# a+bi Form Calculator

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:

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)`

and

`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`

and

`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:

- Complex number calculator;
- Multiply complex numbers calculator;
- Divide complex numbers calculator;
- Imaginary number calculator;
- Complex number to polar form calculator;
- Complex number to trigonometric form calculator;
- Complex number to rectangular form calculator;
- i calculator; and
- Polar form calculator.

## FAQ

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

To convert a complex number from polar to rectangular form:

- Compute
`cos(φ)`

and`sin(φ)`

, where`φ`

is the argument of your number. - Multiply each of these two numbers by
`r`

, where`r`

is the magnitude (modulus) of your number. - The
**real part**of your number is`a = r × cos(φ)`

. - The
**imaginary part**of your number is`b = r × sin(φ)`

. - 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.