Complex number Calculator

Complex Number Calculator is an online calculator that can conduct operations between two sets of complex numbers. This calculator evaluates expressions in complex numbers and performs basic arithmetic on complex numbers.

First number: \(z_1 = a + bi\)
First number: \(z_2 = c + di\)

What are Complex numbers?

An expression of the form \(a + bi\), where a and b are real values, is a complex number. For example, if z is a complex number and \(z = a + bi\), then a and b are the real and imaginary parts of z.

Numerous scientific disciplines, including engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory, require complex numbers.

How to Use the Complex Number Calculator?

The complex number calculator should be used as follows:

  1. Fill in the appropriate input field with the complex numbers.
  2. To obtain the outcome, press the “Calculate” button now.
  3. Lastly, you will get two complex numbers’ multiplication, division, addition, or subtraction.

Basic operations with complex numbers

Working with complex numbers is simple because you may utilize the imaginary unit \(i\) as a variable. And to make complicated expressions simpler, use the formula \(i^2 = -1\). Numerous procedures are identical to those performed on two-dimensional vectors.


Combine the imaginary portions (with i) and the real parts (without i):

Use rule \((a + bi) + (c + di) = (a + c) + (b + d)i\) is equivalent to this.

\((1 + i) + (7 – 5i) = 8 – 4i\)


Subtract the imaginary components (with i) and the real parts again, very simply:

Use rule \((a + bi) + (c + di) = (a – c) + (b – d)i\) is equivalent to this.

\((10 – 5i) – (-6 + 5i) = 16 – 10i\)


You can use this formula for multiplication between two complex numbers, \(z_1\) and \(z_2\)

\(z_1 = a + bj\)

\(z_2 = c + dj\)

Multiplication Formula

\(= z_1 × z_2\)

\(= (a + bj) × (c + dj)\)

\(= ac + adj + bcj + bdj^2\)

But \(j^2 = -1\),

So \(z_1 × z_2 = (ac – bd) + (bc + ad)j\)


How to write complex numbers in standard form?

The complex numbers will be expressed as \(a + bi\) in standard form, where a represents the real part and bi represents the imaginary part. A complex number is something like \(3 + 5i\). The real portion is 3, and the imaginary portion is \(5i\).

What is the modulus of complex numbers?

The distance of a complex number from its origin in the argand plane is known as the modulus of the complex number. If \(z = x + iy\) is a complex number with x and y are real and \(i = \sqrt{-1}\), then the non-negative value \(\sqrt{x^2 + y^2}\) is the modulus of the complex number \(z = x + iy\).

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