Why do we take slope as rise over run?

In summary, slope is defined as rise over run or dy/dx in general. It is a mathematical generalization of the way we measure inclines or pitches. The convention of plotting dependent variables on the y-axis and independent variables on the x-axis makes it easier to read and interpret graphs. The choice of which variable to put on which axis is purely a convention. The convention of using rise over run or dy/dx is convenient in calculus because it ensures that the derivative will never "blow up".
  • #1
Juwane
87
0
Why do we take slope=rise/run (or y/x)?

Is it just a definition, or does it have a special significance?

Why can't we take slope as run/rise (i.e. x/y)?
 
Mathematics news on Phys.org
  • #2
It is the definition. In general it is dy/dx.
 
  • #3
I think it's related to the definition of a function.

A function a unique y for any given x; it does not necessarily have a unique x for any y.
 
  • #4
Slope answers "how fast is y increasing compared with x". It is exactly the same as dividing distance by time to find speed.
 
  • #5
HallsofIvy said:
Slope answers "how fast is y increasing compared with x". It is exactly the same as dividing distance by time to find speed.

Yes but if it were run/rise then we would just make it conventional to plot distance on the x-axis and time on the y-axis.
I believe it's just the way they defined it. We need it to be one or the other, so why not just choose?
 
  • #6
You can think of slope as the "math generalization" of the way we measure the pitch of a roof or the incline of a hill - both those measure rise over run, albeit in different language. Those ideas were generalized and 'abstracted' (if that isn't a word, it should be) to the notion of slope in the plane.
 
  • #7
Mentallic said:
Yes but if it were run/rise then we would just make it conventional to plot distance on the x-axis and time on the y-axis.
I believe it's just the way they defined it. We need it to be one or the other, so why not just choose?
When graphing values, the convention - because it's easier to read and interpret - is to put the consistent value along the x-axis and the dependent value on the y-axis. That way, the graph is "read" left-to-right.
 
  • #8
having slope = dy/dx also makes the equation y = mx + b much prettier.
 
  • #9
high slope = graph goes up really quickly = high speed, acceleration, flow rate, whatever

The other way round:

high slope = graph goes up really slowly = low speed, acceleration, flow rate, whatever

seems counter-intuitive
 
  • #10
Juwane said:
Is it just a definition, or does it have a special significance?

Rise over run is convenient because it "always works" in calculus. The definition of a derivative is the limit of the fraction with f(x+h) - f(x) on top and h on the bottom as h approaches 0. We have no guarantees what f(x+h) - f(x) might be. But we know for damn sure that the denominator, h, will never be equal to zero. And since the only restriction on division is that the denominator can't be zero, we know the derivative will never "blow up".
 
  • #11
Surely just a convention, isn't it? If the the tradition had been to draw graphs with the independent variable on the vertical axis, I bet we'd be able to come up with just as many reasons why that was the most natural and intuitive way. Then run-over-rise would be the one that'd conveniently "always work" in calculus, because a function--by the definition of a function--would never have a horizontal slope. In that bizarro universe, Joe Hx would be telling us how much prettier x = my + b is than y = mx + b, and ideasrule might be saying how much more intuitive it was to represent greater speed, acceleration, etc. with a more forward slanting slope than a sluggish, bunched up one that hardly got off the starting blocks of the vertical axis. Actually the books on relativity that I've seen mostly do follow that convention, putting time on the vertical axis and using the horizontal axis to represent some dimension of space, labelled x.
 

FAQ: Why do we take slope as rise over run?

1. Why do we take slope as rise over run?

Slope is a measure of the steepness of a line. It tells us how much the line rises or falls as we move from left to right. By taking the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run), we can determine the slope of a line.

2. What does rise over run represent in slope?

Rise over run represents the vertical change (rise) divided by the horizontal change (run). This ratio tells us how much the line is rising or falling for each unit of horizontal movement.

3. How does rise over run help us understand the slope of a line?

The rise over run formula allows us to calculate the slope of a line by dividing the change in the vertical direction by the change in the horizontal direction. This helps us understand the steepness of a line and how much it is changing as we move along it.

4. What is the significance of using rise over run instead of other methods to calculate slope?

Rise over run is a standardized method for calculating slope that is consistent regardless of the scale or units used. It also allows us to compare slopes of different lines and determine which is steeper or flatter. Additionally, it can be easily applied to any linear function.

5. Can we use rise over run to measure slope for non-linear functions?

No, rise over run is only applicable for linear functions, where the rate of change is constant. For non-linear functions, the slope is constantly changing, so we cannot use the rise over run formula to accurately measure it. Other methods, such as finding the tangent line at a specific point, must be used to calculate the slope of a non-linear function.

Back
Top