This document introduces complex numbers. It defines the imaginary unit i as the square root of -1, which allows quadratic equations with no real solutions, like x^2=-1, to be solved. Complex numbers have both a real part and an imaginary part in the form a + bi. They can be added, subtracted, multiplied, and divided by distributing terms and using properties of i such as i^2 = -1. Complex numbers are plotted on a plane with real numbers on the x-axis and imaginary numbers on the y-axis.

1. 5.4 Complex Numbers

2. The Imaginary Unit
 Not all quadratics have real number
solutions.
 Example: x2 = -1 has no real solution. (A
number squared can’t be negative.)
 Imaginary units are numbers that can
be represented by equations but could
not physically exist in real life.



3.  Since , i can be used to write
the square root of any negative
number.
 If r is a positive real #, then
◦ Example:
 It follows that
◦ Example:

4. Solving a Quadratic Eqn.
 Solve 3x2 + 10 = -26

5. Your Turn!
 Solve 2x2 + 26 = -10

6. Complex Numbers
 In standard form, a complex number is
a number a + bi where a and b are real
numbers.
 a is the real part.
 bi is the imaginary part.
 If b  0, a + bi is an imaginary number.
 If a = 0 and b  0, a + bi is a pure
imaginary number

7. Examples
 Real numbers (a + 0i):
◦ -1, 5/2, 3,
 Imaginary Numbers (a + bi , b  0):
◦ 2 + 3i, 5 – 5i
 Pure Imaginary Numbers (0 + bi, b  0):
◦ -4i, 6i

8. Plotting Complex Numbers
 Plotted on the complex plane.
 Horizontal axis = real axis
 Vertical axis = imaginary axis
imaginary
Plot:
 2 – 3i
 -3 + 2i real
 4i

9. Adding and Subtracting
 Add (or subtract) real and imaginary
parts separately.
 Examples:
 (4 – i) + (3 + 2i)
 (7 – 5i) – (1 – 5i)
 6 – (-2 + 9i) + (-8 + 4i)

10. Your Turn!
 Write in standard form:
(3 – 5i) – (9 + 2i)

11. Multiplying
 Distribute or use FOIL (remember )
 5i(-2 + i)
 (7 – 4i)(-1 + 2i)

12. Complex Conjugates
 Write in standard form: (6 + 3i)(6 – 3i)
 The product of complex conjugates
(a + bi) and (a – bi) is always a real
number.

13. Dividing
 Multiply the numerator and denominator
by the complex conjugate of the
denominator.
 Example:

14. Your Turn!
Write as a complex number in standard form:
 (-1 + 4i)(3 – 6i)
