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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?

Watch This

http://www.youtube.com/watch?v=htiloYIILqs  James Sousa: Complex Number Operations

Guidance

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,  $4+5i+2-3i=6+2i$ .

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 $i$  are:

• $i=\sqrt{-1}$
• $i^2=-1$
• $i^3=-\sqrt{-1}=-i$
• $i^4=1$
• $i^5=i$
• . . . and the pattern repeats

The complex plane is set up in the same way as the regular $x, y$  plane, except that real numbers are counted horizontally and complex numbers are counted vertically. The following is the number $4+3i$  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 $|4+3i|$  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, $|4+3i|=\sqrt{4^2+3^2}=\sqrt{25}=5$ .

Example A

Multiply and simplify the following complex expression.

$(2+3i)(1-5i)-3i+8$

Solution:  $(2+3i)(1-5i)-3i+8$

$&= 2-10i+3i-15i^2-3i+8 \\&= 10-10i+15 \\&= 25-10i$

Example B

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

$\left(\sqrt{3}+2i\right)^3$

Solution:  $\left(\sqrt{3}+2i\right) \cdot \left(\sqrt{3}+2i\right) \cdot \left(\sqrt{3}+2i\right)$

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

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 $-9\sqrt{3}$

Example C

Simplify the following complex expression.

$\frac{7-9i}{4-3i}+\frac{3-5i}{2i}$

Solution: To add fractions you need to find a common denominator.

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

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

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

Concept Problem Revisited

A better way to think about the absolute value 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.

Vocabulary

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

The complex number plane is just like the regular $x, y$  coordinate system except that the horizontal component is the real portion of the complex number $(a)$  and the vertical component is the complex portion of the number $(b)$

A complex number is a number written in the form $a+bi$  where both $a$  and $b$  are real numbers.  When $b=0$ , the result is a real number and when $a=0$  the result is an imaginary number.

An imaginary number is the square root of a negative number. $\sqrt{-1}$ is defined to be the imaginary number $i$ .

Complex conjugates are pairs of complex numbers with real parts that are identical and imaginary parts that are of equal magnitude but opposite signs.   $1+3i$ and $1-3i$  or $5i$  and $-5i$  are examples of complex conjugates.

Guided Practice

1. Simplify the following complex number.

$i^{2013}$

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

$-12+5i$

3. For  $a=3+4i, b=1-2i$  compute the sum, difference and product of $a$  and $b$ .

1. When simplifying complex numbers, $i$ should not have a power greater  than 1.  The powers of $i$ repeat in a four part cycle:

$i^5 &=i= \sqrt{-1} \\i^6 &=i^2=-1 \\i^7 &=i^3=-\sqrt{-1}=-i \\i^8 &=i^4=1$

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  $i^{2013}=i$ .

2.

The sides of the right triangle are 5 and 12, which you should recognize as a Pythagorean triple with a hypotenuse of 13.   $|-12+5i|=13$

3.

$a+b &=(3+4i)+(1-2i)=4-2i\\a-b &=(3+4i)-(1-2i)=2+6i \\a \cdot b &=(3+4i) \cdot (1-2i)=3-6i+4i+8=11-2i$

Practice

Simplify the following complex numbers.

1.  $i^{252}$

2.  $i^{312}$

3.  $i^{411}$

4.  $i^{2345}$

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

5.  $6-8i$

6.  $2+i$

7.  $4-2i$

8.  $-5i+1$

Let  $c=2+7i$  and  $d=3-5i$ .

9. What is  $c+d$  ?

10. What is $c-d$  ?

11. What is  $c \cdot d$  ?

12. What is  $2c-4d$  ?

13. What is  $2c \cdot 4d$  ?

14. What is  $\frac{c}{d}$  ?

15. What is  $c^2 - d^2$  ?

Vocabulary Language: English

$i$

$i$

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

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 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

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

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

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

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.