Download Complex Numbers Objectives

Objectives
Complex Numbers
Section 2.1
i = the square root of negative 1
• In the real number system, we can’t take the
square root of negatives, therefore the complex
number system was created.
• Complex numbers are of the form, a + bi, where
a = real part & bi = imaginary part.
• If b = 0, a + bi = a; therefore a real number (thus
reals are a subset of complex #).
• If a = 0, a + bi = bi; therefore an imaginary #
(imaginary # are a subset of complex #).
Multiplying Complex Numbers
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Add & subtract complex numbers
Multiply complex numbers
Divide complex numbers
Perform operations with square roots of
negative numbers
Adding & Subtracting Complex Numbers
• Add real to real, add imaginary to imaginary
(same for subtraction)
• Example:
(6+7i) + (3-2i)
(6+3) + (7i-2i) = 9+5i
• When subtracting, DON’T FORGET to
distribute the negative sign!
• (3+2i) – (5 – i)
(3 – 5) + (2i – (-i)) = -2 + 3i
EXAMPLE
Find the product of (2 + 3i) • (3 – 6i)
• Treat as a binomial times binomial.
• BUT what is i times i? It’s -1!! Why??
• Let’s consider i raised to the following powers:
i 2 = ( − 1) 2 = −1
i 3 = i 2 ⋅ i = (−1)i = −i
i 4 = (i 2 ) 2 = (−1) 2 = 1
1
Dividing Complex Numbers
• It is not standard to have a complex number in a
denominator. To eliminate it, multiply by a wellchosen value of 1: ( conjugate/conjugate).
EXAMPLE
Divide. ( 3 – 8i )
( 4 + 3i )
• The conjugate of a + bi = a – bi.
• We use the following fact:
(a + bi ) ⋅ (a − bi ) = a 2 − abi + abi − b 2i 2
= a 2 − b 2 ⋅ (−1) = a 2 + b 2
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