Download Complex Numbers

Section 8.7
Complex Numbers
Overview
• In previous sections, it was not possible to find
the square root of a negative number using real
numbers:
•  81 is not a real number, or “n.a.r.n.”
• In this section we will discover a way to find
such square roots.
• We will also learn about complex numbers and
how to add, subtract, and multiply them.
The imaginary number i
• First, a definition: the imaginary number i is
defined to be the square root of –1:
i  1
• It follows that
i  1
2
Definition
• In general, if a is a real number and a > 0, then
a i a
Examples
• Write each number as a product of a real
number and i:
 49
  121
3
 32
“Take the i out”
• When multiplying and dividing square
roots with a negative radicand, first use
the definition to change the radical.
a i a
Examples: Multiply and Divide
 4   16
 5   11
 3   12
13   2
 72
8
 48
3
Complex numbers
• A number in the form a + bi, where a and b are
real numbers and i   1
is called a complex number.
• Complex numbers can be:
1. added (add the real parts together and the
imaginary parts together)
2. subtracted (subtract the real parts and the
imaginary parts)
3. multiplied (use the FOIL method and the fact
2
that i  1 )
Examples: Add and Subtract
•
•
•
•
(-3 + 2i) + (4 + 7i)
(7 + 10i) – (3 + 5i)
(5 – i) + (-3 + 3i) + (6 – 4i)
(-1 + 12i) – (-1 – i)
Examples: Multiply
• 6i(4 + 3i)
• (6 – 4i)(2 + 4i)
• (3 + 2i)(3 + 4i)
Remember that “conjugate”
thing?
• Dividing complex numbers is very much
like dividing radicals.
• To divide complex numbers, multiply both
the numerator and the denominator by the
conjugate of the denominator.
Examples: Divide
4  2i
1  3i
5  4i
i