Argument of Complex Number Calculator

Formula first

Overview

The argument of a complex number is the angle formed between the positive real axis and the vector representing the number in the complex plane. This value is a crucial component for converting complex numbers from Cartesian coordinates (a + bi) into polar or exponential forms.

Symbols

Variables

a = Real Part, b = Imaginary Part, $\theta$ = Argument (Rad)

Apply it well

When To Use

When to use: This formula is used whenever you need to determine the direction of a complex number relative to the origin. It is a prerequisite for calculations involving De Moivre's Theorem, such as finding the powers or roots of complex values.

Why it matters: In electrical engineering and physics, the argument represents the phase shift of a waveform. It allows professionals to quantify the timing offset between voltage and current in AC circuits or to analyze frequency responses in signal processing systems.

Avoid these traps

Common Mistakes

• Degrees vs Radians.
• Wrong quadrant (adding Ϭ).

One free problem

Practice Problem

Calculate the argument 't' in degrees for a complex number with a real part a = 5 and an imaginary part b = 5.

Solve for: $t$

Hint: When the real and imaginary components are identical and positive, the vector bisects the first quadrant.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Wikipedia: Argument (complex analysis)
2. Britannica: Complex number
3. Stewart, James. Calculus: Early Transcendentals.
4. Halliday, Resnick, and Walker, Fundamentals of Physics
5. Wikipedia: Radian
6. Complex Variables and Applications, 9th ed. by James Ward Brown and Ruel V. Churchill
7. AQA Further Mathematics — Core Pure (Complex Numbers)