1. The imaginary unit i is defined as
Date: 2.5 Complex Numbers (2.5)
Complex Numbers
The imaginary unit i is defined as 81 -
Example

2. Equality of Complex Numbers
The set of all numbers in the form a+bi with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part, of the complex number:
a+bi
Equality of Complex Numbers
a+bi = c+di if and only if a = c and b = d

3. Adding and Subtracting Complex Numbers
(a+bi) + (c+di) =
(a+bi) - (c+di) =
(a+c)
+ (b+d)i
(a-c)
- (b-d)i
Perform the indicated operation, writing the result in standard form.
(-5 + 7i) - ( i)
=(-5-(-11)) - (7-(-6))i
= (-5+11) - (7+6)i
= i

4. i 8 6 2 3 - + 5 - = i 4 - i ) 3 1 )( 2 ( + - 2 = i 6 + i - i 3 - 2 = i
Examples i 8 6 2 3 - + 5 - = i 4 - i ) 3 1 )( 2 ( + -
remember
i2=-1 2 = i 6 + i - i 3 2 - 2 = i 5 + 3 + i 5 + =

5. Conjugate of a Complex Number
The complex conjugate of the number
a+bi is a-bi, and the conjugate of a-bi is a+bi.
distribute
Example:
foil

6. Principal Square Root of a Negative Number
For any positive real number b, the principal square root of the negative number -b is defined by 9 16 - ×
Example

7. Example

8. Solve using the quadratic formula:
standard form for complex numbers

9. The Complex Plane
A complex number z = a + bi is represented as a point (a, b) in a complex plane, shown below. The horizontal axis of the coordinate plane is called the real axis. The vertical axis is called the imaginary axis. The coordinate system is called the complex plane.
Imaginary axis
Real axis a b
z = a + bi

10. Plot in the complex plane:
Text Example
Plot in the complex plane:
a. z = 3 + 4i b. z = -1 – 2i c. z = d. z = -4i
Plot the complex number z = 3 + 4i the same way we plot (3, 4) in the rectangular coordinate system.
Imaginary axis -5 -4 -3 -2 1 2 3 4 5
z = 3 + 4i
Real
axis
z = -1 – 2i -1 -1
The complex number z = -1 – 2i corresponds to the point (-1, -2) in the rectangular coordinate system. -2

11. Plot in the complex plane:
Text Example
Plot in the complex plane:
a. z = 3 + 4i b. z = -1 – 2i c. z = d. z = -4i
Imaginary axis
z = -3 = i, this complex number corresponds to the point (-3, 0). -5 -4 -3 -2 1 2 3 4 5
z = 3 + 4i
z = -3
Real
axis
z = -4i = 0 – 4i, this complex number corresponds to the point (0, -4).
z = -4i -1 -1 -2
z = -1 – 2i

12. The Absolute Value of a Complex Number
The absolute value of the complex number a + bi is

13. Determine the absolute value of z = 2-4i
Example
Determine the absolute value of z = 2-4i

14. Complete Student Checkpoint
Perform the indicated operation.
a. (-5+7i)-(-11+6i)
Divide and express the result in standard form:
=(-5-(-11))-(7-6)i
=6 + i
b. (5+4i)(6-7i)
= i + 24i - 28i2
= i + 28
= i

15. Complete Student Checkpoint
Solve using the quadratic formula:
Perform the indicated operation and write the result in standard form:

16. Complete Student Checkpoint
Determine the absolute value of the following complex number: 2 - 3i
Plot in the complex plane. -1 -2 2 1

17. 2.5 Complex Numbers