So i’m trying to understand more about math. For the operation – – = +, what would be a real life example of me meeting with that situation?
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This is less about having an example and more about the pure math. Ask yourself, in the same way, what’s the difference between “three” and an example of three, like three apples? Can you tell me what “three” is *without* an example?
Like, let’s say you were trying to explain the idea of three-ness without being able to use any examples, so you can’t hold up three fingers, or talk about three apples. You just have to get across the abstract idea of the number three. How would you do it?
This is really the same kind of thing. Your understanding of multiplying two negatives and getting a positive is less to do with a lack of examples and more to do with why negative numbers interact this way in a pure math sense.
To develop an understanding of that, ask yourself how multiplication works in general. Think of what happens to a number line when you multiply by 3, for instance … where do all the numbers go?
When you multiply by three, it’s like grabbing the number line from both ends and stretching it like a rubber band such that zero stays put, and unity (1) gets sent to the multiplier, in this case three. Then you can look at any other number on the entire line (a multiplicand) and see where it landed to get the result of multiplying that number by three. If you look at what happened to the negative numbers, they all grew in magnitude by a factor of three as well, so that’s what happens when you multiply negative numbers by a positive. Everything just gets stretched in magnitude away from zero.
Okay now take the multiplier and reduce it to two. Not too hard. Now visualize sliding that value smaller and smaller to one. Everything stays put, no stretching at all. Keep sliding it down toward zero. Now the number line is getting squished instead of stretched, until you get to zero and everything gets mapped to zero.
Now what happens if you keep on sliding the multiplier below zero? The multiplier is now negative, and what happens to the mappings? Positive numbers that got closer and closer to zero, then hit zero, now zoom right past and head into the negatives. Negative numbers that got closer and closer to zero, then hit it, zoom right past zero into the positives. So, a negative times a negative is a positive.
When you multiply by a negative, it’s equivalent to stretching the number line, but instead of grabbing negative infinity with your left hand and positive infinity with your right, you pinch negative infinity with your *right* hand and positive infinity with your *left*, and then get to pulling as normal, with your left hand landing on your left side (now holding positive infinity) and your right hand landing on your right (holding negative infinity).
BTW, I highly recommend visualizing all operations on a single variable like this. We’re taught the Cartesian plane early in our education and we tend to conflate simple manipulations of a single variable on the x-y plane, but really a function of one variable, f(x), is just a 1D thing, not a 2D thing. The x-y plane is just a construction that is showing you the result of squishing, stretching, and sliding around a the x-axis by projecting it onto the y-axis, but it’s just a convenience because we have an extra dimension on a piece of paper to spare. You should really think about taking the x-axis and squinching and stretching it such that it *becomes* the y-axis in place for any function of a single variable. Or, another way to think about it is to lay the y-axis collinear with the x-axis, just above it, and then think about where each x-value is projected onto y by the function. Or, yet another valuable visualization is to think about every x-value as a vector with the tail at 0, and how it gets distorted by f(x)–if you picture an infinite number of these vectors for every value on x, if you click that vector by selecting its head, you are also shown the corresponding vector it gets mapped to. You could imagine just clicking down on x=0 and then sliding your mouse cursor along the x-axis, picking all these different x-values and being shown the corresponding vectors each one is mapped to. I think these are all useful ways of thinking about a function than the x-y plane, and there’s nothing special about x-y in this set of possible visualizations (and often it’s the worst of them).
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