Am I thinking correctly in this regard or am I just thinking aimlessly, if the direction is correct then if you could can you give a suitable example to illustrate it. I would be pleased if you can also answer this additional question.

  • 3
    What you quote there is **not** a definition of "straight line", but instead defines what is meant by an "_equation_ of a straight line". To make sense of it, you're supposed to **already** know what a "straight line" _by itself_ means. – 2017-01-21
  • 0
    @HenningMakholm - After thinking on what "line" by itself means, I tried to come up with an informal definition but I couldn't come up with a convincing enough definition, all I could think up was in algebraic terms which points me to think that its a locus of all those points which satisfy a given linear equation. And after some more thought to the notion of line made me think that its kind of a basic object(or you can say primitive) analogous to an axiom. Am I thinking correct? – 2017-01-21
  • 1
    x @user: There are several different options. In a modern development of everything from first principles (say, for a computer-assisted proof system) one would almost certainly define a "straight line" as something like any subset of $\mathbb R^2$ that can be written as $\{(x,y)\mid ax+by=c\}$ for some $a,b,c$ where $a$ and $b$ are not both $0$. However you can also _choose_ to take the position that you believe in Euclidean geometry already, in which case a line it whichever Euclid says it is, and then define coordinate systems _in_ Euclidean geometry and ... – 2017-01-21
  • 2
    ... _derive_ from this that every straight line satisfies an equation of that shape. Ideally, if you want to be a mathematician, you should become used to the fact that there are several ways to build definitions for familiar things, and comfortable with _switching_ mentally between different definitions that you know are equivalent, depending on which is more useful for you in the moment. – 2017-01-21
  • 0
    @HenningMakholm - After reading your last comment it made me think rather than my doubt regarding the definition of line, it was my rigidity of thought that gave me the ape feeling(Inspired by Feynman) or the itchy feeling that you get when you dont understand why something is done the way it is. Sometimes this external stimulus does pull me out of that viscious itchy feeling – 2017-01-21

0 Answers 0