Why is a negative times a negative a positive?

OK, I can hear the groans already. There are many contexts for answering this question and they are dubious in varying degrees because the real answer is “because I said so.” That is to say, the rule for multiplying negatives is a convention; adopted for good reasons, but a convention nonetheless. Those good reasons are mathematical: we want to make sure that when we extend multiplication and addition to negative numbers the properties of operations still apply. In particular, we want the distributive property to apply. Meditate on this:
$$
3\cdot(5 + (-5)) = 3\cdot5 + 3 \cdot (-5).
$$
The left side is really $3 \cdot 0$, so it had better be zero. So the right side had better be zero as well. The first term on the right side is 15, so the other term had better be $-15$. So $3 \cdot (-5) = -15$. We want the commutative law to hold, so we had better say $(-5)\cdot 3 = -15$ as well. Now meditate on
$$
(-5)\cdot(3 + (-3)) = (-5)\cdot 3 + (-5)\cdot(-3).
$$
The same reasoning tells us that $(-5)\cdot(-3) = 15$.

Trouble is, all this is really hard to explain to middle schoolers, so people invent contexts. One context I’ve seen has something to do with sending out bills. If you receive 5 bills for 3 dollars then you have $5 \cdot (-3) = -15$ dollars. Sending out is the opposite of receiving, so if you send out 5 bills for 3 dollars, you have $(-5)(-3)$ dollars. But once you receive payment, you have \$15. So $(-5)(-3) = 15$.

One problem with this is that you have to buy more conventions to believe it: the convention about negative amounts of money representing debt, the convention about negative receiving being kinda sorta like sending out. That’s a lot of conventions to prove something that is, as I said, a convention itself. Another problem is that all this context really shows is that $-(-3) = 3$, five times. The multiplication in this context is really just repeated addition; it doesn’t work for numbers that are not integers. You can’t send out 5.6 bills.

There is one context that I think does a better job here, and that is $\mbox{distance} = \mbox{speed} \times \mbox{time}$. This does work with non-integers, and you can make sense of all of the quantities involved as negative numbers. Let’s assume that an object is moving along the number line, and that you measure its position at different times, setting your stop watch to 0 when it passes through the origin. Negative distance is distance to the left; negative speed is speed from right to left; and negative time is time before you started measuring. (Later we use the terms displacement and velocity, but there’s no need to introduce them right away.)

So if the object is moving at $-5$ m/sec, where is it at time $-3$ seconds? Well, it’s moving from right to left and it has 3 seconds before it hits the origin, so it is 15 m to the right of the origin. So $(-5)(-3) = 15$.

Was I cheating there? Is this context subject to the same objections I made about the money context? Didn’t I just make up a whole bunch of conventions about negative distance, time, and speed? I think these conventions pass the cognitive sniff test better. They don’t seem as artificial to me. You can really make quantitative sense of negative distance, speed, and time. It feels more like the real world and less like an accountant’s convention. (No offense to accountants intended.) In a way, we have replaced the mathematician’s desire to have the properties of operations continue to hold with the physicist’s desire to have the laws of physics continue to hold.

So where is the distributive property in all of this? I think it is built into our physical intuition about this context. If I travel for 3 hours, and then for another 2 hours, I can figure out how far I have gone by just adding the times and multiplying by my speed, or I can add the distances traveled in each time period. That’s the distributive property. If you dig into the reasoning I gave for the object moving at $-5$ m/sec in the light of this common sense, questioning each claim, you end up with something not too far from the mathematical reasoning I gave earlier.

By the way, this is the approach we take in the Illustrative Mathematics middle school curriculum. Finding contexts for mathematical ideas that are faithful to the mathematics is difficult and requires real sensitivity to both the mathematics and the way students think. Our brilliant curriculum writing team is up to that challenge.

15 thoughts on “Why is a negative times a negative a positive?

