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Complex-Numbers

last edited by 11 years, 7 months ago

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)

---------------------

(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

http://www.purplemath.com/modules/complex.htm

http://www.uncwil.edu/courses/mat111hb/Izs/complex/complex.html