Minus times minus equals plus. / The reason for this we will not discuss.
— W. H. Auden, recalling a popular verse from his school days
Ever tried mixing together your two least favorite foods? I suspect you haven’t. Nobody mixes two noxious ingredients and expects the results to be tasty. So why should numbers behave differently in the numerical recipe called multiplication? What mystical two-wrongs-make-a-right alchemy removes the taint of negativity and makes the product of two negative numbers positive? It just don’t make no sense!
The pioneering sixteenth-century algebraist Girolamo Cardano had qualms about this alchemy, and toyed with the idea of defining the product of two negative numbers to be negative. In the intervening centuries, legions of schoolchildren have been tempted to follow that road. But the mathematical community staunchly insists that it’s the wrong road. Why?
The law of signs goes back at least as far as the seventh-century Indian mathematician Brahmagupta, who wrote: “The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.” By the end of the thirteenth century if not sooner, Chinese mathematicians had adopted these ideas too, with little fuss. But whereas Asian mathematicians were matter-of-fact about the public mating of signed numbers and the nature of their multiplicative offspring (both endogamous and exogamous), Europeans, and especially the English, were uptight. They knew they could use negative numbers to get the correct positive answers to problems, but they worried, was it right to do so? Negative numbers seemed deeply unnatural to them, and even today, English-speaking mathematicians reserve the term “natural” for the ordinary counting numbers. I’m reminded of the uneasy attitude of magic-scholars in the early part of Susanna Clarke’s wonderful novel Jonathan Strange and Mr. Norrell: in Clarke’s England, the question of whether spells work is subordinated to the question of whether one might cast them without compromising one’s gentility.
In the late seventeenth century, in our England (not Susanna Clarke’s), the mathematician John Wallis tried to dispel some of the smoke surrounding negative numbers: “Yet is not that Supposition (of Negative Quantities) either Unuseful or Absurd when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing: Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense.”
In accord with his physical interpretation of negative quantity, Wallis gave Europe the number line. It’s a great tool for understanding addition of positive and negative numbers but in a subtle way it may contribute to misunderstanding of multiplication: since the laws of the natural world are (with minor exceptions) bilaterally symmetric, and since multiplication is natural, shouldn’t multiplication be symmetrical too? One kind of number goes right, and the other goes left; there’s no reason to privilege right over left, is there?
My main goal this month (and next month) is to shed some light on why mathematicians define multiplication of negative numbers the way they do, and to discuss a deviant alternative to the standard law of signs.
Trigger warning: There will be an evil clown. Also balloons.
LAYING THE GROUNDWORK
The number of times I’ve taught precollege students about negative numbers, while not exactly a negative number itself, comes close. So I don’t really have much relevant experience. But my experience as a mathematician leads me to recognize that the question “Why is −1 times −1 equal to +1?” is a deep one. A question this deep deserves to be answered more than once, and in more than one way. A good educational system will lay groundwork for the answer years before the question is asked.
In my own case, the groundwork for understanding the law of signs was laid by pre-school experiences, such as watching “The Addams Family” — specifically, the episode in which Gomez gets amnesia. The episode played with two common 1960s TV tropes: identity amnesia, and the belief the same thing that causes a condition (in this case, a whack on the head) cures it. Medically, this is nonsense; mathematically, it embodies the idea that in many situations, doing an operation twice has the same effect as not doing it at all. (See, Mom: I told you that the show was educational.)
Also, in the lobby of my grandmother’s Manhattan apartment building there were two mirrors that formed a concave 90 degree angle. If I walked past it, I’d see myself walking past the other way, and if I looked carefully I’d see that double-mirror-Jim parted his hair on the left the way I did (unlike single-mirror-Jim who parted his hair on the right).
Later, when I started getting an allowance, the relationship between less and more was clarified by the practical insight that to spend less is to save more. At that stage I didn’t know enough algebra to understand an equation like a−(b−c) = a−b+c, but on some level I think I had a glimmerings that, years later, would become an understanding of where that plus sign comes from.
Ultimately, I learned about negative numbers, which were cool but mystifying. Unlike the number 2, which makes sense in all contexts where numerical answers are expected, −2 is sometimes meaningful (“What’s the temperature?”) and sometimes not (“How many cookies would you like?”). I no longer remember how I became comfortable with the law of signs. I imagine I was given some partial explanations, but since the best justifications for the law come from algebra, and since I didn’t know algebra yet, I imagine that a leap of faith on my part played a role. It helped to have teachers I trusted. Also, although I was an inquisitive and easily-frustrated child, I wasn’t so rigid as to require an understanding of every step in full detail before moving on to the next. Lastly, I found the law of signs amusing; I wanted it to be true. Even if it felt like magic, it was good magic, and I wanted to see where it led.
