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.
- And I don't know what "describe" means in this context. So, this would help if you could answer that too.