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Complex Numbers

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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 .

Answers:

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  ?

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