Why is the slope of a line defined as the change in y over the change in $x$?

Question

I understand that this is a reasonable definition and it shows how "fast" $y$ values change corresponding to $x$ values because it's a ratio, but what I'm asking is, couldn't the slope have been defined as the angle between the line and the positive $x$-axis for example? And it would have the same meaning; if the angle was large (but less than $90$) then that would mean the line is steep and $y$ values change fast corresponding to $x$ values...etc. Why is the first definition better? Is the second one even correct?

Answer 1

The angle is one thing you might care to think about, sure. The rise-over-run is another thing you might care to think about. The fact that the word "slope" went to the latter is just a caprice of history. On the other hand, the fact that the latter turned out to be an interesting and fruitful object of investigation is no mere quirk (but also, not that surprising). Sometimes we are interested in direct proportionality relations like $Y = mX$, and in those cases, the constant of proportionality $m$ is a natural thing to consider. Ratios are of ubiquitous arithmetic importance, and that's all that "slope" comes down to; investigating ratios.

But there's nothing wrong with thinking about angles, either. Just because we spend a lot of time talking about slopes doesn't mean we're against thinking about angles. Think about both! Think about everything! Math isn't an either-or world; you can think about anything, everything, and see what comes of it.

Answer 2

As Math_QED points out in his comment, the derivative of a function at a point is equal to the slope of the tangent line at that point, so there is a connection between slopes defined as “rise over run” and rates of change. There are other contexts in which the angle of the line is more natural or convenient. Since the slope of a line is equal to the tangent of the angle that it makes with the $x$-axis, the two definitions are equivalent.

Answer 3

In the end, it all comes down to the current definition of slope being more natural than using degrees. Here are a few thoughts on that subject:

Using angles is dependent on the aspect ratio between the axes. You can have a line that looks like it's $30^\circ$ in the picture, but it's really $89.97^\circ$. This can happen with slope as well, but degrees carry a heavier implication of what something looks like. Usually, with geometric figures, if you "squash" them, you accept that the angles change. You could do that with functions. This is not a good idea, because that means that if you draw a graph, it gets one angle, and if you've written that angle down next to the graph, and someone makes a bad photocopy, then what's on there is suddenly invalidated. So the angle has to be inherent in the function alone, which breaks how we tink of angles: We accept that if you draw a square, and squash it in one direction, then it's not a square anymore, and the diagonals are not $45^\circ$ to the sides.

Also, note the difference between a line with slope $89.97^\circ$ and one with slope $89.98^\circ$. Because the natural way to "move" is to move with constant speed along the $x$-axis, and not along the graph, they will separate fast, while two lines of slope $30^\circ$ and $30.01^\circ$ are more or less indistinguishable.

Then consider what happens if you change your units. If the units on the $x$-axis is seconds, and the units on the $y$-axis is meters, then the line represents what your position is at ny given second, and the slope your speed in meters / second. Say you have a line of slope $1$ in that coordinate system. If you change the length units to feet, the slope becomes $1\frac ms \cdot 3.28 \frac{\text{feet}}{m} = 3.28\frac{\text{feet}}{s}$, while the degrees go from $45^\circ$ to... what, exactly? Insert trigonometry here. Related to this, what do the degrees even measure? What's a natural unit for them? I daresay there isn't one.

What about if you add functions? The slope of the sum is the sum of the slopes. What is the angle you get if you add a line with angle $37^\circ$ with a line of angle $62^\circ$? It's not impossible, but it takes a few calculations to figure out.

And then, of course, the derivatives, and all the rules we have for those. Try doing the chain rule with degrees. That's trigonometry galore.

So you could define slope from degrees, but you'd only make life hard for yourself in the long run.

Answer 4

In short: slope is so defined because the conventional "rise-over-run" definition is immensely useful. It will allow us to make lots of connections. For example:

• In calculus: the slope of the graph of $f(x)$ represents the derivative $f'(x)$.
• In mechanics: the slope of the time-distance graph represents the speed.
• In physics: the slope of the time-charge graph represents the electric current.

And this list can go on and on... (In applications, if you need the average speed/current/etc., use the slope of a secant line; if you need the instantaneous speed/current/etc., use the slope of a tangent line. The derivative is the slope of the tangent line to the $f(x)$ graph.)

Of course the key concept here is the derivative; and it has a vast number of applications in physics, technology etc. And for it all to work like a charm, we do need the "rise-over-run" definition of slope (i.e. the change in $y$ over the corresponding change in $x$).

A definition in terms of the angle just would not work, i.e. it would not translate so nicely into a lot of different contexts.