1. 5.4 Complex Numbers

2. Objectives
Solve quadratic equations with complex solutions and perform operations with complex numbers.
Assignment: odd

3. Not all quadratic equations have real-number solutions:
Has no real-number solution because the square of any real number x is never negative.
We create an “expanded” number system by introducing the imaginary unit i.
i =
So, x x

4. Simplify: x x x

5. Solve the equation:

6. Complex Numbers 2 + 3i 5 - 5i a + bi These are complex numbers:
A complex number written in standard form that has a real part a and an imaginary part b.
a + bi
Real part imaginary part
If the real part a = 0, then a + bi is called a pure imaginary number.
These are pure imaginary numbers: -6i, 2i

7. Just as every real number corresponds to a point on the real number line, every complex number corresponds to a point in the complex plane.
-6 + 3i
2 + 3i
(-6,3)
(2,3)
-2i
(0,-2)
Notice these are just like the ordered pairs we are familiar with.

8. (a + bi) + (c + di) = (a + c) + (b + d) i
To add complex numbers (in standard form):
Add the real part to the real part
And the complex part to the complex part:
(4 – i) + (3 + 2i)
= 7 + i
(a + bi) + (c + di) = (a + c) + (b + d) i
Subtraction of complex numbers:
(a + bi) - (c + di) = (a - c) + (b - d) i

9. Multiplying Complex Numbers
Recall that
5i(-2 + i)
(7-4i)(-1+2i)

10. We will call these two complex numbers:
Notice (1+6i)(1-6i)=37
1 - 6i
Complex Conjugates.
The product of two complex conjugates (a+bi) & (a – bi) is always a REAL NUMBER.

11. Dividing two complex numbers
Multiply numerator and denominator by the complex conjugate of the denominator…
Answer left in standard form.

12. Absolute Value of a Complex Number
The absolute value of a complex number z = a + bi, denoted |z|, is a non-negative real number defined as follows:
Geometrically, the absolute value of a complex number is the numbers distance from the origin in the complex plane.
Z = i
Z = i