2.4 Complex Numbers
Complex numbers are expressions that include real and non-real portions. These are used to find solutions to problems where there is no real solution. Standard form of complex numbers is written as "a+bi", where a is the real portion of the answer and bi is the imaginary portion (b is a real coefficient).
i is equal to the square root of -1.
For example, to symplify -5:
Factor out an i from , so you get i times the square root of 9, or 3i.
Now put the answer in standard form with the real portion first. -5+3i
Two complex numbers, a+bi and c+di are equal. Therefore, a=c and b=d.
To add and subtract complex numbers, group the real parts together and the imaginary parts together like so:
(a+bi)+(c+di)=(a+c)+(b+d)i
ex. (3+2i)+(4-i)-(7+i)=3+2i+4-i-7-i
=3+4-7+2i-i-i
=0+0i
=0
Multiplying Complex Numbers:
Example 1:
*=(2i)+(4i) Write the factor in standard form
=8i^2 Multiply out the factors
=8(-1) Change the squared, imaginary number to -1
=-8
Example 2:
(2-i)(4+2i)=8+6i-4i-3i^2 Factor out
= 8+6i-4i-3(-1) Change the squared, imaginary nember to -1
= 8+3+6i-4i Group Like Terms
=11+2i Set In Standard From
Complex Conjugates: occurs with pairs of complex numbers of the form a+bi and a-bi
(a+bi)(a-bi)=a^2 -abi =abi-(B^2)(i^2)
= (a^2)-b^2(-1)
= (a^2)+(b^2)
To find the quotient of a+bi and c+di, multiply the numerator and the denominator by teh conjugate of the denominator:
a+bi a+bi (c-di)
------ = ------ * ------
c+di c+di (c-di)
=(ac+bd)+(bc-ad)i
---------------------
(c^2)+(d^2)
Dividing Complex Numbers:
1 1 (1-i)
------ = ------ * ------ Multiply Numerator and denominator by conjugate of denominator
1+i 1+i (1-i)
= 1-i
------------ Expand
(1^2)-(i^2)
= 1-i
------- i^2 = -1
1-(-1)
= 1-i
----- Write in standard form
2
= 1 1
--- _ ---(i)
2 2
For Additional Help:
http://www.purplemath.com/modules/complex.htm
http://www.uncwil.edu/courses/mat111hb/Izs/complex/complex.html
Comments (1)
Reid Knuth said
at 5:45 pm on Dec 16, 2008
Garrett, if u wanna write in color just use the tool at the top of our page.
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