1. Section 5.4 Imaginary and Complex Numbers

2. Imaginary Numbers The result of a square root of a negative number.
To overcome this problem, the IMAGINARY UNIT “i” was created.

5. Complex Numbers Contains a real number and an imaginary number
Always written in the form a + bi
 a is the real part, b is the imaginary part
Examples:
7 + 2i
2.5 – 3i i
¼ + 2i
5 + 10i

6. Adding and Subtracting Complex Numbers
Add the real parts, then add the imaginary parts
Examples:
(4+7i) + (-8 + 2i)
(7-5i) + (12-4i)
(12+6i) – (15-3i)

7. Practice
Page 277, #18-28 even and #38-46 even

8. Multiplying Complex Numbers
Use the distributive property or FOIL, just as we did with real numbers (remember that i2 = -1)
i(7+i)
2i(10-3i)

9. (4+i)(3+2i)
(2+3i)(2-3i)

12. Practice with Multiplication
Page 278, #47-55 ALL

13. Division with Complex Numbers
Just like with square roots, we do not want to have a complex number in the denominator of a fraction, or division problem. Example: 5 3+4i
To simplify this, we use the Complex Conjugate of the denominator.

14. The Complex Conjugate
of any complex number (a+bi) is (a-bi). Examples: Complex Conjugate 7+4i 7-4i 12-3i 12+3i -9-2i -9+2i

15. 5 3+4i
Multiply by the complex conjugate of the denominator over itself

16. 8
2+4i

17. 5+3i
1-2i

18. -5-4i
6+7i

19. Practice
Page 278, 56-63