I'm not a mathematician, but I think that the initial response by Karim saying that Khan was incorrect about the definition is really nitpicky and petty. Karim is saying that "Rise over run" is how you calculate it, but he claims it's not the definition. Even if it wasn't the precise definition, for most people, rise over run is good enough for a basic understanding.
Khan saying that Karim was wrong, is wrong and actually made him look worse in my eyes. In the video he said "Slope can present rate of change." No, slope is the rate of change. Him saying Karim was wrong about the price vs gigabyte being the inverse if you switch the axes is ridiculous. By definition, you always say the first variable vs the second variable, where the first variable is the y axis and the second is the x. Khan should have left it at that instead of trying to twist things around to make Karim look wrong, it was a poor and transparent attempt at being vindictive.
"Slope" implies that the denominator is a distance. In the context of a graph on paper or screen, it is clear how rise and run are mapped to distances. It's a totally sensible definition: slope only makes sense given a choice of axes.
"Rate" implies that the denominator is a timespan. This does not make sense, as there are many slopes which are not rates and which are not presented as rates.
It's kind of a silly thing to worry about, but Khan's answer is unequivocally better.
In math, is not accurate to say the "definition" and the "calculation" are one in the same? Sometimes we require ambiguous non-math language to start understanding a concept but ultimately, once understanding has happened, the calculation and the definition are identical.
Definition and technique to calculate are not the same thing. Consider integer division, which is defined by an equation, but calculated by an incremental long division algorithm.
Most definition use a phrase like "such that", which says nothing at all about calculating.
Khan saying that Karim was wrong, is wrong and actually made him look worse in my eyes. In the video he said "Slope can present rate of change." No, slope is the rate of change. Him saying Karim was wrong about the price vs gigabyte being the inverse if you switch the axes is ridiculous. By definition, you always say the first variable vs the second variable, where the first variable is the y axis and the second is the x. Khan should have left it at that instead of trying to twist things around to make Karim look wrong, it was a poor and transparent attempt at being vindictive.