  1. I like the contextual argument. I have a small but useful modification of the “standard” approach with which I’ve had some success for pre-service elementary teachers. Namely consider say 16 x 27 = (20 + -4) x ( 30 + -3). If the extension of the distributive law to two binomials (aka “FOIL”) is to hold for negative numbers, then it must be the case that (-4) x (-3) = 12. etc. This can be well-supported by area models.

    I find this better motivated for students than the version using zero, and it reinforces using properties for estimation, and promotes deeper understanding of area models.

    • I like this argument too! I think it is the approach Klein takes in Elementary Mathematics from an Advanced Standpoint.

  2. I wouldn’t call it a convention, Biull, certainly not in the same sense that, say, the order of operations is a convention. Once you accept the axioms of arithmetic, negative times negative equals positive is a theorem, not a convention. Andy Isaacs

    • I agree. Order of operations remains forever a convention, whereas we eventually formalize the principle of preserving properties of operations into axioms from which we can make this deduction. But we aren’t there yet in middle school, so I shy away from suggesting it is a theorem and then giving a bogus explanation. But yes, probably a good idea to avoid the word “convention” as well.

  3. “Negative distance”? What does that do to the geometric definition of absolute value? I’m not at all comfortable with that.

    I’m also wondering how negative time works with interpreting roots of quadratic equations for a projectile launched from non-zero height b at time = 0, where the x-variable is time and the y-variable is height, and claiming that if one root is -3 that we know that the projectile was at ground level 3 seconds before launch: we don’t. It might have been in lots of different places at that time, including at b. We don’t know what was going on at t = -3 or any other negative time, do we?

    • I agree we don’t want to settle on the term “negative distance.” I would make it a class discussion: “How do we interpret distance as a negative quantity?” and then move to talking about position on the line of motion. You choose a reference point and then positive numbers refer to positions to the right and negative numbers refer to positions to the left.

      Time works the same way. You choose a time to start your stop watch, and then positive numbers refer to times after that and negative numbers refer to times before that. It’s a good question which of those times make sense in the context, and it’s not only negative times you have to worry about there, but also, for example, times after the projectile hits the ground. But one thing at a time! First establish an understanding of time as a signed quantity, then start worrying about the domain for a given context.

  4. Interesting idea – very nice!

    A more complete definition for multiplication makes this much less mysterious, as I am sure you know. A number is an arrow from zero. Multiply by “2” means make the arrow twice as long, multiply by -2 means make the arrow twice as long but in the opposite direction.

    Tempting to use your idea with students that are already proficient with negative number arithmetic to create some thought about “d = rt”.

    Here are some other interesting ideas. One similar to d= rt involves a movie playing backwards.

  5. Hi Bill,

    I like your original argument however, I agree with you that from the perspective of a kid, sounds very similar to “it’s just true because it has to be true.”

    I have a related post on subtracting negative numbers which is the result of a conversation with my wife. The idea there is that the goal is the learner understanding the math, not the context itself. As it turns out the “mathematical consistency as derived from a pattern” argument worked the best with my wife.

    It’s also a good instructional strategy to have kids practice something they know, observe a pattern as a result of that practice, and then test applying that pattern to a slightly novel idea. I kind of describe this strategy in this post.

    So that you don’t have to click on the post I’ll summarize it here:

    Have kids calculate the following:

    3 x 4 =
    3 x 3 =
    3 x 2 =
    3 x 1 =
    3 x 0 =

    Ask them what they notice then “What would 3 x -1 be so that the pattern continues?”

    Now the do the reverse (I don’t think we can assume kids naturally see/use commutativity):

    4 x 5 =
    3 x 5 =
    2 x 5 =
    1 x 5 =
    0 x 5 =

    Same questions basically, except now it’s -1 x 5.

    Now ask students to apply what they’ve learned and try out a sequence of problems like this:

    5 x -3 = -15
    4 x -3 = -12
    3 x -3 = -9
    2 x -3 = -6
    1 x -3 = -3
    0 x -3 = 0

    Ask them what they notice. “What would -1 x -3 be so that this pattern continues?”

    I actually tried this strategy with a small group of kids about a year ago and it worked great. Afterward, I asked students to write a reflection and to generalize what they learned to see how it might work with all numbers.

