# Multiply Complex Numbers Calculator

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:

As we can see:

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

#### Multiplying complex numbers in polar form

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

We can see that:

- $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$; and
- $\arg( z_1\cdot z_2) = \varphi_1 + \varphi_2$.

## How to use this multiplying complex number calculator?

The multiply complex numbers calculator is really straightforward to operate:

- 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.

- For the rectangular form, enter the
- Enter the second complex number in a similar manner. You do not have to use the same number form as for the 1st number!
- The calculator multiplies the imaginary numbers.
- It
**displays the result in both the rectangular and polar form**, so you can pick the form that is more convenient for you. - 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)`

:

- Compute
`ac - bd`

. This will be the**real part**of the result. - Compute
`ad + bc`

. This will be the**imaginary part**of the result. - Bring the two parts together, writing down the number
`(ac - bd) + i(ad + bc)`

. **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)`

:

**Multiply the two magnitudes**:`r × s`

. This will be the magnitude of the result.**Add the two phases together**:`φ + ψ`

. This will be the phase of the result.- Write down the result as
`(rs) × exp(i(φ + ψ))`

. - We're
**done**! Multiplying complex numbers in polar form is easy-peasy, isn't it?