1. Sec. 1.5
Complex Numbers

3. Imaginary Unit
i = √-1

4. Complex numbers are made by adding real numbers to real multiples of the imaginary unit
Standard form
a + bi
Every real number can be written as a complex number by using b = 0

5. Complex numbers are equal
a + bi = c + di
If and only if a = c and b = d

6. Addition and Subtraction of complex numbers
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) – (c + di) = (a – c) + (b – d)i
Combine like terms

7. Additive Identity
Is 0 (same as real numbers)

8. Additive Inverse of a + bi
So (a + bi) + (-a – bi)
0 + 0i
Or 0

9. Ex. 1
(3 – i) + (2 + 3i)
2i + (-4 – 2i)
3 – (-2 + 3i) + (-5 + i)

10. P. 125 properties that still apply
To multiply use FOIL
Ex. 2
i(-3i)
(2 – i)(4 + 3i)
(3 + 2i)(3 – 2i)
(3 + 2i)2

11. Complex Conjugates and Division
a + bi
Conjugate is
a – bi
Notice what happens when you multiply a complex by its conjugate

12. Dividing You don’t want to leave a complex number in the denominator
To get rid of it in the denominator multiply both numerator and denominator by the conjugate

13. Ex. 3

14. Ex. 4

15. Complex solution of Quadratic Eq.
When you have a negative under the radical in the quadratic formula, factor out i
Ex.
√-3

16. Principal square root
√-a = i√a
Ex. 5
√-3 √-12
√-48 - √-27
(-1 + √-3)2

17. Ex. 6 Solve
3x2 – 2x + 5 = 0
p odd, 51, 53, 63, 65,