Click here to purchase the entire book in PDF format. Slope Let's go downhill skiing. One of the big questions when you're a beginner downhill skiier is ``how steep is the hill?'' Well, there is a mathematical way to calculate the answer to this question. Essentially, another way to ask the same question is ``how many metres do I drop for every metre that I ski forward?'' The more you drop over a given distance, the steeper the slope of the hill. So, what we're talking about when we discuss the slope of the hill is how much it rises (or drops) for a given run. Mathematically, the slope is written as a ratio of these two values as is shown in Equation 1.4. $$\textrm{slope} = \frac{\textrm{rise}}{\textrm{run}}$$ (2.4) but if we wanted to be a little more technical about this, then we would talk about the ratio of the difference in the y-value (the rise) for a given difference in the x-value (the run), so we'd write it like this: $$slope = \frac{\Delta y}{\Delta x}$$ (2.5) Where $\Delta$ is a symbol (it's the Greek capital letter delta) commonly used to indicate a difference or a change. Let's just think about this a little more for a couple of minutes and consider some different slopes. When there is no rise or drop for a change in horizontal distance, (like sailing on the ocean with no waves) then the value of $\Delta y$ is 0, so the slope is 0. When you're climbing a sheer rock face that drops straight down, then the value of $\Delta x$ is 0 for a large change in $y$ therefore the slope is $\infty$. If the change in x and y are both positive (so, you are going forwards and up at the same time) then the slope is positive. In other words, the line goes up from left to right on a graph. If the change in y is negative while the change in x is positive, then the slope is negative. In other words you're looking at a graph of a line that goes downwards from left to right. If you look at a real textbook on geometry then you'll see a slightly different equation for slope that looks like Equation 1.6, but we won't bother with this one. If you compare it to Equation 1.5, then you'll see that, apart from the $k$ they're identical, and that the $k$ is just a sort of altitude reading. $$y = m x + k$$ (2.6) where $m$ is the slope.