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Complex Numbers Lesson 3.3.

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Presentation on theme: "Complex Numbers Lesson 3.3."— Presentation transcript:

1 Complex Numbers Lesson 3.3

2 It's any number you can imagine
The Imaginary Number i By definition Consider powers if i It's any number you can imagine

3 Using i Now we can handle quantities that occasionally show up in mathematical solutions What about

4 Complex Numbers Combine real numbers with imaginary numbers Examples
a + bi Examples Real part Imaginary part

5 Try It Out Write these complex numbers in standard form a + bi

6 Operations on Complex Numbers
Complex numbers can be combined with addition subtraction multiplication division Consider

7 Operations on Complex Numbers
Division technique Multiply numerator and denominator by the conjugate of the denominator

8 Complex Numbers on the Calculator
Possible result Reset mode Complex format to Rectangular Now calculator does desired result

9 Complex Numbers on the Calculator
Operations with complex on calculator Make sure to use the correct character for i. Use 2nd-i

10 Warning Consider It is tempting to combine them
The multiplicative property of radicals only works for positive values under the radical sign Instead use imaginary numbers

11 Try It Out Use the correct principles to simplify the following:

12 The Discriminant Return of the
Consider the expression under the radical in the quadratic formula This is known as the discriminant What happens when it is Positive and a perfect square? Positive and not a perfect square? Zero Negative ? Complex roots

13 Example Consider the solution to Note the graph
No intersections with x-axis Using the solve and csolve functions

14 Fundamental Theorem of Algebra
A polynomial f(x) of degree n ≥ 1 has at least one complex zero Remember that complex includes reals Number of Zeros theorem A polynomial of degree n has at most n distinct zeros Explain how theorems apply to these graphs

15 Conjugate Zeroes Theorem
Given a polynomial with real coefficients If a + bi is a zero, then a – bi will also be a zero

16 Assignment Lesson 3.3 Page 211 Exercises 1 – 78 EOO

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