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Complex Numbers

Complex Numbers

Problems

Problems

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: (a 1 + b 1 i) + (a 2 + b 2 i) = (a 1 + a 2) + (b 1 + b 2)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: (a 1 + b 1 i) - (a 2 + b 2 i) = (a 1 - a 2) + (b 1 - b 2)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 + 14i 2 = 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 i 2 term ( i 2 = - 1 ) and combine like terms.


(a 1 + b 1 i)(a 2 + b 2 i) = a 1 a 2 + a 1 b 2 i + a 2 b 1 i + b 1 b 2 i 2  
  = a 1 a 2 + (a 1 b 2 + a 2 b 1)i + b 1 b 2(- 1)  
  = (a 1 a 2 - b 1 b 2) + (a 1 b 2 + a 2 b 1)i.  


Examples:

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


  = 10 + 4i + 15i + 6i 2  
  = 10 + 19i - 6  
  = 4 + 19i.  

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


  = 18 + 3i - 24i - 4i 2  
  = 18 - 21i + 4  
  = 22 - 21i.  

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


  = 42 - 14i - 12i + 4i 2  
  = 42 - 26i - 4  
  = 38 - 26i.  

(2 + 3i)2 = ?


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

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


  = 25 - 16i 2  
  = 25 + 16  
  = 41.  

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