Section 2.4

1. Section 2.4 Complex Numbers

2. Complex Numbers Solve x2 + 1 = 0 But what is ? Remember in Algebra II, this was defined as the imaginary unit or i. Therefore, in our problem the final solution is

3. Complex Numbers With the addition of the imaginary unit, a new set of numbers was formed – the complex numbers. Definition of a Complex Number If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form. If b = 0, the number a + bi = a is a real number. If b 0, the number a + bi is called an imaginary number. A number of the form bi, where b  0, is called a pure imaginary number. Complex Numbers Imaginary Numbers Real Numbers

4. Complex Numbers Equality of Complex Numbers Two complex numbers a + bi and c + di, written in standard form, are equal to each other a + bi = c + di if and only if a = c and b = d. Example: Solve for x and y. x – 3 (x – 3) (4 – y)i 4 – y 9 9 – 6i – 6 = + a a b b = = y = 10 x = 12

5. Complex Numbers Operations with Complex Numbers Addition and Subtraction of Complex Numbers If a + bi and c + di are two complex numbers written in standard form, Their sum and differences are defined as follows. Sum: (a + bi) + (c + di) = (a + c) + (b + d)i Difference: (a + bi) – (c + di) = (a – c) + (b – d)i Examples: (4 + 2i) + (7 – 6i) = (4 + 7) + (2 – 6)i = 11 – 4i (7 – 2i) – (3 – 5i) = (7 – 3) + (– 2 + 5)i = 4 + 3i Additive Inverse: Additive Identity (a + bi) + [ -(a + bi)] = 0 (a + bi) + 0 = a + bi

6. Complex Numbers Powers of i i2 = = -1 Every power of i can be simplified into one of four choices: = (-1)i i3 = = – i i4 = = 1 = – 1  – 1 i – 1 – i 1 = 1(i) i5 = = i i6 = = 1(– 1) = – 1 = – i = 1(– i ) i7 = What is i98? = 1(1) = 1 i8 = Think of i98 as (i4)24i2 i98 = (i4)24 i2 = (1)24 – i i98 = – i

7. Complex Numbers Powers of i (cont.) YES! Is there a “quicker” way to simplify a power of i? Since i4 = 1, divide the power of i by 4 and find the remainder. It is the remainder that gives one the simpler power of i to simplify. Example: Simplify i115 Step 3: Simplify the equivalent power of i Step 1: Step 2: Rewrite i115 as its equivalence using the remainder as the new power of i i3 = – i i115 = – i Therefore, i115 = i3 REMAINDER

8. Complex Numbers Multiplication of complex numbers When multiplying follow the same procedures as in algebra, Except simplify all negatives under the square root symbol first And simplify all powers of i. Example: Example: Solution: Solution:

9. Complex Numbers Multiplication of complex numbers (cont.) Example: Example: Example: Solution: Solution: Solution:

10. Complex Numbers Rationalizing the Denominator To rationalize a denominator, remember you are multiplying by a form of 1. That form of one consists of the denominator’s conjugate. Remember a conjugate is identical to its partner, but holds the opposite operation. For example, the conjugate of 2 + 3i is 2 – 3i.

11. Complex Numbers Example: Express the following in a + bi form.

12. Complex Numbers Try these a = 6, b = 5 20 + 29i 1. 5. Solve for a and b. 85 2. 6. 7 3 + 4i 7. 3. 8 – 20i 8. 4.

13. Complex Numbers Graphing a complex number When graphing a complex number in the complex plane, a corresponds to the x-axis and bi corresponds to the y-axis. The x-axis is then referred to the Real axis and the y-axis is referred to as the Imaginary axis. Imaginary axis – 3 + 4i Graph –3 + 4i 4i + bi a Real axis –3

14. Complex Numbers What you should know 1. How to simplify a power of i 2. How to add and subtract complex numbers. 3. How to multiply complex numbers • How to use complex conjugates to write the quotient of two • complex numbers in standard form 5. How to graph a complex number