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Complex Number Calculator

Complex Number Calculator

Write down any arithmetic combination of complex numbers and the calculator will simplify them, with the steps shown.

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An online complex number calculator allows you to perform the basic mathematical operations to simplify the given complex expressions. This free imaginary number calculator will simplify any complex expression with step-by-step calculations quickly. So, keep reading to understand how to simplify complex numbers such as polar form, inverse, conjugate, and modulus.

What is a Complex Number?

In mathematics, a complex number is defined as a combination of real and imaginary numbers. It is expressed as x + yi. Here, “i” is an imaginary number, and “x” and “y” are real numbers. Where “I” is also known as iota, and its value is \(\sqrt{-1}\). Complex numbers calculator can add, subtract, multiply, or dividing imaginary numbers. In short, we can use an expression as z = x + iy, where “x” is the real part and “iy” is the imaginary part.

Basic Operations with Complex Numbers:

When performing simple operations on complex numbers, it is helpful to think of them as vectors. Then, it is very simple to subtract and adding complex numbers with complex solutions calculator.

If the first number is A = x + yi and the second number is B = m + ni, then the sum of two complex numbers is:

$$ A + B = x + yi + m + ni = (x + m) + (y + n) * I $$

Where Re (A + B) = x + m is part of the sum of real numbers, And Im(A + B) = y + n is part of sum of imaginary number.

If the numbers are: A – B = x + yi – m + ni = (x – m) + (y – n) * I, then Re (A – B) = x – m and Im (A – B) = y – n. The complex number calculator, add (or subtract) each pair of given components separately!

However, an online Scientific Notation Calculator allows you to add, subtract, multiply, and divide numbers in scientific notation.

Multiplying and Dividing Complex Numbers:

Let’s take a look at how complex numbers are multiplied together by simplify complex numbers calculator. According to the notation in the previous section:

  • $$ A * B = (x + yi) * (m + ni) = x * m + x * n * i + y * m * i + y * n * i * i = (x * my * n) + (x * n + y * m) * i $$
  • This time, the real part can be written as Re(A * B) = x * m – y * n, and the imaginary part as Im(A * B) = x * n + y * m.
  • Remember that complex number calculators use a negative sign in the real part because, at some point, we are faced with the product of two numbers i * i, which by definition is -1.
  • Multiplying by complex numbers is not difficult with the complex calculator. Let’s take a look at the calculation of theorem: A / B = (x + yi)/(m + ni) =, expand the numerator and denominator by combining the complex numbers of the numerator and denominator.
  • (x + yi)*(m – ni)/((m + ni) * (m – ni))= Perform multiplication of complex number in standard form
  • (x * m – x * n * i + y * m * i – y * n * i * i) / (c^2 – (ni)^2)=, again using the fact that i * i = -1
  • (x * m + y * n + (y * mx * n) * i) / (m^2+n^2) we get the following result:
  • Re (A / B) = (a * c + b * d) / (m^2 +n^2), Im(A x B)=(y * m – x * n) / (m^2 + n^2). Of course, division is only possible when B ≠ 0.
  • We can also use polar coordinate notation to consider the above operations, such as A = |Z_1| * exp (iφ_1), B = | z^2| * exp (iφ^2).
  • To multiply complex numbers the imaginary number calculator use formula as: F * G = | z_1| * exp (iφ_1) * | z^2| * exp (iφ^2) = | z_1 * z^2| * exp (i(φ_1 + φ^2)), we see: A * B  = | z_1 * z^2| and arg (A * B) = φ_1 +φ^2.
  • The division of complex numbers with this notation is almost the same: A / B = | z_1| * exp (iφ_1)/ | z^2| * exp (i)= | z_1/z^2| * exp (i(φ_1 – φ^2) ),Rewrite the result as: A / B = | z_1 / z^2| and arg (A / B) = φ_1 – φ^2.

Complex Power and Logarithm:

  • First, the imaginary numbers calculator finds a general formula for the complex power of two numbers, given as A * B.
  • AB = (x + yi) (m + ni) = Since it is not clear how to extend this expression, the complex calculator use F as the polar form of a complex number.
  • (|z_1| * exp (iφ_1)) (c + di) = , now the product of any power multiplied by the sum.
  • | z_1|^c * exp (iφ_1* c) * | z_1|^{nx} * exp (-φ_1 * d) = We can use the well-known exponential property: xn = exp (n * ln (x)), where ln is natural logarithm.
  • | z_1| Exp * exp(-φ_1 * d) * exp (i(φ_1 * m + n * ln |z_1|)).
  • Therefore, the absolute value is: AB = | z_1| exp * exp (-φ_1* d), and the independent variable is: arg(AB) = φ m + n * ln | z_1|
  • The logarithm of a complex number (also known as the complex logarithm) can be computed as follows: ln (F) = ln (|z_1| * exp (iφ_1)) = ln (|z_1|)+iφ_1.

However, use an online Composite Function Calculator that solves the composition of the functions from entered values of functions f(x) and g(x) at specific points.

How Complex Number Calculator Works?

The imaginary number calculator makes the given expression simple with these steps:


  • First, enter an expression with real and imaginary numbers.
  • Click the calculate button.


  • The complex number calculator provides inverse, conjugate, modulus, and polar forms of given expressions.
  • The calculator displays a stepwise solution of multiplication and other basic mathematical expressions.


Is every number complex?

Every real number is a complex number, but it’s not compulsory each complex number is a real number. The set of all complex numbers is represented by Z ∈ C. The set of all imaginary numbers is expressed as Z ∈ C – R.

In real life, where are complex numbers used?

Complex numbers are also used to calculate the voltage, current, or resistance in an AC circuit (AC means alternating current).

How does a complex number work?

A complex number is the sum of an imaginary number and a real number, expressed as a + bi. So, an intersection point of the real part is on the horizontal axis, and the imaginary part found on the vertical axis.


Use this online complex number calculator to perform basic operations like multiplication and division with complex numbers. You can write both the imaginary and real parts of two numbers. If a number is purely imaginary or purely real, then set the other part equal to 0. After that, you will get the polar form of a given complex expression.


From the source of Wikipedia: Notation, Visualization, Cartesian complex plane, Polar complex plane, Modulus and argument, Complex graphs.

From the source of Brilliant: Complex Plane, The Imaginary Unit i, Complex Numbers Arithmetic, Multiplication of Complex Numbers, Complex Conjugates.

From the source of Varsity Tutors: Complex Numbers, complex plane, purely imaginary, imaginary unit, Cartesian Plane.