Transcript Section 2-5

Section 2-5
Complex Numbers
Section 2-5
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complex numbers and i
operations with complex numbers (+,-, x)
complex conjugates and division
solving quadratic equations with complex
solutions
• plotting complex numbers
• absolute value of complex numbers
Complex Numbers
• we learned back in Algebra 1 that the
square root of a negative number is not a
real number
• there is a way to work with these numbers
using the imaginary unit, i
• we use this simple definition: i  1
• for example:
9  3i and
8  2i 2
Complex Numbers
• all numbers we work with are part of the
set of complex numbers
• this set consists of all real numbers and
all imaginary numbers (contain i)
• all complex numbers can be written in the
form a + bi
• a is the real part, b is the imaginary part
Operations With Complex Numbers
• to add complex numbers, add their like
parts (same for subtraction)
(a  bi)  (c  di)  (a  c)  (b  d )i
(a  bi)  (c  di)  (a  c)  (b  d )i
• to multiply complex numbers, use FOIL
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• use the fact that i  1
Division of Complex Numbers
• if a + bi is a complex number, then its
complex conjugate is a – bi
• in order to simplify the division of two
complex numbers, multiply the top and
bottom of the fraction by the conjugate of
the denominator
• use FOIL on both top and bottom; the
bottom will no longer contain i
Solving Quad.’s
• now, when you solve a quadratic equation
for which the discriminant is negative, you
can find its complex solutions
• the solutions will be complex conjugates
Plotting Complex Numbers
• complex numbers cannot be plotted on a
single number line because they have
both a real and imaginary part
• instead, we plot them on a complex plane
which looks a lot like a coordinate plane
we use for ordered pairs
• the axes of this plane are the real axis
and the imaginary axis
Plotting Complex Numbers
b
-5 + 3i
a
3 – 6i
Absolute Value of Complex
Numbers
• remember that absolute value means
distance from 0 on a number line
• for complex numbers, it’s the distance
from the origin
• we use the distance formula to compute it
a  bi  a  b
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