    One student basically, without special prompting from me, the “rules” that are often taught as the starting place for operations with signed numbers.

    Once I think this argument is understood, then I would build toward an argument like yours which as you noted isn’t limited to operations on integers. I really like that aspect.

    • Thanks David. I agree this is a good approach, although I’ve gotten into arguments with people who dismiss it as patternology. But in fact this is another approach that has the distributive property hiding in the background. If you dig into why the pattern holds, it’s because $3 \times (n-1) = 3\times n – 3\times 1$. So by continuing the pattern you are, once again, demanding that the distributive property continue to hold. This would be a good connecting observation in building towards the general argument.

      • On patternology:

        “For all of human history, we have relied on patterns that we observe in order to make initial conjectures that may or may not be true about numbers and relationships, but in my class, we will deduce everything from first principles.”

        This just seems like a ridiculous position to assert, given our history as humans doing mathematics.

        FYI, there is a textbook based on this approach that derives all of elementary school mathematics from the Peano axioms. It’s 4000 pages long.

  6. Interesting discussion about an issue that plagues many teachers.

    I side with the posts that have “because we said so” under the hood. I’ve always tried to convey in my (HS) classes that the fact that $(-3)(5) = -15$ is not an act of congress or a rule of nature—it’s imposed on us by the desire to preserve the “rules of arithmetic” when we extend the systems in which we calculate (it’s the same motivation when extending exponentiation beyond whole number exponents). So a motivating context that stays close to the underlying mathematics should lead to student questions about why the rules need to be what they are.

    Here’s one context (albeit mathematical) that emphasizes the extension theme and that has worked well for us and that finds use in our other HS courses (algebra 2, for example).

    Start with the ordinary addition and multiplication tables, oriented in an unusual way:

    https://go.edc.org/arithmetic-tables

    Look at “addtable” and “multable”, for example. We first ask kids to fill in the blanks, just to get a rhythm going. Then turn them loose on finding as many patterns as they can. Most look at diagonals first. Sooner or later, many kids start noticing the corners of rectangles, up and over patterns, and so on. These can (now or later) be proved using the basic rules of algebra and things like $(a-1)(a+1)$. Many people (including adults) get engrossed in what they find in the multiplication table. Roger Howe has created for his students a nice collection of algebraic identities that arises here. I put a set of notes in the dropbox that we used for an all-day seminar for teachers, that includes some of Roger’s stuff, and that shows just how far you can push all this.

    Next, extend the tables in all directions as in the other two figs in the dropbox. Here’s where the patterning comes in, but by now, the tables are contexts in their own right. You hear all kinds of ways to extend the patterns, most of them producing the usual extensions. But someone always has a different one like going across a row decreasing to 0 and then going back up. Here’s the chance to ask if the usual rules of arithmetic extend via this new extension. They don’t, and this is a good chance to introduce a derivation similar to what Bill posted.

    BTW, the arrangment of the tables leads to other things. Circle all the 12s in the mult table. It looks like it can be fit with a nice curve. What’s its equation? Why? Stuff like that.

    Al Cuoco

  7. I like this explanation a lot for middle school. In the geogebra app, you can really see how it works.

    Once there’s a better understanding of transformations, I like to think of it in terms of reflections of the number line. Multiplying by a negative corresponds to reflecting and dilating the number line. A negative times a negative is two reflections, which is the same effect as multiplying by a positive.

  8. Thank-you for the honesty. I think we need to extend that honesty to our middle schoolers. They already innately believe that the real reason is “because I said so”. Let’s admit it and explore the good reasons for adoption of this convention.

    I have used “the opposite of”.

    If 3=three
    Then -3 is the opposite of 3 which is of course negative 3
    And -(-3) is the opposite of negative 3 which is positive 3

    The pattern continues when applied to operations.

    • Well, the “because I said so” is not arbitrary or capricious. It’s because we want the properties of operations to continue to hold for negative numbers. And it makes sense in terms of patterns in the multiplication table. I think we can explain why we do it.

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