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A complex number’s argument describes its direction from the origin, while a complex sequence converges when its real and imaginary parts both converge. These ideas are linked by the complex plane: arguments describe where individual numbers point, and convergence describes where a sequence of such points settles.
How do you find the argument of a complex number?
Write a complex number as z = a + bi. It corresponds to the point (a, b) in the complex plane. Its modulus, or distance from the origin, is |z| = √(a² + b²). For a nonzero number, an angle θ measured anticlockwise from the positive real axis gives its direction, so it can be written in polar form as z = r(cos θ + i sin θ), where r = |z|. Delft University of Technology’s polar-form explanation uses this convention.
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Use both coordinates to determine the quadrant
For z = a + bi, tan θ = b/a when a ≠ 0. But the tangent value alone does not distinguish angles in different quadrants. Check the signs of both coordinates before choosing an angle; a two-argument arctangent, commonly written atan2(b, a), does this when its convention is known.
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One free scan finds every outdated or missing driver and matches the right update for your exact hardware.Free scan · exact hardware match- If a > 0, the point is on the right half-plane.
- If a < 0, it is on the left half-plane, so an angle based only on arctan(b/a) may need adjustment by π.
- If a = 0 and b > 0, the direction is π/2; if a = 0 and b < 0, it is −π/2 under the interval convention below.
- If b = 0 and a > 0, the direction is 0; if b = 0 and a < 0, it lies on the negative real axis.
At z = 0, the modulus is zero but the direction is undefined: the origin has no angle from itself.
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What is the difference between an argument and the principal argument?
For any nonzero z, an argument is any angle θ that represents its direction. Because adding a full turn leaves the point unchanged, if θ is an argument then so is θ + 2kπ for every integer k. Thus the argument is not unique.
A principal argument selects one representative from those equivalent angles. There is no single endpoint convention used everywhere, so state the interval when using it. Delft’s text describes its principal value as between −π and π. If that is interpreted as the open interval (−π, π), the negative real axis has no representative in the interval; a common alternative is (−π, π], which assigns the negative real axis the value π. Follow the convention used in your course or software, especially at that axis. The notation arg(z) is often used for the set of arguments, while Arg(z) denotes a chosen principal value, but notation also varies.
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 all terms satisfy |zn − ζ| < ε. Geometrically, every sufficiently late term lies inside the open disk of radius ε centered at ζ. This is the usual metric definition with the complex modulus; see Complex Analysis’s treatment of sequences and series.
Test the real and imaginary parts separately
Write each term as zn = xn + iyn. Then zn converges if and only if the real sequence (xn) and the real sequence (yn) both converge. If xn → x and yn → y, the complex limit is ζ = x + iy. The NIST Digital Library of Mathematical Functions, §1.9 states this component criterion.
Example
Consider zn = 1/n + i(2 − 1/n). Its real part tends to 0 and its imaginary part tends to 2, so the sequence converges to 2i. Equivalently, its distance from 2i is √((1/n)² + (−1/n)²) = √2/n, which tends to zero.
When convergence fails
A complex sequence fails to converge if either component fails to converge. For example, zn = (−1)n + i/n has an imaginary part tending to zero, but its real part alternates between −1 and 1. Therefore the complex sequence has no limit.
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