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Dr. Keith Devlin


How do you read "-3"?

Posted: 03/12/2012 5:15 pm

How do you say "-3": "negative three" or "minus three"?

It sounds like a simple enough question. But a recent group discussion on LinkedIn generated over 60 contributions when I last checked. People seem to have very clear preferences as to what is "right." Unfortunately, those preferences differ.

In expressions such as "5 - 3" there is general agreement: You say "Five minus three." In this case, the symbol "-" denotes the binary arithmetic operation of subtraction, expressed as "minus", and the expression "5 - 3" means "subtract 3 from 5," or "5 minus 3."

It is when the expression "- 3" appears on its own that the fun begins. Traditionalists (the kind of people who rely on the Chicago Manual of Style and insist I should have put the colon and question mark inside the quotes in my opening question) will say it should be read as "negative three." (Your last math teacher probably said that too.) But in everyday situations, most people say "minus three." For example, I doubt you have ever heard the TV weather person say that the temperature will fall to "negative three (degrees)." No, she or he will have said "minus three."

My sense (and it is nothing more than that) is that almost all professional mathematicians will probably come down on the side of "minus three." The reason is that those of us in the math biz put the minus sign in front of numbers all the time, and those numbers may themselves be positive or negative. For example, we frequently find ourselves referring to numbers such as "- N" where N might turn out to be -3. Since the result in such a case is in fact a positive number, it seems totally wrong (to us) to refer to it using "negative," which we take as indicating the sign of a number. In other words, we view "negative" as an adjective, which tells us that the number is less than 0.

We generally view "-", on the other hand, not as a symbolic adjective but an operation. Usually it is a binary operation, but sometimes we think of it as a unary one. In such cases, it simply changes the sign (or reflects it in the origin on the number line, if you want to use geometric language). In other words, we do not view "-N" as indicating that the number is positive or negative, rather that it has the opposite sign to N. Quite simply, "minus" can function as a sign-changing, unary operator.

You might claim that we could use "negative" similarly, so that "negative negative three" means "three," but to me at least (and I know I am not alone) it sounds bizarre (in a way that "minus minus three" does not).

If the symbol "-" were only ever put in front of positive numbers, it would be fine to read it as "negative". In fact, it could be advantageous to do so, as it would tell us the sign of the number. Since many people outside of mathematics, science, and engineering may in fact never encounter a double-negative (except to wonder how it works out to be positive), that might help explain why the "negative three" camp is well occupied.

But for anyone dealing with numbers in a scientific context, for whom "-" is frequently applied to a negative (sic) quantity or to one whose sign is not known, rather than use "negative" on some occasions and "minus" on others, the latter is the default reading on all occasions.

That, I think, makes the case for professional scientists and mathematicians always using "minus three." But why do so many non scientists use the same terminology? My guess is that they either pick it up from their science teachers at school (but perhaps not their math teachers), or maybe from the TV weather person. TV weather forecasters may well not be trained scientists, but they do have to read scientific reports from professional meteorologists, and that could account for the domination of "minus."

So, if you want a ruling from a qualified mathematician, I'll give one: Always read "-N" as "minus N." Feel free to use my name to try to settle any dispute. But if you ask me to get personally involved, I'll give you an answer right out of Top Gun: "That's a negative."


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