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)?
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.
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?
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.
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.
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
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".
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.