What should a teacher say when the question of why negative times negative equals positive explicitly arises? The most satisfying sort of answer to give would be a concrete demonstration, the way one can with formulas like 6 times 1/2 equals 3. Discussing the law of signs, mathematician David Mumford writes: “One difficulty in arguing for this rule is that there are not many simple cases of quantities in the world where the units of the two multiplicands allow us to infer the multiplication rule using our physical intuition about the world.”
There aren’t many, but there are some. Here are some physical examples of the law of signs.
Clowns: You’re at a circus, holding two helium balloons, each of which provides 3 ounces of lift. A clown hands you two 3-ounce flowers. By how much does your weight increase? He has given you 2 flowers, each of which weighs 3 ounces, so your weight increases by 2 times 3 ounces, or 6 ounces. But now the treacherous clown reveals that his seeming act of generosity was a mere pretext for theft disguised as trade, and he snatches away your two balloons. By how much does your weight increase? The clown has given you −2 balloons, each of which weighs −3 ounces, so your weight increases by −2 times −3 ounces, or 6 ounces.
Movies: If you play a video at double speed, and make a video of that and play it at triple speed, the result is a speed-up by a factor of six. So multiplication is a good way to describe speed-up of videos (and sound recordings for that matter). Multiplication by 1/2 means slowing things down by a factor of 2. But what does it mean to play a movie at negative speed? Why, to play it backwards of course! (This answer might have been more intuitive to someone who grew up playing with a phonograph in the days of vinyl records, when you could affect the playback speed by turning the disk by hand, clockwise or counterclockwise. For that matter, some kinetoscopes would let you play movies at a whole range of speeds, both positive or negative.) If you take a movie of a movie being played backwards and play it backwards, you get the original movie played forward.
Lenses: If you blow up an image by a factor of two and blow up the result by a factor of three, you get a blow-up factor of six. So multiplication is a good way to describe blow-up of images. Multiplication by 1/2 means shrinking by a factor of 2. But what does it mean to magnify an image by a negative factor? Here the answer is less obvious, but the laws of optics suggest that a negative magnification results in an inverted image, that is, one that is rotated by an angle of 180 degrees. (Look in a concave mirror and move your head so that it passes from one side of the focal point to the other and you’ll see what I mean.) If you invert an inverted image, you get a non-inverted image.
Showers: This isn’t as quantitative as the others, but maybe you’ll like it. True story: A few weeks ago when I was starting to write this essay, I was taking a shower in my parents’ apartment and realized mid-shower that the water was too hot. I’d turned on the two shower taps oblivious to which tap was which and to which way I’d turned them (when a shower tap is off, it can be turned only one way, so one can do it on autopilot). I wanted a cooler shower, but I didn’t know which knob to turn or which way to turn it. If I reduced the flow of hot water or increased the flow of cold water, I’d be okay; but if I reduced the flow of cold water or increased the flow of hot water I’d scald myself. Fortunately I guessed doubly right (or doubly wrong). Life is full of situations like this, if we’re alert to them.
My examples of clowns, movies, lenses, and showers won’t satisfy everyone. Some students may not want that kind of answer at all; instead, they are saying “This way of thinking hurts my brain, so can you please tell me why it’s useful?” Those students may like the way in which the law of signs, in combination with the compound distributive law (a+b)(c+d) = ac+ad+bc+bd, gives us shortcuts for doing calculations. Want to square 999 in your head? Think of it as (1000 + (−1)) × (1000 + (−1)). Then we can view it as 1000 × 1000, plus 1000 × (−1), plus (−1) × 1000, plus (−1) × (−1), or 1,000,000 plus −1000 plus −1000 plus 1, or 1,000,000 minus 1000 minus 1000 plus 1, or 998,001. That’s a lot easier than doing it the standard way, multiplying all those 9’s and adding together all those 81’s. This is a utilitarian rationale for the law of signs.
Other students may be more reassured to see that, with the ordinary law of signs, the tapestry of the multiplication table is very orderly, with the entries in each row and column showing steady increase or decrease. Figure 1 shows a partial table for the ordinary multiplication operation × ; Figure 2 shows the corresponding table for the deviant multiplication operation, which I’ll write as ❎. (In both figures, addends increase from left to right and from top to bottom, so that the ordinary multiplication for 0,1,2 times 0,1,2 appears at the lower right.) Alongside the table in Figure 1, I’ve indicated the amount of increase in each row as we read from left to right and the amount of increase in each column as we read from top to bottom. I’ve tried to do the same with Figure 2, but with some of the rows and columns it can’t be done, as the amount of increase isn’t consistent. This is an aesthetic rationale for the standard definition of multiplication.
