Negative numbers squared

Background

Mathematical conventions represent the linguistic aspect of mathematics. One of the strengths of modern mathematics is the way in which we can represent some fairly complex ideas in a shortened, rigorous symbol set. However, as a result of these abbreviations, there are some ambiguities that are generally settled democratically: Some group decides that the ambiguity will be settled in one fashion, and eventually enough of the majority decide that this is the way that things ought to be that such conventions become mathematical “facts”.

I think it is important that we ever distinguish mathematical facts from mathematical conventions. A fact is something that has an underlying truth; a convention is something that’s been decided, either formally or by habit, and has been accepted as the proper way to do things.

Negation

Before I get into the specific example that inspired this entry, I want to give a little bit of background on negation. Standard arithmetic has three uses of the negative sign, although only two are particularly common prior to algebra:

  1. To indicate subtraction of one value from another, as in \(4 – 3 = 1\). This a binary function, and the standard sign is a hyphen; typeset sources use a longer dash, the Unicode entity − (−).
  2. To indicate a negative value, as in -3. This sign is part of the value; a (very) few typeset sources use a separate symbol for this negation. In HTML, the separate symbol for negation can be created using a superscript hyphen (3 vs. -3) or by using the specific Unicode entity ⁻ (⁻).
  3. To indicate the additive inverse of a value, that is, the result of \(0 – x\), where the 0 is implied. In some cases, the zero symbol is left off merely for convenience purposes; in others, there is some practicality to using a unary rather than binary operator (although cases where there’s an absolute need for a unary operator elude my mind right now).

One thing that complicates matters is that subtraction is sometimes presented as the addition of the inverse. For instance, \(4 – 3 = 4 + \;^-3\). It is indeed a mathematical fact that the value of subtracting x from y is equivalent to the value of adding the additive inverse of x to y, but that does not mean that the value of x is equivalent to the value of its additive inverse. Functions with different inputs can yield the same value output; we even emphasize this in other cases, such as teaching that 95 + 107 = 100 + 102 (that is, we can make addition problems easier by adjusting the values in opposite, balanced directions).

To read Internet opinions on negation, it feels like there’s a weak concept of actual negative values even among some mathematically minded sorts; the link I provided above distinguishes only binary and unary operators, while on one page, Dr. Math likewise speaks in terms of unary operators with regards to the matter at hand below.

This is crucial, which is why I’m taking such pains to indicate the difference: \(^-3\) is an inherently negative value, while \(-3\) reflects the application of the unary “additive inverse” operator on the value \(3\). In standard typing, including most typeset sources, they are displayed in exactly the same way, and are either disambiguated entirely through the text, or not at all. Since \(^-3\) and \(-3\) have the same operational value, it generally doesn’t matter, so disambiguation is not needed.

Negative numbers squared

There is one fairly common place where it does matter, however. Consider: \[-3^2\]

Is this the square of negative three, or is the negation of the square of three? If we used separate signs for the operator and the negation, there would be no such concern: \(^-3^2 = 9\) and \(-3^2 = -9\). The unary operator is the result of multiplying a positive value by -1, and hence the exponentiation takes precedence over the multiplication. Thought of another way, \(-3\) is shorthand for \(0-3\), and so \(-3^2\) is shorthand for \(0-3^2\). There’s simply no cogent argument that \(0-3^2 = 0+9\), which is what would have to be the case if \(-3^2 = 9\).

At the same time, \(^-3\) is a valid mathematical value in its own right, not simply the result of a unary operator on a positive value. As such, it has a square value, that being \(9\).

By convention, -32 = -9, while (-3)2 = 9. This is how the ambiguity is usually resolved. As a convention, that’s fine, although based on my own experience with high school students, it’s not an ideal situation.

The problem comes in when mathematicians commit to this convention to the point of treating it as a fact. Here’s a quote from Dr. Math: “If you ever see the expression -3^2 evaluated as 9, that’s incorrect. The exponentiation is always done before the negation unless there are parentheses there to indicate otherwise” (emphasis mine).

The phrase “before the negation” suggests that the author of this particular Dr. Math entry (and other people I’ve talked to) haven’t considered -3 as a negative value, but rather solely as a negated value. It’s not that the concept is beyond the ken of the author; two paragraphs later, he gives the example, “‘If I take negative three and square it, I get nine'” (emphasis mine).

A negated value is the result of the application of the unary additive inverse operator on a value; a negative value reflects an inherent state. Numerically, negated 3 and negative 3 have the same value, but that doesn’t mean they’re identical concepts.

I’ll repeat: Most of the time it just doesn’t matter. It’s picking fly dust out of pepper. We could represent all negative values as -1 times their (positive) additive inverse, just as we represent all imaginary numbers as i times their real counterparts. But just because we choose to represent 3i as the result of a multiplication operator, that doesn’t mean that the value represented by 3i doesn’t exist, any more than it would mean that if we chose to represent 4 as 2 + 2, 4 has no value.

Indeed, the ultimate result of this “turtles all the way down” approach is that there are only five values needed to represent all possible values (within the complex plane): 0, 1, b, a, and i. b represents the base (which would be 2), a represents the additive inverse, and i represents the imaginary multiplicand (binary representations go one step yet further by reserving a place value slot for and characterizing it as 1, but that’s only reasonable when the number of place values is strictly limited).

Microsoft Excel’s intransigence and confusing calculators

While the convention -3^2 = -9 is reasonable in contexts where there’s only one negative/negation/minus sign, some calculators have a \(^-\) sign on their number keypad and a \(-\) sign on their operator pad. Since the \(^-\) sign is on the number pad, it’s reasonable for students to conclude that it’s meant to be part of the number, not an operator. However, all calculators I’ve seen, regardless of whether they have the extra button or not, interpret 3^2 as -9. I disagree with that.

On the other hand, Microsoft Excel has firmly and apparently defiantly kept to interpreting =-3^2 in a cell as -9. When I learned of this, I applauded… at first. Sadly, the situation is not as halcyon as it seems.

First of all, let’s say A1 contains 5. If A2 contains =-A1^2, then A2 will return 25. There is no question but that =-A1^2 reflects the application of the unary additive inverse function, and by the standard order of operations, =-A1^2 should return -25, as argued above. A1 is not a value, it’s a variable; as such -A1 cannot be a value, nor can it be a variable; rather, it’s a variable to which a function is being applied.

Secondly, Excel VBA, being a programming language, behaves as programming languages tend to behave in this regard:

Sub test()
 x = 5
 y = -5
 Debug.Print x ^ 2; y ^ 2; -5 ^ 2
End Sub

returns 25 25 -25. y ^ 2 is treated as a negative (or already-negated) value, while -5 ^ 2 performs the exponentiation first. Just as with Excel’s inconsistent rounding (3.45 is rounded as 3.5 in a cell and as 3.4 in VBA), this opens the door to potential confusion when someone used to working with cell formulas decides to write some macros.

I’ll leave off with a reference to WolframAlpha: If you enter “-3^2”, it returns -9; if you enter “negative three squared”, it returns 9. So someone, somewhere, appears to get it.

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