11.2. Arithmetic with Complex Numbers http://www.ck12.org
Example C
Simplify the following complex expression.
(^74) −− (^93) ii+ 3 − 2 i 5 i
Solution:To add fractions you need to find a common denominator.
( 7 − 9 i)· 2 i
( 4 − 3 i)· 2 i+
( 3 − 5 i)·( 4 − 3 i)
2 i·( 4 − 3 i)
=^148 ii++^186 +^12 −^208 ii+− 69 i−^15
=^158 i−+^156 i
Lastly, eliminate the imaginary component from the denominator by using the conjugate.
=(^15 ( 8 i−+^156 )i·)·( 8 (^8 i−i− 66 ))
=^120 i−^90100 +^120 +^90 i
=^30100 i+^30
=^3 i 10 +^3
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
Theabsolute value of a complex numberis the distance from the complex number to the origin.
Thecomplex number planeis just like the regularx,ycoordinate 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).
Acomplex numberis a number written in the forma+biwhere bothaandbare real numbers. Whenb=0, the
result is a real number and whena=0 the result is an imaginary number.
Animaginary numberis the square root of a negative number.√−1 is defined to be the imaginary numberi.
Complex conjugatesare pairs of complex numbers with real parts that are identical and imaginary parts that are of
equal magnitude but opposite signs. 1+ 3 iand 1− 3 ior 5iand− 5 iare examples of complex conjugates.