Signed zero

Key takeaways

Signed zero belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Signed zero to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Signed zero from memory before moving on to harder problems.

Reference excerpt

Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, and −0, +0 and 0 are three ways of writing the same number. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 (negative zero) and +0 (positive zero), regarded as equal by the numerical comparison operations but with possible different behaviors in particular operations. This occurs in the sign–magnitude and ones' complement signed number representations for integers, and in most floating-point number representations. The number 0 is typically expressed as +0. The IEEE 754 standard for floating-point arithmetic (presently used by most computers and programming languages that support floating-point numbers) requires both +0 and −0. Its arithmetic can be considered to model a variant of the extended real number line (the real numbers including additional numbers +∞ and −∞). In IEEE 754 arithmetic, ⁠1/−0⁠ = −∞ and ⁠1/+0⁠ = +∞; the divisions ⁠±0/±0⁠ and ⁠±∞/±∞⁠ return "Not a Number" (NaN). Negatively signed zero echoes the mathematical analysis concept of approaching 0 from below as a one-sided limit, which may be denoted by x → 0−, x → 0−, or x → ↑0. The notation "−0" may be used informally to denote a negative number that has been rounded to zero. The concept of negative zero also has some theoretical applications in statistical mechanics and other disciplines. 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 usual assumption made in mathematics that negative zero is the same value as zero. Representations that allow negative zero can be a source of errors in programs, if software developers do not take into account that while the two zero representations behave as equal under numeric comparisons, they yield different results in some operations.

Representations Binary integer formats can use various encodings. In the widely used two's complement encoding, zero is unsigned. In a 1+7-bit sign-and-magnitude representation for integers, negative zero is represented by the bit string 1000 0000. In an 8-bit ones' complement representation, negative zero is represented by the bit string 1111 1111. In all these three encodings, positive or unsigned zero is represented by 0000 0000. However, the latter two encodings (with a signed zero) are uncommon for integer formats. The most common formats with a signed zero are floating-point formats (IEEE 754 formats or similar), described below.

In IEEE 754 binary floating-point formats, zero values are represented by the biased exponent and significand both being zero. Negative zero has the sign bit set to one. One may obtain negative zero as the result of certain computations, for instance as the result of arithmetic underflow on a negative number (other results may also be possible), or −1.0 × 0.0, or simply as −0.0. In IEEE 754 decimal floating-point formats, a negative zero is represented by an exponent being any valid exponent in the range for the format, the true significand being zero, and the sign bit being one.

Properties and handling The IEEE 754 floating-point standard specifies the behavior of positive zero and negative zero under various operations. The outcome may depend on the current IEEE rounding mode settings.

Notation In systems that include both signed and unsigned zeros, the notation 0 + {\displaystyle 0^{+}} and 0 − {\displaystyle 0^{-}} is sometimes used for signed zeros.

Arithmetic Addition and multiplication are commutative, but there are some special rules that have to be followed, which mean the usual mathematical rules for algebraic simplification may not apply. The = {\displaystyle =} sign below shows the obtained floating-point results (it is not the usual equality operator). The usual rule for signs is always followed when multiplying or dividing:

( − 0 ) ⋅ | x | = − 0 {\displaystyle (-0)\cdot \left|x\right|=-0\,\!} (for x {\displaystyle x} different from ±∞)

− 0 | x | = − 0 {\displaystyle {\frac {-0}{\left|x\right|}}=-0\,\!} (for x {\displaystyle x} different from 0)

( − 0 ) ⋅ ( − 0 ) = + 0 {\displaystyle (-0)\cdot (-0)=+0\,\!}

There are special rules for adding or subtracting signed zero:

x + ( ± 0 ) = x {\displaystyle x+(\pm 0)=x\,\!} (for x {\displaystyle x} different from 0)

( − 0 ) + ( − 0 ) = ( − 0 ) − ( + 0 ) = − 0 {\displaystyle (-0)+(-0)=(-0)-(+0)=-0\,\!}

( + 0 ) + ( + 0 ) = ( + 0 ) − ( − 0 ) = + 0 {\displaystyle (+0)+(+0)=(+0)-(-0)=+0\,\!}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Signed zero

Start with the simplest possible case. Write down what Signed zero claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Signed zero before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Signed zero ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Signed zero

In research: Signed zero appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Signed zero in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Signed zero is common in secondary-school and first-year university syllabi. It links to neighbouring topics 0 (number), Computer arithmetic, Sign (mathematics), so understanding it makes those chapters shorter.
In everyday life: Look for Signed zero outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Signed zero in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Signed zero means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Signed zero out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Signed zero in simple terms?

Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, and −0, +0 and 0 are three ways of writing the same number.

Why does Signed zero matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Signed zero?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Signed zero.