## Introduction to the Polar Form of Complex Numbers

Complex numbers are commonly written in **x+iy** format, where **i** stands for **√(-1)**, **x** is the **real number part** and **y** is the **imaginary** coefficient.

In mathematics, we can plot complex numbers in both the **Cartesian (x,y)** and **Polar (r,θ)** coordinate systems. In this article, we will focus just on the Polar form of complex numbers.

For example, let’s assume a complex number,

**z=3+4i**

If this number is plotted in a Cartesian coordinate system, it will display as follows:

If a line is added from **(0,0)** to **(3,4)**, it will be the hypotenuse of a right-angle triangle whose base and vertical are **x** and **y** respectively.

So hypotenuse **r= √(x2+y2)**, according to the **Pythagorean theorem**.

If the hypotenuse makes an angle θ, then **θ= tan-1(y/x) or arctan(y/x)**.

Now, **z=x+iy**.

So for the complex number, **z=3+4i**, **x=3,** and **y=4**.

If we substitute **x** and **y** in terms of **r**, **x** will be **rcosθ** and **y** will be **rsinθ**.

So **z=rcosθ+yisinθ**

**⇒ z=r(cosθ+isinθ)**

∴ **r=√(x2+y2)=√(25)=5**

And **θ= arctan(y/x)=arctan(4/3)=53°**.

So the Polar form of the number is:

**5(cos53°+isin53°)**

## How to Convert Complex Numbers to Polar Form in Excel

Here we will convert complex numbers to Polar form by means of the **COMPLEX**, **IMABS**, and **IAMARGUMENT** functions, using the following dataset containing some x and y values.

We will convert them into complex numbers in Rectangular form first, then into Polar form.

### Step 1 – Generate Complex Numbers Using the COMPLEX Function

- Use the following formula to convert input numbers into some complex numbers.

`=COMPLEX(B5,C5,"i")`

The syntax of the **COMPLEX** function is: **=COMPLEX(real_num,i_num,[suffix])**.

The value in **cell B5** is the real number part, the value in **cell C5** is the imaginary coefficient part, and the last argument is optional (enclosed by []). You can fix either** i** or **j** as the imaginary number indicator.

- Drag the
**Fill Handle**icon to obtain the outputs for the rest of the column.

### Step 2 – Convert Complex Numbers to Polar Form

- Use the following formula to convert the complex numbers generated in the
**Step 1**to Polar form:

`=ROUNDDOWN(IMABS(D5),2)&"(cos "&ROUNDDOWN(DEGREES(IMARGUMENT(D5)),2)&CHAR(176)&" +isin "&ROUNDDOWN(DEGREES(IMARGUMENT(D5)),2)&CHAR(176)&")"`

- Drag the
**Fill Handle**icon to auto-fill the other outputs.

**⧪**** Formula Explanation:**

The first output is: **9.05(cos 83.65° + isin 83.65°)**.

This formula is made using the **IMABS** & **IMARGUMENT** functions. The rest of the functions used here (**ROUNDDOWN**, **DEGREES**, and **CHAR**) are used to make the output look better.

- The
**IMABS**function returns the modulus (r) of a complex number. So**IMABS(D5)**will return the modulus of the complex number in**cell D5**. The**ROUNDDOWN**function formats the output to two-decimal places. - The
**IMARGUMENT**function returns the argument or θ angle of a complex number’s Polar form. Since Excel uses radian unit for angles by default and there is no way to change this default setting, we use the**DEGREES**function to make them look better. To display the degree symbol (°), we add**CHAR(176)**inside the larger formula.

- Alternatively, the following formula can be used to avoid the extra functions used here:

`=IMABS(D5)&"(cos "&IMARGUMENT(D5)&" + isin "&IMARGUMENT(D5)&")"`

The output will be: **9.05538513813742(cos 1.460139105621 + isin 1.460139105621)**.

## How to Convert Complex Numbers from Polar to Rectangular Form in Excel

Let’s deal with a reverse case, ie we have some complex numbers in Polar form (angles in radian unit) and want them in Rectangular form.

** Steps:**

- Extract the magnitude or modulus from the Polar form using the following formula:

`=VALUE(LEFT(B5,5))`

- Extract the argument θ using the following formula:

`=VALUE(MID(B5,FIND("cos",B5)+3,8))`

- Use the following formula to return the Rectangular complex number formats:

`=IMEXP(COMPLEX(LN(C5),D5,"i"))`

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