Students often struggle with the concept of multiplying negative numbers, particularly with the notion that multiplying two negative numbers results in a positive. I’ve seen numerous attempts by teachers and teacher educators to explain why, conceptually, it is that the product of two negative numbers is a positive number. However, what if the underlying presupposition (we can multiply two negative numbers) is wrong?
I’m not saying that \(-3 \times -4 = +12\) is a mathematically invalid equation. It is valid: It is true. I’m saying that our mathematical language is imprecise and, as a result, we wind up saying something that’s misleading.
For the sake of simplicity, we tend to talk about numbers as if they’re one sort of thing. In reality, we have several sorts of “numbers”. In this discussion, I’ll limit myself to three: Quantities, distances, and areas.
The first thing we teach children in mathematics is quantity: Five books, three butterflies, twelve dogs. We may use money or time to start teaching the basics of fractions (a dime is 1/10 of a dollar, fifteen minutes is 1/4 of an hour).
However, for the sort of quantity mathematics children normally do, they don’t need negatives. Negative values don’t really make sense with quantities: If you have five apples and I demand that you give me eight apples, you don’t at some point possess negative three apples. You may owe me a debt of three apples, so that the next time you have some, you’re going to settle your debt. And indeed in high school and beyond we often talk about negative values in terms of debt. But money is an abstract and a cultural concept; “negative money” can make sense. “Negative apples” doesn’t.
Said philosophically: Addition and multiplication are said to be “closed” in the set of natural numbers (whether or not zero is included) because the sum of any two natural numbers, or the product or any two natural numbers, is a natural number. The same is not true for subtraction or division, and so we need to expand the set of numbers for each operator.
Said more casually: If you try to subtract a bigger number from a smaller number, the answer won’t make sense if you’re limiting yourself to (positive) quantities. So you need negative quantities. (You also need fractions for division.)
When I was trying to each my then-five-year-old son about negative numbers, I tried the quantity method. He had initially told me that 3 – 5 = 0, because once you give up all you have, you’re done. This makes sense to a child who doesn’t spend time thinking about debt.
How I convinced him about negatives was using distance instead. I said: If you’re three steps in front of Daddy and you take five steps toward Daddy, how far from Daddy will you be then? He understood that he would then be two steps behind me. This is how the “negative” numbers are used in terms of distance.
However, “negative” and “positive” values on the number line are arbitrary. They sound like quantities because we’re using the same terminology as we do with quantities, and we’ve somewhat arbitrarily decided that one direction is “positive” and the other is “negative”, but that’s as much a matter of convenience for the sake of discussion and for the sake of mapping numbers as it is anything that’s meaningful. We could just as easily refer to “sinister” and “dexter” numbers: “Sinister” numbers being to the left of the origin and “dexter” numbers being to the right. We could have drawn the number line vertically and called them “super” and “sub” numbers.
But we didn’t. We called them “positive” and “negative”, because it was convenient to map them onto the quantity terms that we were already using.
However, when we see numbers in terms of locations on the number line (a higher level of abstraction, but still not the echelon that professional mathematicians generally reside in), “positive” and “negative”, as well as “addition” and “subtraction”, are not about quantity, they’re about distance and movement with regards to the origin.
Somewhere before high school (it’s a Common Core 6th Grade standard), we teach absolute value, i.e., the distance of a number from the origin. But students come away with the understanding that what we’re really doing is converting negatives to positives (“the absolute value of a positive is positive; the absolute value of a negative is negative”). What we’re really doing is taking a signed value and converting it to an unsigned value. This process is opaque because we don’t mark positive values, and so “positive” numbers and “unsigned” numbers are visually (and in many cases functionally) identical.
So, let’s talk about addition and subtraction along the number line. In terms of movement along the traditional horizontal number line, “addition” means to face to the right and begin moving; “subtraction” means to face to the left. Note that I haven’t said how the movement is to proceed, because that depends on the value being added or subtracted.
