JavaScript’s two zeros

[2012-03-23] numbers, dev, javascript, jslang

JavaScript has two zeros: −0 and +0. This post explains why that is and where it matters in practice.

The signed zero

Numbers always need to be encoded to be stored digitally. Why do some encodings have two zeros? As an example, let’s look at encoding integers as 4-digit binary numbers, via the sign-and-magnitude method. There, one uses one bit for the sign (0 if positive, 1 if negative) and the remaining bits for the magnitude (absolute value). Therefore, −2 and +2 are encoded as follows.

Binary 1010 is decimal −2
Binary 0010 is decimal +2

Naturally, that means that there will be two zeros: 1000 (−0) and 0000 (+0).

In JavaScript, all numbers are floating point numbers, encoded in double precision according to the IEEE 754 standard for floating point arithmetic. That standard handles the sign in a manner similar to sign-and-magnitude encoding for integers and therefore also has a signed zero. Whenever you represent a number digitally, it can become so small that it is indistinguishable from 0, because the encoding is not precise enough to represent the difference. Then a signed zero allows you to record “from which direction” you approached zero, what sign the number had before it was considered zero. Wikipedia nicely sums up the pros and cons of signed zeros:

It is claimed that the inclusion of signed zero in IEEE 754 makes it much easier to achieve numerical accuracy in some critical problems, in particular when computing with complex elementary functions. On the other hand, the concept of signed zero runs contrary to the general assumption made in most mathematical fields (and in most mathematics courses) that negative zero is the same thing as zero. Representations that allow negative zero can be a source of errors in programs, as software developers do not realize (or may forget) that, while the two zero representations behave as equal under numeric comparisons, they are different bit patterns and yield different results in some operations.

JavaScript goes to some lengths to hide the fact that there are two zeros.

Hiding the zero’s sign

In JavaScript, you’ll usually write 0, which always means +0. But it also displays −0 simply as 0. The following is what you see when you use any browser command line or the Node.js REPL:

    > -0
    0

The reason is that the standard toString() method converts both zeros to the same "0".

    > (-0).toString()
    '0'
    > (+0).toString()
    '0'

The illusion of a single zero is also perpetrated by the equals operators. Even strict equality considers both zeros the same, making it very hard to tell them apart (in the rare case that you want to).

    > +0 === -0
    true

The same holds for the less-than and greater-than operators – they consider both zeros equal.

    > -0 < +0
    false
    > +0 < -0
    false

Where the zero’s sign matters

The sign of the 0 rarely influences results of computations. And +0 is the most common 0. Only a few operations produce −0, most of them just pass an existing −0 through. This section shows a few examples where the sign matters. For each example, think about whether it could be used to tell −0 and +0 apart, a task that we will tackle in Sect. 4. In order to make the sign of a zero visible, we use the following function.

    function signed(x) {
        if (x === 0) {
            // isNegativeZero() will be shown later
            return isNegativeZero(x) ? "-0" : "+0";
        } else {
            // Otherwise, fall back to the default
            // We don’t use x.toString() so that x can be null or undefined
            return Number.prototype.toString.call(x);
        }
    }

Adding zeros

Quoting Sect. 11.6.3 of the ECMAScript 5.1 specification, “Applying the Additive Operators to Numbers”:

The sum of two negative zeros is −0. The sum of two positive zeros, or of two zeros of opposite sign, is +0.

For example:

    > signed(-0 + -0)
    '-0'
    > signed(-0 + +0)
    '+0'

This doesn’t give you a way to distinguish the two zeros, because what comes out is as difficult to distinguish as what goes in.

Multiplying by zero

When multiplying with zero with a finite number, the usual sign rules apply:

    > signed(+0 * -5)
    '-0'
    > signed(-0 * -5)
    '+0'

Multiplying an infinity with a zero results in NaN:

    > -Infinity * +0
    NaN

Dividing by zero

You can divide any non-zero number (including infinities) by zero. The result is an infinity whose sign is subject to the usual rules.

    > 5 / +0
    Infinity
    > 5 / -0
    -Infinity
    > -Infinity / +0
    -Infinity

Note that -Infinity and +Infinity can be distinguished via ===.

    > -Infinity === Infinity
    false

Dividing a zero by a zero results in NaN:

    > 0 / 0
    NaN
    > +0 / -0
    NaN

Math.pow()

The following is a table of the results of Math.pow() if the first argument is zero:

pow(+0, y<0) → +∞
pow(−0, odd y<0) → −∞
pow(−0, even y<0) → +∞

Interaction:

    > Math.pow(+0, -1)
    Infinity
    
    > Math.pow(-0, -1)
    -Infinity

Math.atan2()

The following is a table of the results that are returned if one of the arguments is zero.

atan2(+0, +0) → +0
atan2(+0, −0) → +π
atan2(−0, +0) → −0
atan2(−0, −0) → −π

atan2(+0, x<0) → +π
atan2(−0, x<0) → −π

Hence, there are several ways to determine the sign of a zero. For example:

    > Math.atan2(-0, -1)
    -3.141592653589793
    > Math.atan2(+0, -1)
    3.141592653589793

atan2 is one of the few operations that produces −0 for non-zero arguments:

atan2(y>0, +∞) → +0
atan2(y<0, +∞) → −0

Therefore:

    > signed(Math.atan2(-1, Infinity))
    '-0'

Math.round()

Math.round() is another operation that returns −0 for arguments other than −0 and +0:

    > signed(Math.round(-0.1))
    '-0'

Here we have the effect that we talked about at the beginning: The sign of the zero records the sign of the value before rounding, “from which side” we approached 0.

Telling the two zeros apart

The canonical solution for determining the sign of a zero is to divide one by it and then check whether the result is -Infinity or +Infinity:

    function isNegativeZero(x) {
        return x === 0 && (1/x < 0);
    }

The above sections showed several other options. One original solution comes from Allen Wirfs-Brock. Here is a slightly modified version of it:

function isNegativeZero(x) {
    if (x !== 0) return false;
    var obj = {};
    Object.defineProperty(obj, 'z', { value: -0, configurable: false });
    try {
        // Is x different from z’s previous value? Then throw exception.
        Object.defineProperty(obj, 'z', { value: x });
    } catch (e) {
        return false
    };
    return true;
}

Explanation: In general, you cannot redefine a non-configurable property – an exception will be thrown. For example:

    TypeError: Cannot redefine property: z

However, JavaScript will ignore your attempt if you use the same value as the existing one. In this case, whether a value is the same is not determined via ===, but via an internal operation that distinguishes −0 and +0. You can read up on the details in Wirfs-Brock’s blog post (freezing an object makes all properties non-configurable).

Conclusion

We have seen that there are two zeros, because of how the sign is encoded for JavaScript’s numbers. However, −0 is normally hidden and it’s best to pretend that there is only one zero. Especially, because the difference between the zeros has little bearing on computations. Even strict equality === can’t tell them apart. Should you, against all expectations or just for fun, need to determine the sign of a zero, there are several ways to do so. Note that the slightly quirky existence of two zeros is not JavaScript’s fault, it simply follows the IEEE 754 standard for floating point numbers.