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a+bi Form Calculator

Created by Anna Szczepanek, PhD
Reviewed by Komal Rafay
Last updated: Jan 18, 2024

Welcome to Omni's a+bi form calculator, where you can quickly convert a complex number from its polar to rectangular form. Keep reading if you want to learn or recall what these two forms of a complex number are and how to write the a+bi form of a polar form complex number.

What is the a+bi form of a complex number?

The two forms of complex numbers are: rectangular (a + bi) form and polar (r × exp(φi)) form. The rectangular form describes z as the point (a, b) on a complex plane. The polar form describes z in terms of distance r from (0,0) to z and of the angle φ between the horizontal axis and the radius connecting (0,0) and z.

Let us summarize the two pairs of coordinates:

  • a is the real part; and
  • b is the imaginary part of z.

And the polar form:

  • r is the modulus (or the magnitude)
  • φ is the argument of z.

Observe where these value appear in the complex plane:

Complex plane

Let us now discuss how to convert the polar form to rectangular form.

How do I go from polar to rectangular form?

When you want to write the a+bi form of a complex number in polar form z = r × exp(iφ) use the formulas:

a = r × cos(φ)


b = r × sin(φ).

To see why these formulas are correct, look at the picture above and recall the basic trigonometric formulas:

cos(φ) = a / r


sin(φ) = b / r.

Solve for a and b and you'll get the formulas given above.

Omni's a+bi calculator uses the same formulas as well.

How to use this a+bi form calculator?

Our a+bi calculator is very easy to operate: to convert a polar form to a rectangular form, you need to input the polar form by filling in the fields magnitude and phase. Note that for the phase, you can choose between radians and degrees - pick whatever is more convenient for you!

Our a+bi form calculator immediately displays the two coordinates of the rectangular form: the real part a and the imaginary part b. You can now write the a + bi form easily.

Omni calculators for complex numbers

Satisfied with this a+bi form calculator? Omni can help you discover various interesting aspects of complex numbers! Take a look and pick the next thing you want to learn:


How do I write the a+bi form of complex number?

To convert a complex number from polar to rectangular form:

  1. Compute cos(φ) and sin(φ) , where φ is the argument of your number.
  2. Multiply each of these two numbers by r, where r is the magnitude (modulus) of your number.
  3. The real part of your number is a = r × cos(φ).
  4. The imaginary part of your number is b = r × sin(φ).
  5. Write the a + bi form of your number.

What is the rectangular form of exp(iπ/4)?

The answer is √2/2 + (√2/2)i. To derive this result, observe that the modulus of exp(iπ/4) is 1. Next, compute cos(π/4) = √2/2 and sin(π/4) = √2/2. In consequence:

  • The real part is a = 1 × cos(π/4) = √2/2.
  • The imaginary part is b = 1 × cos(π/4) = √2/2.

If you struggle or want to verify your calculations, don't hesitate to use an online a+bi calculator.

Anna Szczepanek, PhD
Polar form r × exp(iφ)
Magnitude (r)
Phase (φ)
Rectangular form a + bi
Real part (a)
Imaginary part (b)
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