In certain physics and engineering applications, it’s easier to perform certain computations with **complex numbers **(numbers that give a negative result when squared), rather than with **real numbers **(“normal” numbers)**. **In this section, we’ll learn how to convert **Rectangular Form **coordinates and equations to **Polar (Trigonometric)** **Complex Form**, in order to perform these computations.

## Review of Complex Numbers

We learned that **Imaginary (Non-Real) and Complex Numbers** exist so we can do certain computations in math, even though conceptually the numbers aren’t “real”.

To work with **complex numbers and trigonometry**, we need to learn about how they can be represented on a coordinate system (**complex plane**), with the “$ x$”-axis being the** real** part of the point or coordinate, and the “$ y$”-axis being the **imaginary** part of the point. For example, the representation of the complex number $ z=x+yi$ is $ (x,y)$ in the complex plane.

Here is a visual representation of a point in the complex plane, with its **magnitude**. The **magnitude **(sometimes called** modulus**) of a complex number is like the **hypotenuse** of a triangle, with lines drawn to the $ x$- (real) and $ y$- (imaginary) coordinates as the sides of the triangles. Thus, the **magnitude** of $ (4,3)$ or $ 4+3i$ is $ \sqrt{{{{4}^{2}}+{{3}^{3}}}}=\sqrt{{25}}=5$.

This form of a trig number is sometimes written as $ z=a+bi$ instead of $ z=x+yi$ (in this case, $ z=4+3i$). And remember that $ i=\sqrt{{-1}}$, although we won’t really need to use this in this section.

You can also put complex numbers in your graphing calculator, and even perform complex conversions on the graphing calculator, as shown after each section.

## Polar (Trigonometric) Form of a Complex Number

Again, we can write the **rectangular form** of a **complex number** in the form $ z=x+yi$, or more commonly, $ z=a+bi$. We can also write this in a **trig** **polar form**, where $ x=r\cos \theta $ and $ y=r\sin \theta $. Notice we can use the abbreviation **cis** or **CIS** (for **c**os plus *i***s**in), since the angles measurements ($ \theta $) are the same. “$ r$” is called the **magnitude** or **modulus** of $ z$, like we saw earlier, and is sometimes written as $ \left| z \right|$. The angle $ \theta $** **is called the** argument.**

$ \displaystyle z=x+yi=\left( {r\cos \theta } \right)+\left( {r\sin \theta } \right)i$ or

$ \displaystyle z=r\cdot \text{cis}\left( \theta \right)$ or $ \displaystyle z=r\cdot \text{cis}\theta $

Note that $ \displaystyle \left( {r\cos \theta } \right)+\left( {r\sin \theta } \right)i$ actually turns out to be the equivalent of $ r{{e}^{{\theta \,i}}}$ (Euler’s equation); in fact, we’ll see when we use the calculator to check answers. (We won’t prove this here, or use it any more).

Here is a visual example; note that we get the cosine and sine of **45°** from the **Unit Circle**, where we learned about here in the **Angles and the Unit Circle** section:

## Converting Complex Rectangular Form to Polar Form

When converting from **Rectangular** to **Polar** Form, to get $ r$ and $ \theta $, we have to use the same equations we did here in the **Polar Coordinates, Equations and Graphs** section here:

$ \displaystyle r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}\,\,\,\,\,\text{(this will be positive)}$

$ \displaystyle \theta ={{\tan }^{{-1}}}\left( {\frac{y}{x}} \right)\,\,\,\text{(must be correct quadrant)}$

Note that when using the **calculator** to get $ {{\tan }^{{-1}}}$, you’ll have to add the following degrees or radians when your point is in the following quadrants. This is because the $ {{\tan }^{{-1}}}$ function on your calculator only give you answers back in the interval $ \displaystyle \left( {-\frac{\pi }{2},\frac{\pi }{2}} \right)$.

Here are some examples (I’ve left the argument in degrees). Remember that $ x$ is the real number and $ y$ is the imaginary number (coefficient of “$ i$”).

### Rectangular-to-Polar Conversions in the Graphing Calculator

You can use the graphing calculator to convert back and forth between **polar** to **rectangular** coordinates (like we saw in the **Polar Coordinates, Equations and Graphs** section__)__, and even find **complex powers** and **roots**.

For **rectangular-polar conversions **(in either **DEGREE** or **RADIAN** mode), use **2 ^{nd} apps (angle) 5** [

**R→Pr(**] and

**2**[

^{nd}apps (angle) 6**R→P**

**Ɵ(**], typing in the rectangular form without the “$ i$” and using a comma (above the

**7**button) between the real and imaginary parts. Note that you have to solve for the $ r$ (which may be in decimal and not exact form) and $ \theta $ separately.

