Q1. Find the modulus and argument of z = 1 + i.
|z| = √(1² + 1²) = √2
θ = tan⁻¹(1/1) = tan⁻¹(1) = 45°
So z = √2 (cos45° + i·sin45°)
Complex Numbers and Quadratic Equations — Class 11
Every quadratic has a solution once you allow i = √−1. Compute the modulus and argument of any complex number instantly.
Imaginary uniti = √−1, so i² = −1. A complex number is z = a + bi, with a the real part and b the imaginary part.
Modulus|z| = √(a² + b²) — the distance of z from the origin on the Argand plane.
Argumentθ = tan⁻¹(b/a) — the angle z makes with the positive real axis.
ConjugateThe conjugate of z = a + bi is z̄ = a − bi; z·z̄ = a² + b² = |z|².
Complex number calculator Interactive
Enter the real and imaginary parts of z = a + bi to get its modulus and argument.
Modulus |z| = √(a²+b²)—
Argument θ (degrees)—
A quadratic with a negative discriminant has complex roots of the form p ± qi — see the Quadratic Equations solver in Class 10 for the full working.
Solved examples
Q2. Solve x² + 4 = 0.
x² = −4
x = ±√(−4) = ±2i
The roots are x = 2i and x = −2i.
Formula sheet
| Quantity | Formula |
|---|---|
| i squared | i² = −1 |
| Modulus | |z| = √(a²+b²) |
| Argument | θ = tan⁻¹(b/a) |
| Complex roots of a quadratic | when D = b²−4ac < 0 |
Common mistakes & exam wins
- i² = −1, i³ = −i, i⁴ = 1 — the powers of i cycle every 4.
- A negative discriminant (D < 0) in a quadratic means the roots are a complex conjugate pair.
- z · z̄ = |z|² is a quick way to compute the modulus squared.
Frequently asked questions
What is a complex number?
A number of the form a + bi, where a and b are real and i = √−1, called the real and imaginary parts respectively.
What is the modulus of a complex number?
|z| = √(a² + b²), the distance of the point (a, b) from the origin on the Argand plane.
When does a quadratic equation have complex roots?
When its discriminant D = b² − 4ac is negative.