PEMDAS and negative numbers

By standard mathematical convention, \(-1^2=-1\). At the same time, students are taught to refer to \(-1\) as a negative number. A friend recently led me to realize that these two conventions are at logical odds with each other, as discussed below. Furthermore, while the first convention is generally taught in high school, the mathematical argument for it relies on philosophical distinctions not often made until calculus or later, leaving students faced with an apparent paradox and no clear way to resolve it.

So let’s take a look at what’s going on. I’ll do my best to make it comprehensible.

First, let’s start with the argument in favor of \(-1^2=1\), that is, that \(-1^2=(-1)^2\). This relies on the notion that the negative sign is part of the number, that is, that \(-1\) is an intrinsically different number than \(1\) is. Based on how numbers are taught in secondary school, this certainly makes a good deal of sense. We don’t write \((1)^2\) or \((\pi)^2\) or even \((i)^2\), so why would we need parentheses in the case of a negative number (and note the terminology here: we call it a negative number, not a negated number). If we see numbers in terms of a static coordinate plane, which is I believe students are generally encouraged to do, it is obvious to see how students would conclude that negative numbers are intrinsically different from positive ones.

However, terminology notwithstanding, is the negative sign a part of the number? Part of the point of view that it is not is introduced in algebra: \(3-2 = 3+(-2)\) is not unusual as part of a discussion of negative numbers, for instance. Both of these expressions, after all, equal one. Students are taught about number lines that go left (for negative) and right (for positive), in lessons that treat real numbers in terms of their distance from zero.

The distinction between a signed value and its unsigned equivalent becomes crucial in calculus: It represents, for instance, the difference between velocity (signed) and speed (unsigned). If we say that a car is traveling at a speed of 55 miles per hour, we’re not making any statement about what direction that car is traveling; if it goes 55 miles per hour for an hour, then 55 miles per hour for another hour, it could be anywhere from 0 to 110 miles from where it started. However, if we say that a car’s velocity is 55 miles per hour, we’re saying (in strictest mathematics sense) that it’s traveling in a specific direction; if it then turns around and comes back to that speed, it’s traveling at a velocity of -55 miles per hour.

Returning to number theory, I think it’s fairer to refer to numbers as being negated rather than as negative. If we think of how real numbers are used pragmatically, this makes sense. If a cash register shows -$1.54, that doesn’t mean that I still owe -$1.54, that means that the store owes me $1.54. The negative sign isn’t an intrinsic part of the value, in other words, but rather shows the direction of the numeric imbalance, with the value itself representing the distance from zero.

In more technical terms, this means that the negative sign is a unary operator: It’s a function that takes a single variable. It does raise a potential question: Why isn’t there a unary operator to indicate a positive distance from zero? The answer is: There is, the positive sign, which is rarely shown because it’s taken to be the default.

If we take the negative sign to be a unary operator rather than an intrinsic part of a number, it becomes clearer why we technically need parentheses with \((-1)^2\): Subtraction (whether binary or unary) takes lower priority than exponentiation.

There’s no particular reason why we couldn’t have decided, as a convention, to resolve the potential ambiguity of \(-1^2\) the opposite way, by representing the negation of one squared as \(-(1^2)=-1\). However, note that this disambiguation still relies on the recognition of a unary operator, so from the perspective of the mathematical philosopher, this may not be the preference. If we have to recognize a unary negation operator, why not simply go all the way and treat so-called negative numbers as negated numbers as a complete class?

My response to that is that, while it’s nice to be consistent and lay groundwork, it’s also important to know when to pick one’s philosophical battles. If students consider negative numbers to be intrinsically negative, and thus write \(-1^2\) to represent the square of a negative number, the teacher is then left to decide whether the rigor of conventional accuracy is more important or the rudiments of student understanding is more important. At the high school level, I tend to lean towards the latter: If I can tell what a student meant, and the student gets the correct final answer from the input conditions, I’m disinclined from being a stickler on parentheses in this specific case.

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