If you add or subtract a positive value, you move in the direction you’re facing; if you add or subtract a negative value, you move in the opposite direction.
So, \(-2 + 5\) says that you start at a distance two units to the left of the origin, face to the right, and move five units forward (in the direction you’re moving). \(5 – 2\) says that you start at a distance five units to the right of the origin, face to the left, and move two units forward. \(5 + -2\) says that you start at a distance five units to the right of the origin, face to the right, and move two units backward. In each case, you will end at 3, but you will take three different movements to get there.
Multiplication is conceptually different. We tend to teach multiplication as repeated addition, much to Keith Devlin’s chagrin, but how can you repeat an addition negative times? Both \(5 \times -3\) and \(-5 \times 3\) can be seen in terms of repeated addition once students understand that multiplication is commutative (\(5 \times -3 = -3 + -3 + -3 + -3 + -3\) and \(-5 \times 3 = -5 + -5 + -5\)), but what of \(-5 \times -3\)?
Another way of seeing multiplication is in terms of area: \(5 \times 3\) is the area of a rectangle with sides of 5 and 3. But distance is always positive, so then no multiplication on negative numbers makes sense.
When we get to complex numbers, our method is to say that \(\sqrt{-4} = 2i\). We do not have a special symbol for every point on the imaginary axis (let alone the complex plane). Instead, we use \(i\) to indicate that we are positioning the point in terms of the imaginary axis, and \(2\) to indicate that we are two units from the origin. \(2i\) is no more the “name” for this point than \(\sqrt{-4}\) is; they are two different representations for a point that has no explicit name.
The same is true for negative numbers, and this is something I have yet to see made explicit at the secondary school level. \(-2\) is not the name of the point two units to the left of the origin; in the echelon that professional mathematicians work in, that point has no name (in part because the number line/plane concepts are simplifications not used so casually in that world). \(-2\) is an instruction for how to get to that point: Start at the origin, turn to the left (negative), and go two units forward.
Because we use a special, identifiably discrete symbol for imaginary numbers, it’s easy to see that \(2i\) is not itself a number. But because we start using negative numbers at a much earlier stage in mathematics, it’s harder to see that the same is true for \(-2\).
And, of course, it’s even harder to realize that \(2\) as a point on the number line or the complex plane is also not the proper name of a point. We’ve chosen “dexter” numbers to be the default and “sinister” numbers to be the special case, just for the sake of convenience, so dexter numbers are generally unmarked. But we could just as easily have decided that signed numbers are always marked, whether or not they’re negative, so that we have \(|+2| = |-2| = 2\) but \(+2 \neq 2\). We didn’t, but we could have.
All this said, multiplication involving negatives works the same way as multiplication involving imaginary numbers: It’s two operations, not one. First, we have a rule for multiplying the paraunits \(i\) and sign, and then we multiply the absolute values. So \(-5 \times 3\) is really \(5 \times 3\) and \(- \times +\). If this is taught correctly, this is good preparation for the imaginary numbers, which can’t be done any other way:
| \(\times\) | + | – |
| + | + | – |
| – | + | + |
| \(\times\) | 1 | \(i\) |
| 1 | 1 | \(i\) |
| \(i\) | \(i\) | -1 |
By the point that we start talking about imaginary numbers, we’ve already been distinguishing “negative” from “opposite”, and that’s what’s really going on in multiplication: Two opposites cancel each other out. When we seek to prove that two negatives multiply to a positive, we generally use some version of this.
So what’s really going on? When we multiply two numbers with the same sign, the result is positive; when we multiply two numbers with opposite signs, the result is negative. This is simple. The complexity is because of current efforts to make everything in mathematics education conceptually driven; there’s no real, honest way to have this make conceptual sense. It’s the rule that we’ve settled on because it makes the mathematics work, and that’s it.
Note that this pattern is not identical to how it works with imaginary numbers: Two imaginary numbers, when multiplied, do make a real number, but that number has the opposite sign. Two “positive” imaginary numbers, for instance, will yield a “negative” real number.