Why slope is written m, and why vertical lines have none
Slope is just rise over run: how far a line climbs for each step sideways. Nobody is sure why mathematicians label it m, but the letter appears in English as early as 1844, and the same simple ratio underpins road grades, roof pitches and the core of differential calculus.
Take any two distinct points on a line, divide the change in height by the change in horizontal position, and the answer is the slope. It is neither a distance nor an angle but a ratio of the two changes, often written with the Greek delta, which mathematicians use for difference. Positive slope means the line rises from left to right, negative means it falls, zero means it is flat, and larger absolute values mean steeper lines. A vertical line breaks the formula, since its run is zero, so its slope is called undefined.
A worked example makes it concrete. A line through the points (1, 2) and (13, 8) climbs 6 while moving 12, giving a slope of one half, a gentle incline under 45 degrees. A line through (4, 15) and (3, 21) has slope minus 6, a steep descent. Through (2, 8) and (3, 20) the slope is 12, which corresponds to an angle of roughly 85.2 degrees above the horizontal.
Trigonometry links slope to angle: slope equals the tangent of the angle of inclination, so a line rising at 45 degrees has slope 1 and one falling at 45 degrees has slope minus 1. Lines sharing a slope, such as y = −3x + 1 and y = −3x − 2, never meet and are parallel. Engineers and geographers use the same idea as the grade of a road.
Calculus extends the concept to curves. The slope at a point is the slope of the tangent line there, which can be approximated by the secant through two nearby points, and calculus supplies exact formulas when the curve is the graph of an algebraic expression. As for the letter, O'Brien wrote the line as y = mx + b in 1844, and Todhunter used y = mx + c in 1888.
Source: Slope
