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Multiply Complex Numbers Calculator

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


Welcome to Omni's multiply complex numbers calculator, where you can quickly compute the product of any two complex numbers, no matter if they are in the rectangular or in the polar form! To make your life easier, the calculator delivers the result in both forms as well!

If you want to learn more on how to multiply complex numbers, read on to discover the formulas for multiplication of complex numbers that we've implemented in our tool! In particular, we'll see how nice it is to multiply complex numbers in polar form :)

Formulas for multiplying complex numbers

Multiplying complex numbers in a+ib form

To multiply two complex numbers in rectangular form (a+ib form), use the formula:

z1z2=(a+bi)(c+di)=ac+iad+ibc+bd=(acbd)+i(ad+bc)\small \begin{split} z_1 \cdot z_2 &= (a + bi) \cdot (c + di) \\ &= ac + \mathrm{i}ad + \mathrm{i}bc + bd \\ &= (ac - bd) + \mathrm{i}(ad + bc) \end{split}

As we can see:

  • Re(z1z2)=acbd\operatorname{Re}(z_1 \cdot z_2) = ac - bd; and
  • Im(z1z2)=ad+bc\operatorname{Im}(z_1 \cdot z_2) = ad + bc.

Multiplying complex numbers in polar form

To multiply two complex numbers in polar form (r×exp(iφ) form), use the formula:

z1z2=z1exp(iφ1)z2exp(iφ2)=z1z2exp[i(φ1+φ2)]\small \begin{split} z_1\cdot z_2 &= |z_1|\exp( \mathrm{i}\varphi_1) \cdot |z_2|\exp( \mathrm{i}\varphi_2) \\ &= |z_1| |z_2| \exp[ \mathrm{i}(\varphi_1+\varphi_2)] \end{split}

We can see that:

  • z1z2=z1z2|z_1 \cdot z_2| = |z_1| \cdot |z_2|; and
  • arg(z1z2)=φ1+φ2\arg( z_1\cdot z_2) = \varphi_1 + \varphi_2.

How to use this multiplying complex number calculator?

The multiply complex numbers calculator is really straightforward to operate:

  1. Enter the 1st number. You can choose between the rectangular form and the polar form:
    • For the rectangular form, enter the real and imaginary parts of your complex number.
    • For the polar form, enter the magnitude and phase of your complex number.
  2. Enter the second complex number in a similar manner. You do not have to use the same number form as for the 1st number!
  3. The calculator multiplies the imaginary numbers.
  4. It displays the result in both the rectangular and polar form, so you can pick the form that is more convenient for you.
  5. By hitting the advanced mode button below the calculator, you can adjust the precision of computations (number of decimal places). The default is three, but you can go up to ten if you wish.

Other Omni tools concerning complex numbers

Since complex numbers are omni(:D)present in science and engineering, we've created a whole collection of Omni calculators concerning them. Once you're done with this multiply complex numbers calculator, make sure to take a look at:

FAQ

What is i times 2i?

The answer is -2. This is because from the definition of i we know that multiplying it by itself yields -1. Hence, we get i × (2i) = 2 × i² = 2 × (-1) = -2.

Is there a multiplicative inverse of i?

Yes, for every complex number (apart from 0 of course) there exists a multiplicative inverse, i.e., a number which multiplied by the original number gives 1. The multiplicative inverse of i equals -i, because: i × (-i) = -i² = -(-1) = 1.

How do I multiply complex numbers in rectangular form?

To find the product (a+ib) × (c+id):

  1. Compute ac - bd. This will be the real part of the result.
  2. Compute ad + bc. This will be the imaginary part of the result.
  3. Bring the two parts together, writing down the number (ac - bd) + i(ad + bc).
  4. That's it! Multiplying complex numbers is not that hard, is it?

How do I multiply complex numbers in polar form?

To find the product of r×exp(iφ) and s×exp(iψ):

  1. Multiply the two magnitudes: r × s. This will be the magnitude of the result.
  2. Add the two phases together: φ + ψ . This will be the phase of the result.
  3. Write down the result as (rs) × exp(i(φ + ψ)).
  4. We're done! Multiplying complex numbers in polar form is easy-peasy, isn't it?
Anna Szczepanek, PhD
Related calculators
First number
Form
a + bi
Real part (a₁)
Imaginary part (b₁)
Second number
Form
a + bi
Real part (a₂)
Imaginary part (b₂)
Product of your numbers
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