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I've read the whole Wikipedia page on the slope, and it says the slope of a line is a number that describes1 both the steepness and the direction of the line. It is a measure of how steep a line is. Then the article proceeds to state the formula for calculating the slope of a line — the-rise-over-run formula.

Is the above definition of slope complete? That is when I learn Advanced Mathematics, will I have to listen to the same-old saying "actually this is not entirely true"? Assuming it is complete, why is rise-over-run a measurement of it? In other words, how did mathematicians concluded that, okay, this is the measure of the steepness of a line?

Then there is the rate of change of a function. If we find the rate of change of a linear function, then we see that it is actually the slope of that line. Is this an accident? How do the slope of a straight line and the rate of change of a function relate together? I want to know the rigorous answer. Thank you.

  1. And I don't know what "describe" means in this context. So, this would help if you could answer that too.
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  • $\begingroup$ "Rise over run" and "rate of change" are basically entirely synonymous. $\endgroup$
    – Arthur
    Apr 15, 2021 at 8:48
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    $\begingroup$ "Is the above definition of slope complete?" Yes "when I learn Advanced Mathematics, will I have to listen to the same-old saying "actually this is not entirely true"?" No. "why is rise-over-run a measurement of it?" Because steepness is a comparison to how much something rises compared to how much horizontal change there is. "How do the slope of a straight line and the rate of change of a function relate together?" Steepness is the rate of change in one's altitude. They are exactly the same thing. $\endgroup$
    – fleablood
    Apr 15, 2021 at 9:15

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Assume you would want to tell someone how steep a road is, in quantitative terms. As that information is independent of the length of the road, it is essentially an angle, such as $\alpha$ on the picture.

enter image description here

But in many situations, measuring the angle is inconvenient and it is easier to measure displacements, and ratios of displacements are independent of the length. Hence equivalently, steepness can be characterized by one of

$$\alpha,\frac{CB}{AB}, \frac{CB}{AC}.$$

(I don't mean that these have equal values, but one gives the others.) $\dfrac{CB}{AC}$, called the slope, is often preferred and indeed represents a rate of change: how much do you displace vertically for a given horizontal displacement ?

In the case of a nonlinear function, the vertical displacement is not proportional to the horizontal one. To work around this, we consider the local slope, i.e. the slope of the tangent to the curve at the point of interest.

enter image description here

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  • $\begingroup$ Also note that angles and hypothenuses make no sense when the $x$ axis and the $y$ axis don't measure the same thing. So of those three steepness measurements, $\frac{CB}{AC}$ is orders of magnitude more generalisable. $\endgroup$
    – Arthur
    Apr 15, 2021 at 9:05
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    $\begingroup$ @Arthur: I did not refer to coordinates axis $x$ and $y$ on purpose (unfortunately, they appear on the figure), and used the term displacements with the implicit understanding that they are isotropic. In calculus, the graphical representation (and axis moduli) is immaterial. When the quantities are in different units, rate-of-change is indeed a more appropriate term. $\endgroup$
    – user65203
    Apr 15, 2021 at 9:10
  • $\begingroup$ So, you're agreeing with the definition that a slope is actually a number that is a measure of the steepness of a straight line? And you are also saying that we can measure the steepness — the slope — of a straight line using the angle it creates with the positive direction of the x-axis. If α is the angle and α = 42.53°, we could've just said, okay, the slope of that line is.. well., 42.53 something. Right? Then you're saying angle measurement is difficult in some cases, so we sought other ways to measure the steepness of a straight line. *I was still writing my comment when you already did! $\endgroup$ Apr 15, 2021 at 9:21
  • $\begingroup$ But my question was how do you know, α, BC/AC, BC/AB are ways to measure the steepness of the straight line? How do you know these really represent the steepness of a straight line? The α-one kind of makes sense, but the ratio-ones don't, at least to me! $\endgroup$ Apr 15, 2021 at 9:29
  • $\begingroup$ And, are you saying that the rate of change of a linear function "turns out" to be the slope of the straight line? $\endgroup$ Apr 15, 2021 at 9:31

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