You can also use **math CMPLX** **→ 7 (Polar)**, after typing in the rectangular $ a+bi$ form. Note that the answer is in the form $ r{{e}^{{\,\theta \,i}}}$; when converting back this way, put your calculator in **RADIAN** mode and enter the angle in this mode (see below).

Try converting $ -1-i$ to Polar. Notice that you may have to add **360°** if you get a negative angle.

## Converting Polar Form to Rectangular Form

Let’s convert **polar complex** numbers to **rectangular form**. Get the** sin** and **cos** of the angle, and then push through (distribute) the number in front (the “$ r$”). Again, this is similar to what we did in the **Polar Coordinates, Equations and Graphs** section here much easier going this way! (These match up with the conversions above, and **some are in radian mode** instead of degree mode).

### Polar-to-Rectangular Conversions in the Graphing Calculator

For **polar-rectangular conversions **(in either **degrees** or **radians** mode), use **2 ^{nd} apps (angle) 7** [

**P→Rx(**] and

**2**[

^{nd}apps (angle) 8**P→Ry(**], typing in the rectangular form using a comma (above the

**7**button) between $ r$ and $ \theta $. Note that you have to solve for the $ x$ and $ y$ separately, and also note again that you won’t get the answers with the roots in them (you’ll get decimals that aren’t “exact”). You also won’t get the “$ i$” in the $ y$-part of the answer.

You can also just put the polar number in $ r{{e}^{{\,\theta \,i}}}$ form, using radians. Put the calculator in **re^(θ i)** (polar) mode first if you want the output in polar mode, like in the second output on the right screen. Use

**2**(under the

^{nd}.**2**button) for

*i*.

Try converting $ \displaystyle \sqrt{2}\,\text{cis}\left( {\frac{{5\pi }}{4}} \right)$ to Rectangular:

## Products and Quotients of Complex Numbers in Polar Form

The polar form of complex numbers can be used to find **products** and **quotients** of complex numbers; you’ll basically want to memorize these formulas.

When you **multiply** two polar complex numbers, you **multiply the magnitudes** (numbers in front), but **add the angle measurements**. When you **divide** two polar complex numbers, you **divide the magnitudes** (numbers in front), but **subtract the angle measurements**. Pretty weird, huh?

$ \displaystyle \begin{array}{c}{{z}_{1}}={{r}_{1}}\left( {\cos {{\theta }_{1}}+i\sin {{\theta }_{1}}} \right)=\,{{r}_{1}}\,\text{cis}\left( {{{\theta }_{1}}} \right)\\\,{{z}_{2}}=\,\,{{r}_{2}}\left( {\cos {{\theta }_{2}}+i\sin {{\theta }_{2}}} \right)={{r}_{2}}\,\text{cis}\left( {{{\theta }_{2}}} \right)\end{array}$

$ \displaystyle \begin{align}\text{Product:}\,\,\,{{z}_{1}}{{z}_{2}}&={{r}_{1}}{{r}_{2}}\left[ {\cos \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)} \right]\\&=\,{{r}_{1}}{{r}_{2}}\,\text{cis}\left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)\end{align}$

$ \displaystyle \begin{align}\text{Quotient:}\,\,\,\,\frac{{{{z}_{1}}}}{{{{z}_{2}}}}&=\frac{{{{r}_{1}}}}{{{{r}_{2}}}}\left[ {\cos \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)} \right]\,\\&=\frac{{{{r}_{1}}}}{{{{r}_{2}}}}\,\text{cis}\left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)\end{align}$

Here are some examples; both types of notation are used for complex polar numbers. We’ll leave our answers in **Polar Form** between **0** and **360°** or **0** and $ 2\pi $:

### Products and Quotients of Complex Numbers using the Graphing Calculator

To **multiply and divide complex numbers**, you can put these in the calculator in **Polar Form** using **re^(θ i)** (

**RADIAN**mode only, with calculator in

**re^(θ**mode if you want the output in polar mode), or

*i*)**Rectangular Form**in either

**DEGREE**or

**RADIAN**mode. Use

**2**(under the

^{nd}.**2**button) for

*i*.

Try $ \displaystyle 4\,\text{cis}\left( {\frac{\pi }{8}} \right)\cdot 3\,\text{cis}\left( {\frac{{9\pi }}{{16}}} \right)$ (in mode **RADIAN**), and $ \displaystyle \frac{{\left( {1-i} \right)}}{{\left( {1+\sqrt{3}i} \right)}}$ (in mode **DEGREE**):

Note that we might have to add **360°** to a negative angle to get the correct angle. And we can always convert the product or quotient back from Polar to Rectangular by using** MATH CMPLX** **→Rect**.