One of the great satisfactions of mathematics, and one of the great mysteries, is the way in which aesthetics and utility impel us in the same direction.
Here’s another thought on pedagogy: I suspect that many students will appreciate the fact that multiplying by −1 is the same operation as one that they’ve already mastered, namely negation; hearing one’s teacher say “You already know how to multiply by minus one, even if you didn’t know that you knew it” is likely to be encouraging. And if you’re already comfortable with the formula −(−a) = a, then, combining it with the formula (−1) × a = −a, you’re likely to accept that (−1) × (−1) × a = (−1) × (−a) = −(−a) = a = 1 × a, which makes −1 × −1 = 1 easier to swallow.
What worked for you? And if you’re a teacher, what works for your students?
Here are some resources that I think could be helpful:
- A web page and a two-part video from master teacher James Tanton
- The mathisfun.com web-page on multiplying negatives
- Mike Lawler’s way of explaining the law of signs to his son
- Math-ed guru Henri Picciotto’s “Two Negatives” activity-sheet
I have more to say (including an answer to the question “Why does the law of signs treat right and left differently?”), but that’ll have to wait until next month.
Thanks to Joerg Arndt, John Baez, Sandi Gubin, Tom Karzes, Mike Lawler, Alberto Martínez, David Mumford, Henri Picciotto, Mike Stay, James Tanton, and Glen Whitney.
Next month (Oct. 17): Going Negative, part 2.
#1: I discussed the example of (1000 + (−1))(1000 + (−1)), in connection with the compound distributive law (a+b)(c+d) = ac+ad+bc+bd, as a rationale for defining (−1)(−1) = +1. It’s also possible to mentally square 999 without using lots of 9s and without using negative numbers: just use the variant distributive law (a−b)(c−d) = ac−ad−bc+bd. Indeed, as Martin Gardner points out in his essay “Negative Numbers”: “Greek mathematicians knew that (10−4)(8−2) equals (10×8)−(4×8)−(2×10)+(2×4). To recognize such an equality is to accept implicitly what later was called the law of signs: The product of any two numbers with like signs is positive and the product of any two numbers with unlike signs is negative. It was just that the Greeks preferred not to call −n a number.” The Mike Lawler video I mentioned shows you how to think about (a−b)(c−d) = ac−ad−bc+bd geometrically, and demonstrates the power of the area model of multiplication.
#2: A great example of the power of “negative thinking” is the classic problem described by Martin Gardner in his essay “The Monkey and the Coconuts”. It turns out (spoiler alert!) that one aid to solving the problem is to allow the variable that represents the unknown number of coconuts to take on a negative value, even though this has no physical meaning. The physicist Paul Dirac is often credited with this trick, and it’s tempting to imagine that he came up with it because of the same predisposition toward negative thinking that led him to predict the existence of antiparticles, but Dirac said he got the trick from another mathematician, who said that the trick wasn’t original with him, and there the trail runs cold.
And where did the problem itself come from? We don’t know, but problems of this kind were discussed by the 9th century Indian mathematician Mahavira. I can’t help wondering whether Mahavira (who surely had read Brahmagupta) knew about the negative solutions to such problems and their usefulness in finding positive solutions.
#3: Another great example of a puzzle about ordinary numbers that can be simplified by allowing negative numbers to enter the scene is the staircase sum problem, also known as the trapezoidal number problem, whose pedagogical virtues have been exploited by Henri Picciotto, James Tanton, and Paul Zeitz (and probably other teachers too). See Picciotto’s description, listed in the References.
Martin Gardner, “The Monkey and the Coconuts” (chapter 9 in “The Second Scientific American Book of Mathematical Puzzles and Diversions”). See also the online discussions of the problem by Gary Antonick and Eric Weisstein .
Martin Gardner, “Negative Numbers” (chapter 11 of “Penrose Tiles to Trapdoor Ciphers, and the Return of Dr. Matrix”).
David Mumford, “What’s So Baffling about Negative Numbers?: A Cross-Cultural Comparison”.
Henri Picciotto, “Staircase Sums” (lesson 5.9 of his “Algebra”). See also his version for teachers.
It’s interesting that you mention Dirac and negative coconuts, because he once wrote:
Feynman went much further with negative probabilities, and I wrote about that here:
• John Baez, Negative probabilities, Azimuth, 19 August 2013.
You might like this post, because it mentions generating functions and also an imaginary object called the “half-coin”. Two flips of this equal one flip of an ordinary coin!
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