SparkNotes: Free Study Guides No Fear Shakespeare: The Bard made easy SparkCharts: Just the facts TestPrep: SAT, ACT, and more 101s: College texts condensed Subject Finder: Browse by subject SparkCollege: Get in! SparkLife: 100% study-free home_bottom home_top BN_link
 
◄ PREVIOUS
Problems
NEXT ►
Problems
 

Complex Numbers

 
 

Complex Numbers

 

Complex Numbers

 
A complex number is a number of the form a + bi, where i = and a and b are real numbers. For example, 5 + 3i, - + 4i, 4.2 - 12i, and - - i are all complex numbers. a is called the real part of the complex number and bi is called the imaginary part of the complex number. In the complex number 6 - 4i, for example, the real part is 6 and the imaginary part is -4i.
 

Adding and Subtracting Complex Numbers

 
To add two complex numbers, add their real parts and add their imaginary parts: (a1 + b1i) + (a2 + b2i) = (a1 + a2) + (b1 + b2)i.
 

Examples:

(12 + 6i) + (11 + 5i) = (12 + 11) + (6 + 5)i = 23 + 11i
(5 - 7i) + (4 + i) = (5 + 4) + (- 7 + 1)i = 9 - 6i.
(2 - 4i) + (- 6 - 5i) = (2 - 6) + (- 4 - 5)i = - 4 - 9i.
 
To subtract two complex numbers, subtract their real parts and subtract their imaginary parts: (a1 + b1i) - (a2 + b2i) = (a1 - a2) + (b1 - b2)i.
 

Examples:

(4 + 5i) - (2 + 3i) = (4 - 2) + (5 - 3)i = 2 + 2i.
(3 - 7i) - (4 + 6i) = (3 - 4) + (- 7 - 6)i = - 1 - 13i
(- 4 + 2i) - (3 - 11i) = (- 4 - 3) + (2 - (- 11))i = - 7 + 13i
(6 - 9i) - (- 3 - 4i) = (6 - (- 3)) + (- 9 - (- 4))i = 9 - 5i
 

Multiplying a Complex Number by a Scalar

 
To multiply a complex number by a scalar, multiply the real part by the scalar and multiply the imaginary part by the scalar: c(a + bi) = ca + cbi.
 

Examples:

4(2 + 5i) = 4(2) + 4(5i) = 8 + 20i
(6 - 9i) = (6) + (- 9i) = 2 - 3i.
-2(11 - 2i) = - 2(11) + (- 2)(- 2i) = - 22 + 4i.
2i(5 + 7i) = 2i(5) + 2i(7i) = 10i + 14i2 = 10i + 14(- 1) = - 14 + 10i.
 

Multiplying Complex Numbers

 
To multiply two complex numbers, use the FOIL method and treat each complex number as an ordinary binomial. Then simplify the i2 term (i2 = - 1) and combine like terms.


(a1 + b1i)(a2 + b2i) = a1a2 + a1b2i + a2b1i + b1b2i2  
  = a1a2 + (a1b2 + a2b1)i + b1b2(- 1)  
  = (a1a2 - b1b2) + (a1b2 + a2b1)i.  

 

Examples:

(2 + 3i)(5 + 2i) = ?


  = 10 + 4i + 15i + 6i2  
  = 10 + 19i - 6  
  = 4 + 19i.  

(3 - 4i)(6 + i) = ?


  = 18 + 3i - 24i - 4i2  
  = 18 - 21i + 4  
  = 22 - 21i.  

(7 - 2i)(6 - 2i) = ?


  = 42 - 14i - 12i + 4i2  
  = 42 - 26i - 4  
  = 38 - 26i.  

(2 + 3i)2 =?


  = (2 + 3i)(2 + 3i)  
  = 4 + 6i + 6i + 9i2  
  = 4 + 12i - 9  
  = -5 + 12i.  

(5 + 4i)(5 - 4i) = ?


  = 25 - 16i2  
  = 25 + 16  
  = 41.  

 
 
Help | Feedback | Make a request | Report an error | Send to a friend

◄ PREVIOUS
Problems
NEXT ►
Problems
 
 
 
 
 
 
 
Test Prep Books
Take the next step in test prep.
  • SAT Subject Test: Math Level 1
  • SAT Subject Test: Math Level 2
  •  
    Test Prep Centers
    Take a practice exam. Do better.
  • SAT Subject Test: Math Level 1 Test Center
  • SAT Subject Test: Math Level 2 Test Center
  •  
    SparkCharts
    A textbook's worth of information on an easy-to-read chart.
  • Math Basics
  • Algebra II
  •  
     
     
    Contact Us | Privacy Policy | Terms and Conditions | About | Sitemap
    ©2008 SparkNotes LLC, All Rights Reserved.