## De Moivre’s Theorem: Powers of Complex Numbers

**De Moivre’s Theorem** (named after the French mathematician Abraham De Moivre), is a formula for **raising a complex number to a power** (greater than or equal to **1**).

When you raise a complex number to a power, you **raise the magnitude to that power**, but **multiply** **the angle measurements**. (Later we’ll see that when you **take the root of a complex number**, you **take the root of the magnitude**, but **divide** **the angle measurements**). See how it’s similar to multiplying and dividing complex numbers? Here is **De Moivre’s Theorem**:

Here are some examples; both types of notation are used for complex polar numbers. We’ll write our answers in **Standard **or** Rectangular Complex Form **$ a+bi$:

### Powers of Complex Numbers using the Graphing Calculator

To raise **complex numbers to powers** (De Moivre’s Theorem) using the calculator, if the complex number is in polar mode, input it using **re^(θ i)** (

**RADIAN**mode only), like in the first example, $ {{\left[ {2\,\text{cis}\left( {80{}^\circ } \right)} \right]}^{{\,3}}}$. (To have kept it in Polar mode, you could have put the calculator in

**re^(θ**mode first). In the second example, $ {{\left( {1-i} \right)}^{7}}$, just type in the complex number, and the rectangular mode will show up, unless the calculator is in

*i*)**re^(θ**(polar) mode:

*i*)## Roots of Complex Numbers

To get **roots** of complex numbers, we do the opposite of raising them to a power; we take the $ n$** th** root of the magnitude, and then divide the angle measurements by $ n$. The only thing that’s a little tricky is there are typically

**many roots for a complex number**, so we have to find all of these by the following formula, with $ k$ going from

**0**to $ (n-1)$. It turns out these are

**evenly spaced on the unit circle**, all with the

**same magnitude:**

$ \displaystyle \begin{align}\sqrt[n]{z}\,\,&=\,\,\sqrt[n]{r}\left[ {\cos \left( {\frac{\theta }{n}+\frac{{2\pi k}}{n}} \right)+i\sin \left( {\frac{\theta }{n}+\frac{{2\pi k}}{n}} \right)} \right]\,\,\,\\&=\sqrt[n]{r}\,\text{cis}\left( {\frac{\theta }{n}+\frac{{2\pi k}}{n}} \right)\,\,\,\\k&=0,\,\,1,\,\,2,\,…,n-1\end{align}$

This looks really complicated, so let’s go through some examples (and get answers in either degrees or radians, per indicated). Let’s first work with roots with either the real or imaginary part of the complex number (but not both). Note that if a problem asks for a root of “**unity**”, this is just **1** $ (1+0i)$.

For those roots that end up with special values (values on the **Unit Circle**), put the answers in $ a+bi$ form; for non-special values, leave in **polar form**.

Here are some more examples; we’ll keep all answers in **trig form** (**radians)**. Notice how we want to find common denominators when adding the fractions with $ 2\pi $ in them. Also note with the second problem how we turned $ \sqrt{{32}}$ into $ \sqrt{{{{2}^{5}}}}$ into $ {{\left( {\sqrt{2}} \right)}^{5}}$ so we could simplify when we take the fifth root.

Here’s another type of problem you might see. Note we could have solved the second part of this problem more easily with the **Complex Conjugate Root Theorem**, or** Conjugate Zeros Theorem **that we learned about here in the **Imaginary (Complex) Numbers** Section.

### Roots of Complex Numbers using the Graphing Calculator

When getting **roots of complex numbers** in the calculator, you’ll only get one root, so this won’t be too helpful. But you can check all the roots you get by putting them in the calculator and taking the reverse power.

For example, to get the roots for $ \sqrt[3]{{-27}}$, we only get **–3**. To check our answers that we got above, cube them to see if we get **–27**. Also, to check to see if **216°** and **288°** are fifth roots of unity (**1**), convert them to radians and raise them to the **5 ^{th}** power (with calculator in

**RADIAN**and

**REAL**modes). (Note that the coefficient of $ i$ is a very tiny number and should be in fact “

**0**”.):

**Hint**: You can always check the complex roots by entering, for example, cube root of **-27** in **Wolfram Alpha**.

**Understand these problems, and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

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On to** Introduction to Calculus and Study Guides** – you’re ready!