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Why do we take slope as rise over run? 
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#1
Dec3109, 01:44 PM

P: 87

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)? 


#2
Dec3109, 03:14 PM

Sci Advisor
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It is the definition. In general it is dy/dx.



#3
Dec3109, 03:19 PM

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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
Jan110, 06:35 AM

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Why do we take slope as rise over run?
Slope answers "how fast is y increasing compared with x". It is exactly the same as dividing distance by time to find speed.



#5
Jan110, 06:42 AM

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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
Jan110, 09:31 AM

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P: 1,372

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
Jan110, 12:50 PM

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#8
Jan510, 10:15 PM

P: 3

having slope = dy/dx also makes the equation y = mx + b much prettier.



#9
Jan610, 01:16 AM

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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 counterintuitive 


#10
Jan710, 04:30 PM

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#11
Jan710, 05:43 PM

P: 1,402

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 runoverrise would be the one that'd conveniently "always work" in calculus, because a functionby the definition of a functionwould 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.



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