<< Chapter < Page Chapter >> Page >
In this section you will:
  • Add and subtract complex numbers.
  • Multiply and divide complex numbers.
  • Simplify powers of i .
A visual representation of the Mandelbrot set

Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple.

In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it.

Expressing square roots of negative numbers as multiples of i

We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number . The imaginary number i is defined as the square root of −1.

−1 = i

So, using properties of radicals,

i 2 = ( 1 ) 2 = −1

We can write the square root of any negative number as a multiple of i . Consider the square root of −49.

−49 = 49 ( −1 ) = 49 −1 = 7 i

We use 7 i and not −7 i because the principal root of 49 is the positive root.

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a + b i where a is the real part and b is the imaginary part. For example, 5 + 2 i is a complex number. So, too, is 3 + 4 i 3 .

The complex number 5 + 2i is displayed.  The 5 is labeled as: Real part and the 2i is labeled as: Imaginary part

Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.

Imaginary and complex numbers

A complex number    is a number of the form a + b i where

  • a is the real part of the complex number.
  • b is the imaginary part of the complex number.

If b = 0 , then a + b i is a real number. If a = 0 and b is not equal to 0, the complex number is called a pure imaginary number. An imaginary number    is an even root of a negative number.

Given an imaginary number, express it in the standard form of a complex number.

  1. Write a as a −1 .
  2. Express −1 as i .
  3. Write a i in simplest form.

Expressing an imaginary number in standard form

Express −9 in standard form.

−9 = 9 −1 = 3 i

In standard form, this is 0 + 3 i .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Express −24 in standard form.

−24 = 0 + 2 i 6

Got questions? Get instant answers now!

Plotting a complex number on the complex plane

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number, we need to address the two components of the number. We use the complex plane    , which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs ( a , b ) , where a represents the coordinate for the horizontal axis and b represents the coordinate for the vertical axis.

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Please prove it
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
the 28th term is 175
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?