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A complex number’s argument is its angle from the positive real axis, while its principal argument is one chosen angle under a stated interval convention. To test whether a complex sequence converges, split each term into real and imaginary parts: the sequence converges exactly when both component sequences converge.
How do you find the argument of a complex number?
Write a nonzero complex number as z = a + bi, where a is its real part and b its imaginary part. Geometrically, it is the point (a, b) in the complex plane. Its modulus, or distance from the origin, is |z| = √(a² + b²). Its argument is an angle θ measured anticlockwise from the positive real axis, so that z = r(cos θ + i sin θ), with r = |z|. Delft University of Technology presents this polar form and direction convention in its polar-form explanation.
The argument is not a single unique angle: adding any integer multiple of 2π gives the same direction and therefore the same complex number. For example, π/4 and π/4 + 2π describe the same angle. The notation arg(z) is often used for the collection of valid arguments, while Arg(z) commonly denotes a selected principal value; notation varies, so check the convention being used.
Choose the quadrant, not just an inverse tangent
Since tan θ = b/a when a ≠ 0, it may be tempting to calculate arctan(b/a) and stop. That inverse tangent returns a restricted range of angles and does not, by itself, distinguish opposite quadrants. Delft’s text warns that this direct substitution does not generally give the correct argument without adjustment.
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- If a > 0, the point is on the right side of the plane; use the sign of b to distinguish above or below the real axis.
- If a < 0, the point is on the left side. The sign of b determines whether it is above or below; adjust the angle to the corresponding second or third quadrant.
- If a = 0 and b > 0, the angle is π/2; if a = 0 and b < 0, it is −π/2 (or an equivalent angle).
- If b = 0 and a > 0, the angle is 0. If b = 0 and a < 0, it lies on the negative real axis, where the principal-value endpoint convention matters.
A correctly defined two-argument arctangent, often written atan2(b, a), uses both signs to choose the quadrant. The zero complex number is a separate case: z = 0 has modulus zero, but no direction, so its argument is undefined.
What is the principal argument?
A principal argument picks one representative from the infinitely many angles for a nonzero complex number. There is no single endpoint convention used everywhere. The Delft text chooses values between −π and π; when using this convention, specify whether the interval is open or closed at each endpoint. One common choice is (−π, π], which assigns π to numbers on the negative real axis. Another convention may exclude π and include −π. Both represent the same direction, but mixing the conventions can cause apparent disagreements at that axis.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How do you tell whether a complex sequence converges?
A complex sequence (zₙ) converges to a complex number ζ if, for every ε > 0, there is a point after which every term is within distance ε of ζ: |zₙ − ζ| < ε. In other words, its terms eventually lie inside every disk centered at the proposed limit. The definition and disk viewpoint are described in the Complex Analysis text’s section on sequences and series.
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Write each term as zₙ = xₙ + i yₙ, where xₙ and yₙ are real sequences. Then:
- (zₙ) converges if and only if (xₙ) converges and (yₙ) converges.
- If xₙ → x and yₙ → y, then zₙ → x + iy.
- If either component fails to converge, the complex sequence does not converge.
This equivalence is stated by the National Institute of Standards and Technology’s Digital Library of Mathematical Functions, §1.9. It turns a complex limit into two ordinary real limits.
Example: split a limit into its components
Consider zₙ = 1/n + i(2 − 1/n). The real component 1/n tends to 0, and the imaginary component 2 − 1/n tends to 2. Therefore zₙ → 2i. The component test also explains why a sequence whose real part approaches one value while its imaginary part keeps oscillating cannot have a complex limit: both parts must settle to limits.
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