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 - y1 = m(x - x1)

where m is the slope of the line and (x1, y1) 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