Calculate with complex numbers in rectangular form a + bi. Enter values for z₁ and z₂, choose an operation, and get instant results in readable math format.
What is an imaginary number?
An imaginary number is built from the unit i, where i² = -1. Since no real number squares to a negative value, mathematicians introduced i to extend arithmetic beyond the real number line.
When a real part and an imaginary part are combined, you get a complex number in the form a + bi. For example, 5 + 2i, -3 - i, and 7i are all complex numbers.
How to read complex numbers
Rectangular form
Most calculators use rectangular form: a + bi, where:
- a = real component
- b = imaginary coefficient
- i = square root of -1
Polar form
Complex numbers can also be represented by magnitude and angle: r∠θ. This form is common in electrical engineering and signal processing.
What this imaginary number calculator does
This tool supports the most useful complex number operations for students, engineers, and self-learners:
- Addition, subtraction, multiplication, and division
- Conjugate, magnitude, argument, and reciprocal
- Principal square root of a complex number
- Formatted output so answers are easy to read and verify
Worked examples
Example 1: Multiplication
Let z₁ = 3 + 2i and z₂ = 1 - 4i.
(3 + 2i)(1 - 4i) = 3 - 12i + 2i - 8i² = 3 - 10i + 8 = 11 - 10i.
Example 2: Division
Let z₁ = 4 + 6i and z₂ = 2 - i.
Multiply numerator and denominator by the conjugate of z₂:
(4 + 6i)/(2 - i) × (2 + i)/(2 + i) = (2 + 16i)/5 = 0.4 + 3.2i.
Common mistakes to avoid
- Forgetting that i² = -1 during multiplication.
- Dropping signs in expressions like a - bi.
- Dividing by a complex number without using the conjugate.
- Confusing magnitude |z| with the real part.
Where imaginary numbers are used in real life
Imaginary and complex numbers are not just abstract algebra topics. They are essential in:
- AC circuit analysis and impedance calculations
- Control systems and stability analysis
- Signal processing, Fourier transforms, and communications
- Quantum mechanics and wave equations
- Computer graphics, rotations, and applied modeling
Quick formula reference
- (a + bi) + (c + di) = (a + c) + (b + d)i
- (a + bi) - (c + di) = (a - c) + (b - d)i
- (a + bi)(c + di) = (ac - bd) + (ad + bc)i
- (a + bi)/(c + di) = [(ac + bd) + (bc - ad)i] / (c² + d²)
- |a + bi| = √(a² + b²)
- conj(a + bi) = a - bi
Final thoughts
If imaginary numbers once felt intimidating, a good calculator can make the structure clear. Enter values, test patterns, and compare multiple operations quickly. Over time, the algebra becomes intuitive—and very useful in higher math and practical engineering problems.