2. What are imaginary and complex numbers? Do Now: Solve for x: x 2 + 1 = 0 ? What number when multiplied by itself gives us a negative one? No such real number Graph it parabola does not intersect x-axis - NO REAL ROOTS

3. Definition: Imaginary Numbers If is not a real number, then is a non-real or imaginary number. i A pure imaginary number is any number that can be expressed in the form bi, where b is a real number such that b ≠ 0, and i is the imaginary unit. In general, for any real number b, where b > 0: b = 5

4. Powers of i i –1 i 2 = If i 2 = – 1, then i 3 = ? = i 2 i = –1( ) = –i i 2 i 2 = (–1)(–1) = 1 = i 4 i = 1( ) = i i 4 i 2 = (1)(–1) = –1 = i 6 i = -1( ) = –i i 6 i 2 = (–1)(–1) = 1 i 0 = 1 i 1 = i i 2 = –1 i 3 = –i i 4 = 1 i 5 = i i 6 = –1 i 7 = –i i 8 = 1 i 9 = i i 10 = –1 i 11 = –i i 12 = 1 What is i 82 in simplest form? 82 ÷ 4 = 20 remainder 2 equivalent to i 2 = –1 i 82 i3i3 i 4 = i 6 = i 8 = i5i5 i7i7 –1 i 2 =

5. Properties of i i Addition: 4i + 3i = 7i Subtraction: 5i – 4i = i Multiplication: (6i)(2i) = 12i 2 = –12 Division:

6. Complex Numbers Definition: A complex number is any number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. a + bi pure imaginary number Any number can be expressed as a complex number: 7 + 0i = 7 0 + 2i = 2i real numbers a + bi

7. Complex Numbers Real Numbers Rational Numbers Integers Whole Numbers Irrational Numbers Counting Numbers The Number System i i i i 75 -i 47 i i -i i3i3 i9i9 i 2 + 3i-6 – 3i 1/2 – 12i

8. Graphing Complex Numbers reals pure imaginaries 1 23456 -5-4-3-2 0 i -i 2i 3i 4i 5i -4i -3i -2i -5i -6i (0 + 0i) (0 + 3i) (0 – 4i) (3 – 2i) (4 + 5i) (–5 – 2i) Complex Number Plane (x, y) a + bi

9. Vectors reals pure imaginaries 1 23456 -5-4-3-2 0 i 2i 3i 4i 5i -4i -3i -2i -i -5i -6i Vector - a directed line segment that represents directed force notation: OP O (3 + 4i) P The length of vectors is found by using the Pythagorean Theorem & is always positive. The length of a vector representing a complex number is

10. Model Problems Add: Express in terms of i and simplify: = 10i = 4/5i Write each given power of i in simplest terms: i 49 i 54 i 300 i 2001 = i= -1= i= 1 Multiply: Simplify:

11. Model Problems 1) (-1.5 + 3.5i) x 1 23456 -5-4-3-2 0 i 2i 3i 4i 5i -4i -3i -2i -i -5i -6i yi Which number is included in the shaded region? 2) (1.5 – 3.5i) 3) (3.5) 4) (4.5i) (1) (2) (3) (4)

12. How do we add and subtract complex numbers? Do Now: Simplify:

13. Adding Complex Numbers (2 + 3i) + (5 + i) = (2 + 5) + (3i + i) = 7 + 4i In general, addition of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i Find the sum of Combine the real parts and the imaginary parts separately. convert to complex numbers combine reals and imaginary parts separately

14. Subtracting Complex Numbers (1 + 3i) – (3 + 2i) = (1 + 3i) + (-3 – 2i) = -2 + i Subtract What is the additive inverse of 2 + 3i? -(2 + 3i) or -2 – 3i Subtraction is the addition of an additive inverse In general, subtraction of complex numbers: (a + bi) – (c + di) = (a – c) + (b – d)i change to addition problem combine reals and imaginary parts separately

15. x 1 23456 -5-4-3-2 0 i 2i 3i 4i 5i -4i -3i -2i -i -5i -6i yi Adding Complex Numbers Graphically (2 + 3i) (2 + 3i) + (3 + 0i) (3 + 0i) (5 + 3i) = (2 + 3) + (3i + 0i) = = 5 + 3i vector: 2 + 3i vector: 3 + 0i vector: 5 + 3i

16. Adding Vectors Vector - a directed line segment that represents directed force notation: OS R The vectors that represent the applied forces form two adjacent sides of a parallelogram, and the vector that represents the resultant force is the diagonal of this parallelogram. O P S resultant force

17. x 1 23456 -5-4-3-2 0 i 2i 3i 4i 5i -4i -3i -2i -i -5i -6i yi Subtracting Complex Numbers Graphically (1 + 3i) (1 + 3i) – (3 + 2i) (3 + 2i) = (1 + 3i) + (-3 – 2i) = -2 + i (-3 – 2i) (-2 + i) The vector representing the additive inverse is the image of the vector reflected through the origin. Or the image under a rotation about the origin of 180 0.

18. Model Problems Add/Subtract and simplify: (10 + 3i) + (5 + 8i) (4 – 2i) + (-3 + 2i) Express the difference of in form a + bi = 15 + 11i = 1