Understanding modulus and argument
The modulus and argument together describe a complex number's position in the complex plane in polar form, complementary to the rectangular a + bi form.
| Quantity | Meaning | Formula |
|---|---|---|
| Modulus |z| | Distance from the origin to the point (a, b) — the number's magnitude | √(a² + b²) |
| Argument arg(z) | Angle (in degrees here) from the positive real axis to the point (a, b) | atan2(b, a) |
| Conjugate z̄ | Reflection of a + bi across the real axis | a − bi |
- The argument is reported using the two-argument arctangent (atan2), which correctly places the angle in the right quadrant for any sign combination of the real and imaginary parts, unlike a plain arctan(b/a).
- Division by zero (0 + 0i) is mathematically undefined; this calculator returns no result in that case rather than an arbitrary value.
- Results are rounded to 6 decimal places for the real/imaginary parts and to 4 decimal places for the argument in degrees.
What is a complex number?
A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = −1. Complex numbers extend the real number line into a two-dimensional plane (the complex plane), with the real part plotted on the horizontal axis and the imaginary part on the vertical axis. They arise naturally when solving polynomial equations that have no real roots, such as x² + 1 = 0.
Every complex number a + bi has two additional standard descriptors: its modulus |z| = √(a² + b²), the distance from the origin to the point (a, b) in the complex plane, and its argument arg(z) = atan2(b, a), the angle the line from the origin to (a, b) makes with the positive real axis. Together, modulus and argument give the polar form of a complex number, an alternative to the rectangular a + bi form that is often more convenient for multiplication, division and powers.
Complex numbers have practical applications well beyond pure mathematics: electrical engineers use them to represent alternating-current impedance and phase, control theory uses them to analyze system stability via the locations of poles in the complex plane, and signal processing relies on complex exponentials for the Fourier transform.
How to use this complex number calculator
- Enter the real part (a) and imaginary part (b) of the first complex number, a + bi.
- Select the operation: addition, subtraction, multiplication or division.
- Enter the real part (c) and imaginary part (d) of the second complex number, c + di.
- Read the result in a + bi form, along with its modulus (magnitude) and argument (angle in degrees). Division is undefined when the second number is 0 + 0i.
The formulas behind complex arithmetic
Addition and subtraction combine real parts with real parts and imaginary parts with imaginary parts, exactly like combining like terms in ordinary algebra: (a + bi) ± (c + di) = (a ± c) + (b ± d)i.
Multiplication expands like a binomial product and simplifies using i² = −1: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i. Worked example with the default values a=3, b=2, c=1, d=−4: (3 + 2i)(1 − 4i) = (3×1 − 2×(−4)) + (3×(−4) + 2×1)i = (3 + 8) + (−12 + 2)i = 11 − 10i.
Division multiplies numerator and denominator by the complex conjugate of the denominator (c − di) to eliminate the imaginary part from the denominator: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²). This is undefined when c² + d² = 0, i.e. when the divisor is 0 + 0i.
The modulus of the result is |z| = √(x² + y²) where x and y are the real and imaginary parts of the result, and the argument is arg(z) = atan2(y, x), converted to degrees. For 11 − 10i, the modulus is √(11² + 10²) = √221 ≈ 14.866 and the argument is atan2(−10, 11) ≈ −42.274°.
Common mistakes
- Forgetting that i² = −1 when multiplying — this is the step that converts the i²bd term into a real (not imaginary) contribution of −bd.
- Dividing complex numbers directly instead of multiplying by the conjugate of the denominator first — direct division leaves an imaginary term in the denominator, which is not a valid final form.
- Confusing the modulus (a magnitude, always ≥ 0) with the real part — the modulus uses both the real and imaginary parts via √(a² + b²).
- Reading the argument as a plain arctan(b/a) — this gives the wrong angle in the second and third quadrants (where a is negative); atan2(b, a) is required for the correct quadrant.
- Mixing up which pair of inputs belongs to which complex number — the calculator treats (a, b) as the first number and (c, d) as the second, applied in that order for subtraction and division, which are not commutative.
الأسئلة الشائعة
How do you multiply two complex numbers?
Expand the product like a binomial and replace i² with −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For example, (3 + 2i)(1 − 4i) = (3×1 − 2×(−4)) + (3×(−4) + 2×1)i = 11 − 10i. The real part of the product is ac − bd, and the imaginary part is ad + bc.
How do you divide complex numbers?
Multiply the numerator and denominator by the complex conjugate of the denominator, which eliminates i from the denominator: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²). This works because (c + di)(c − di) = c² + d², a real number. Division is undefined if the denominator is 0 + 0i.
What is the modulus of a complex number?
The modulus, written |z|, is the distance from the origin to the point representing the complex number in the complex plane, calculated as |a + bi| = √(a² + b²). It generalizes absolute value from real numbers to complex numbers and is always a non-negative real number.
What is i squared?
By definition, i² = −1. The imaginary unit i is defined specifically so that its square is negative one, which is what allows complex numbers to represent solutions to equations like x² + 1 = 0 that have no real solution.
What is the argument of a complex number?
The argument, arg(z), is the angle between the positive real axis and the line from the origin to the point (a, b) representing the complex number a + bi, measured counterclockwise. It is computed as atan2(b, a) to ensure the correct quadrant, and this calculator reports it in degrees.
Why do we use the complex conjugate to divide?
Multiplying a complex number c + di by its conjugate c − di produces a purely real result, c² + d², because the cross terms cancel: (c + di)(c − di) = c² − (di)² = c² + d². Multiplying both the numerator and denominator of a division by the conjugate of the denominator therefore clears the imaginary part from the denominator, producing an answer in standard a + bi form.
المراجع
- Churchill RV, Brown JW. Complex Variables and Applications. 9th ed. McGraw-Hill, 2013.
- NIST Digital Library of Mathematical Functions (DLMF), §1.9 Calculus of a Complex Variable. dlmf.nist.gov.
- Stewart J. Calculus: Early Transcendentals. 8th ed. Cengage Learning, 2015. (Appendix on complex numbers.)