Complex Numbers
Operations in a+bi form
About This Calculator
Complex numbers extend the real number line to a plane by introducing the imaginary unit i, where i² = −1. Every complex number has the form a + bi, where a is the real part and b is the imaginary part. Complex numbers are essential in electrical engineering, quantum mechanics, signal processing, and the mathematics of wave phenomena.
Formula
Standard form: z = a + bi, where i = √(−1)
Addition: (a+bi) + (c+di) = (a+c) + (b+d)i
Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i
Modulus: |z| = √(a²+b²); Conjugate: z* = a−bi
Example Calculation
Compute (3+2i)(1−4i).
- (3+2i)(1−4i) = 3(1) + 3(−4i) + 2i(1) + 2i(−4i)
- = 3 − 12i + 2i − 8i²
- Since i²=−1: = 3 − 10i − 8(−1) = 11 − 10i
(3+2i)(1−4i) = 11 − 10i
Complex Number Operations
| Operation | Formula | Example (z=3+4i) |
|---|---|---|
| Modulus | |z| = √(a²+b²) | √(9+16) = 5 |
| Conjugate | z* = a−bi | 3−4i |
| Argument | θ = arctan(b/a) | arctan(4/3) ≈ 53.13° |
| Inverse | 1/z = z*/|z|² | (3−4i)/25 |
Frequently Asked Questions
Why do we need complex numbers?
Some polynomial equations have no real solutions. For example, x²+1=0 has no real roots, but has complex roots ±i. Complex numbers also describe oscillations, electrical impedance, and quantum states — phenomena that genuinely require two components.
What is the complex conjugate used for?
The conjugate (a−bi) is used to divide complex numbers (multiply numerator and denominator by the conjugate to eliminate i from the denominator) and to find the modulus (|z|² = z × z*).
What is Euler's formula?
Euler's formula e^(iθ) = cos(θ) + i·sin(θ) links exponential and trigonometric functions through complex numbers. The special case e^(iπ) + 1 = 0 (Euler's identity) is considered one of the most beautiful equations in mathematics.
What is the Argand diagram?
The Argand diagram (complex plane) plots complex numbers with the real part on the x-axis and imaginary part on the y-axis. Addition looks like vector addition; multiplication rotates and scales the vector.