### 1. Definitions

1.
Infinity is not a real number. Attempts to define arithmetic with
infinity will meet with difficulties. We can say that infinity and
zero are reciprocals of each other. Since zero is equal to its own
negative, in this case we will consider that positive infinity and
negative infinity are the same thing. Topologists call this the
"Alexandrov one point compactification of the real line". As a result
of the topologists work, it is possible to say that numbers with
large absolute values, both positive and negative, are closer to
infinity than numbers with small absolute values.

2. There
are several ways to express a linear equation. One way would be to
transpose all unknown terms to the left and all known terms to the
right. If we then remove parenthese and combine like terms, our
equation will look like

rx + sy = t
where r, s, and t are real numbers. The advantage of expressing
the equation in this form, which algebra teachers call the
**standard form of the equation** is that any equation in at most
two unknowns can be expressed this way. An equation where the only
unknown is x can be expressed in this form by letting s be 0. In an
equation where the only unknown is y, r is 0.

However, there is another form which is very useful as in Theorem
1.1. If s is not zero then it will be possible to solve the equation
for y and express it as

y = mx + b
where m = -r/s and b = t/s. This form of the equation is called
the **slope-intercept form of the equation**. The advantage of the
point-slope form of the equation is that the slope of the line can be
read off from the equation. The disadvantage is that equations with
no y term cannot be fit into this form and have to be treated
separately.

If the line is not vertical, there is another form of the equation
known as the **point-slope form **

y - y_{1} = m(x - x_{1})
where m is the slope of the line and (x_{1},
y_{1}) is a point on the line. See
Theorem 1.3 for the realtion
between the point-slope form and the slope-intercept form of the
equation.

3. As a
result of Theorems 1.1 and 1.2 the slope of the line is well defined
independently of the choice of the two points to use for the
computation.

4.
For purposes of this definition, 0 and infinity are considered to be
negative reciprocals of each other. That way the fact that horizontal
and vertical lines are perpendicular fit into this definition.

Analytic Foundations of
Geometry