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# Complex Numbers

## a + bi, the sum of a real and an imaginary number.

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Arithmetic with Complex Numbers

The idea of a complex number can be hard to comprehend, especially when you start thinking about absolute value.  In the past you may have thought of the absolute value of a number as just the number itself or its positive version.  How should you think about the absolute value of a complex number?

### Arithmetic Operations with Complex Numbers

Complex numbers follow all the same rules as real numbers for the operations of adding, subtracting, multiplying and dividing.  There are a few important ideas to remember when working with complex numbers:

1. When simplifying, you must remember to combine imaginary parts with imaginary parts and real parts with real parts.  For example, \begin{align*}4+5i+2-3i=6+2i\end{align*}.
2. If you end up with a complex number in the denominator of a fraction, eliminate it by multiplying both the numerator and denominator by the complex conjugate of the denominator.
3. The powers of \begin{align*}i\end{align*} are:
• \begin{align*}i=\sqrt{-1}\end{align*}
• \begin{align*}i^2=-1\end{align*}
• \begin{align*}i^3=-\sqrt{-1}=-i\end{align*}
• \begin{align*}i^4=1\end{align*}
• \begin{align*}i^5=i\end{align*}
• . . . and the pattern repeats

Take the following complex expression.

\begin{align*}(2+3i)(1-5i)-3i+8\end{align*}

First, multiply the two binomials and then combine the imaginary parts with imaginary parts and real parts with real parts.

\begin{align*}&= 2-10i+3i-15i^2-3i+8 \\ &= 10-10i+15 \\ &= 25-10i\end{align*}

Note that a power higher than 1 of \begin{align*}i\end{align*} can be simplified using the pattern above.

The complex plane is set up in the same way as the regular \begin{align*}x, y\end{align*} plane, except that real numbers are counted horizontally and complex numbers are counted vertically. The following is the number \begin{align*}4+3i\end{align*} plotted in the complex number plane.  Notice how the point is four units over and three units up.

The absolute value of a complex number like \begin{align*}|4+3i|\end{align*} is defined as the distance from the complex number to the origin.  You can use the Pythagorean Theorem to get the absolute value.  In this case, \begin{align*}|4+3i|=\sqrt{4^2+3^2}=\sqrt{25}=5\end{align*}.

### Examples

#### Example 1

Earlier, you were asked how to  think about the absolute value of a complex number. A good way to think about the absolute value for all numbers is to define it as the distance from a number to zero.  In the case of complex numbers where an individual number is actually a coordinate on a plane, zero is the origin.

#### Example 2

Compute the following power by hand and use your calculator to support your work.

\begin{align*}\left(\sqrt{3}+2i\right)^3\end{align*}

\begin{align*}\left(\sqrt{3}+2i\right) \cdot \left(\sqrt{3}+2i\right) \cdot \left(\sqrt{3}+2i\right)\end{align*}

\begin{align*}&= \left(3+4i \sqrt{3}-4\right)\left(\sqrt{3}+2i\right) \\ &= \left(-1+4i \sqrt{3}\right)\left(\sqrt{3}+2i\right) \\ &= -\sqrt{3}-2i+12i-8 \sqrt{3} \\ &= -9 \sqrt{3}+10i\end{align*}

A TI-84 can be switched to imaginary mode and then compute exactly what you just did.  Note that the calculator will give a decimal approximation for \begin{align*}-9\sqrt{3}\end{align*}

#### Example 3

Simplify the following complex expression.

\begin{align*}\frac{7-9i}{4-3i}+\frac{3-5i}{2i}\end{align*}

To add fractions you need to find a common denominator.

\begin{align*}& \frac{(7-9i) \cdot 2i}{(4-3i) \cdot 2i}+\frac{(3-5i) \cdot (4-3i)}{2i \cdot (4-3i)} \\ &= \frac{14i+18}{8i+6}+\frac{12-20i-9i-15}{8i+6} \\ &= \frac{15-15i}{8i+6}\end{align*}

Lastly, eliminate the imaginary component from the denominator by using the conjugate.

\begin{align*}&=\frac{(15-15i) \cdot (8i-6)}{(8i+6) \cdot (8i-6)} \\ &= \frac{120i-90+120+90i}{100} \\ &= \frac{30i+30}{100} \\ &= \frac{3i+3}{10}\end{align*}

#### Example 4

Simplify the following complex number.

\begin{align*}i^{2013}\end{align*}

When simplifying complex numbers, \begin{align*}i\end{align*} should not have a power greater  than 1.  The powers of \begin{align*}i\end{align*} repeat in a four part cycle:

\begin{align*}i^5 &=i= \sqrt{-1} \\ i^6 &=i^2=-1 \\ i^7 &=i^3=-\sqrt{-1}=-i \\ i^8 &=i^4=1\end{align*}

Therefore, you just need to determine where 2013 is in the cycle.  To do this, determine the remainder when you divide 2013 by 4.  The remainder is 1 so \begin{align*}i^{2013}=i\end{align*}.

#### Example 5

Plot the following complex number on the complex coordinate plane and determine its absolute value.

\begin{align*}-12+5i\end{align*}

The sides of the right triangle are 5 and 12, which you should recognize as a Pythagorean triple with a hypotenuse of 13.  \begin{align*}|-12+5i|=13\end{align*}

### Review

Simplify the following complex numbers.

1. \begin{align*}i^{252}\end{align*}

2. \begin{align*}i^{312}\end{align*}

3. \begin{align*}i^{411}\end{align*}

4. \begin{align*}i^{2345}\end{align*}

For each of the following, plot the complex number on the complex coordinate plane and determine its absolute value.

5. \begin{align*}6-8i\end{align*}

6. \begin{align*}2+i\end{align*}

7. \begin{align*}4-2i\end{align*}

8. \begin{align*}-5i+1\end{align*}

Let \begin{align*}c=2+7i\end{align*} and \begin{align*}d=3-5i\end{align*}.

9. What is \begin{align*}c+d\end{align*} ?

10. What is \begin{align*}c-d\end{align*} ?

11. What is \begin{align*}c \cdot d\end{align*} ?

12. What is \begin{align*}2c-4d\end{align*} ?

13. What is \begin{align*}2c \cdot 4d\end{align*} ?

14. What is \begin{align*}\frac{c}{d}\end{align*} ?

15. What is \begin{align*}c^2 - d^2\end{align*} ?

To see the Review answers, open this PDF file and look for section 11.2.

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### Vocabulary Language: English

$i$

$i$ is an imaginary number. $i=\sqrt{-1}$.

Absolute Value

The absolute value of a number is the distance the number is from zero. The absolute value of a complex number is the distance from the complex number on the complex plane to the origin.

Complex Conjugate

Complex conjugates are pairs of complex binomials. The complex conjugate of $a+bi$ is $a-bi$. When complex conjugates are multiplied, the result is a single real number.

i

$i$ is an imaginary number. $i=\sqrt{-1}$.

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.

rectangular coordinates

A point is written using rectangular coordinates if it is written in terms of $x$ and $y$ and can be graphed on the Cartesian plane.

rectangular form

The rectangular form of a point or a curve is given in terms of $x$ and $y$ and is graphed on the Cartesian plane.