11/9/2023 0 Comments Negative minus negative numbers![]() So now, we can d- actually I'll do this one first. Let's say we had a negative two, lets say we had negative two, let me do it a different color, let's say we had a negative two, I already used this color negative two times negative three. Now why is that the case? Well we can just build from this example right over here. Now lets take to the really un-intuitive one and measure negative times a negative, and all of a sudden negatives kind of cancel to give you a positive. Now, so you are already starting to feel better about this part right over here negative times a positive, or a positive times a negative is going to give you a negative. And either way, you can conceptualize right over here, you are going to get negative six negative six is the answer. But since here is two times negative three you could also imagine you are going to subtract two, three times So instead of up here, I could written two plus two plus two because this is a positive two right over here, but since we're doing this over negative three we could imagine subtracting two, three times, so this would be subtracting two (repeated) subtract another two right over here, subtract another two and then you subtract another two notice you did it, once again, you did it three times, so this is a negative three, so essentially you are subtracting two, three times. I'll try to color code it negative three and then another negative three or you could say negative three minus three or, and this is the interesting thing, instead of over here there's a two times positive three you added two, three times. Well, one way you could view this is the same analogy here, it's negative three twice so it would be negative. Let's do two times negative three, I want to make the negative into a different color. Now let's try to make one of these negatives and see what happens. This is going to be equal to six, fair enough! Now, you knew this before you even tried to tackle negative numbers. this deal does not make complete concrete sense to you, you want to have a slightly deeper institution than just having to accept its consistent with the distributive property and whatever else and so you try another thought experiment, you say "well what is just a basic multiplication way of doing it?" So if I say, two times three, one way to to conceptualize is basic multiplication is really repeating addition, so you could view this as two threes so let me write three plus three and notice there are two of them, there are two of these or you could view this as three twos, and so this is the same thing as two plus two plus two and there are three of them, and either way you can conceptualize as you get the same exact answer. So you, as the ancient philosopher in mathematics have concluded in order for the multiplication of positive and negative numbers to be consistent with everything you've been constructing so far with all the other properties of multiplication that you know so far that you need a negative number times a positive number or a positive times a negative to give you a negative number and a negative times a negative to give you a positive number and so you accept it's all consistent so far. In wood working if you no longer have to build and give 3 items to three people you can keep those items yourself and you'll be up 9 items. In money the money you no longer have to pay back in debt payment becomes yours and is therefore kept by you as a positive amount. It works the same with money and I believe it applies in possibly all areas. This example is the same as you might think about balance scales it's the same principle. If I take away 3 groups of 3 holes (-3 x -3) I will be removing a total of 9 holes therefore leaving me with, in regards to my example, 1 hole and 10 hills or 9 hills equilibrium so +9. If I make three holes and make three groups of them (-3 x 3) I will get -9 or 9 more holes in the ground so I will have -9 or 9 more holes than hills.ģ. If I then start and I make three more groups of three hills (3 x 3) I will get 9 extra hills. Let say I have 10 hills and 10 holes to start so I'm balanced and I'm at 0 equilibrium.ġ. ![]() ![]() For example I spent a fair amount of time thinking about this and I came up with this holes and hills analogy.Ī hill is positive, meaning a grassy hill and hole is negative, a hole I dug in the ground. I believe it would help to think of 0 not as nothing but as an equilibrium or balanced amount, having the same on both sides.
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