ELI5: Why do double minuses become positive, and two pluses never make a negative?

Read the Story

Show Top Comments

Image you’re facing me. I instruct you to turn around and then walk backwards. This is a negative (turned around) multiplied by a negative (walking backwards) But you’re *getting closer* to me. Negative times negative has given you positive movement. What if you just faced me and walked forwards? Still moving towards me from positive times positive. Any multiplication of positives will always be positive. Even number multiplication sequences of negatives will also be positive as they “cancel out” – flipping the number line over twice.


Try thinking of money. Someone gives me 3 $10 bills: 3 x 10= 30. I am $30 richer Someone takes 3 $10 bills away from me: -3×10= -30. I am $30 poorer Someone saddles me with 3 $10 debts: 3 x -10= -30. I am $30 poorer Someone takes 3 $10 debts away from me: -3 x -10= 30. I am $30 richer


The difference between positive and negative is that positive actually occurs naturally. You can have 5 apples, but never -5 apples. The minus is something mathematicians made up. It means “opposite of”. So -5 apples is opposite of 5 apples. It’s hard to picture what *this* would mean (5 apples made of antimatter?), but there are cases where it’s more logical – opposite of receiving 5 dollars is paying 5 dollars (or receiving -5 dollars, if you will), opposite of 5 ships arriving is 5 ships leaving (or -5 ships arriving, if you will), etc. Double minus is double opposite. The opposite of opposite is what you started with. If 5 ships do the opposite of opposite of arriving, what they do is… arrive.


Think about the negative sign as “not”. If you say “I’m not not going to go to the park” then you are actually saying you are going to the park. Now let’s say “very” is positive. “I’m very very happy.” That means the same thing as “I’m very happy”. This holds true for numbers. -(-2) or not(not2) is 2.


TL;DR: The rule is an arbitrary choice. We defined it that way because that rule made common calculations for problems we care about convenient. There’s a lot of answers in here trying to give some kind of intuitive underpinning of how to understand – * – = + by describing some analogy. But these answers are all incorrect as to why it is actually the case. In fact, they are making the same mistake that many professional mathematicians made in the 1800’s and earlier when negative numbers were first encountered. For the longest time, mathematicians didn’t accept negative numbers at all. They were working in algebraic systems of symbolic calculations, and if a negative number popped out as an answer, many would regard that result as an indication that the problem was improperly set up in the first place. After all, you can’t have something that is less than nothing. You can’t have a length that has a negative magnitude. Some would argue that a negative sign on an answer could represent a magnitude in the opposite direction or an amount owed rather than an amount you had. But these explanations only apply in certain contexts. And they are still making a fundamental mistake. These explanations are attempting to provide a physical meaning to a system of symbols and rules as if there is only one true system of symbols and rules. What was finally and slowly realized in the late 1800’s and the early 1900’s is that there isn’t one true algebra. Algebra is just a made up system of symbols and rules. And there’s nothing stopping anyone from making up their own systems of symbols and their own new rules that behave differently. This is exactly how quaternions were invented. William Hamilton liked using imaginary numbers for representing 2d spaces, but he wanted a new algebra that could do the same kind of thing for 3d spaces, so in addition, he tried adding a j where i^2 = j^2 = -1 but i != j so that they’d have 3 axes in their representation: x + yi + zj. However, he found that when he tried to do some basic operations with these new numbers, he found inconsistencies. His new algebra led to contradictions with how he’d defined the rules for i and j. But with some more tinkering, he found that by adding a third kind of imaginary number, k such that i^2 = j^2 = k^2 = ijk = -1; he got a perfectly consistent system that in some ways modeled 4 dimensional spaces, but could also be useful in representing rotations in 3d spaces. He’d made up a new algebra with different rules than the one people were familiar with: the quaternions. With this realization, symbolic algebra really took off. Later also called “Abstract Algebra” concerned itself with things called Groups, Rings, and all other sorts of structures with a multitude of different sets of rules governing them. And so, the real and true reason that a negative times a negative is positive: ***The rule is an arbitrary choice. We defined it that way because that rule made common calculations for problems we care about convenient.*** But you could define your own algebra where this is not the case if you wanted. You could make your own consistent system where -1 * -1 = -1 and +1 * +1 = +1. But then you have to decide what to do with -1 * +1 and +1 * -1. To resolve that and keep a consistent system, you might have to do away with the commutativity of multiplication. The order in which you multiply terms together might now matter. One way to do it is to say the result takes the same sign as the first term so that -1 * +1 = -1 and +1 * -1 = +1. This would make positive and negative numbers perfectly symmetric rather than the asymmetry the algebra most people are familiar with. Now, whether this new set of rules is convenient for the kinds of real world problems you want to solve via calculation, whether this system is a good model for the things you care about is another question. But that convenience is the only reason we use the rule -1 * -1 = -1 There’s a great book that covers all of this along with more of the history, more of the old arguments about negative numbers, imaginary numbers, and the development of new algebras along with an exploration of a new symmetric algebra where -1 * -1 = -1 called “Negative Math” by Alberto A. Martinez.