Calculate the result of addition, subtraction, multiplication, or division between two complex numbers.
The Easy Converters Complex Numbers Calculator is a free, accurate, and user-friendly tool designed to help you perform operations on complex numbers with ease. Whether you're solving quadratic equations, analyzing electrical circuits, or simplifying expressions in algebra or engineering, our calculator handles all the complexities of real and imaginary number arithmetic in one place.
A **complex number** is a number that combines a real part and an imaginary part. It is written in the form:
a + bi
where:
A complex number can also be represented in **polar form** as:
r∠θ = r(cos θ + i sin θ)
where:
You can extract real or imaginary components using mathematical functions, especially in calculus or complex analysis.
(a + bi)(c + di) = (ac – bd) + (ad + bc)i
(a + bi) ÷ (c + di) = [(ac + bd) + (bc – ad)i] / (c² + d²)
The calculator performs all the steps automatically and shows results in simplified form.
Our Complex Numbers Calculator works perfectly across all devices – desktop, tablet, or mobile. Whether you’re in class or on the go, solving complex math problems is now a tap away.
Q: Can I input complex numbers with negative real or imaginary parts?
A: Yes, the calculator supports negative values like –3 + 5i or 6 – 4i.
Q: What is the imaginary unit 'i'?
A: The imaginary unit i is defined as √–1. It is not a real number but is used to solve equations that have no real roots.
Q: Can I convert between rectangular and polar form?
A: Yes. Our calculator automatically displays both formats for supported operations.
Q: Does it handle division by zero?
A: No. Like with real numbers, division by zero is undefined and the calculator will display an error.
Complex numbers are a critical part of higher mathematics and engineering. From solving equations with no real roots to modeling alternating current, complex numbers unlock a new dimension in problem-solving. The Easy Converters Complex Numbers Calculator helps you master this topic with clarity and speed.
Try it now to simplify your complex calculations, convert to polar forms, or just explore the beauty of imaginary math.
One of the most important formulas involving complex numbers in polar form is De Moivre’s Theorem. It states:
[r(cos θ + i sin θ)]ⁿ = rⁿ[cos(nθ) + i sin(nθ)]
This formula is widely used to:
Our calculator will soon integrate De Moivre’s Theorem to compute nth powers and roots of any complex number.
A complex number can be represented in multiple formats:
Our tool will soon support seamless conversion between these formats, helping you understand how each represents the same quantity in different ways.
The nth roots of a complex number can be found using:
z1/n = r1/n [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] where k = 0 to n – 1
This formula yields n distinct roots of a complex number arranged symmetrically in the complex plane.
Example: The cube roots of 1 are: 1, –½ + √3/2 i, and –½ – √3/2 i
Complex numbers can be visualized in a 2D plane called the **Argand Plane**, where:
Every complex number is a point in this plane or a vector from the origin. The length of the vector is the modulus, and the angle with the positive real axis is the argument.
Using this geometric interpretation, multiplication becomes rotation and scaling, and division becomes rotation in the opposite direction with scaling.
(4 + 3i)(4 – 3i) = 25 – product of conjugates gives a real number1/(2 + i) = (2 – i)/5 = 0.4 – 0.2i – rationalize denominator|–3 + 4i| = √(9 + 16) = 5 – modulus calculationWe welcome suggestions from students, teachers, and professionals! Our upcoming updates may include:
Complex numbers are powerful tools in math and engineering — and with Easy Converters' Complex Numbers Calculator, solving them has never been easier. From basic arithmetic to advanced operations like polar transformations and conjugates, everything is covered in one simple interface.
Start using the calculator now and let it be your go-to companion for mastering complex numbers, acing your exams, or powering through engineering problems.