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Section 1.5 Complex Numbers

Section 1.5 Complex Numbers. What you should learn. How to use the imaginary unit i to write complex numbers How to add, subtract, and multiply complex numbers How to use complex conjugates to write the quotient of two complex numbers in standard form

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Section 1.5 Complex Numbers

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  1. Section 1.5 Complex Numbers

  2. What you should learn • How to use the imaginary unit i to write complex numbers • How to add, subtract, and multiply complex numbers • How to use complex conjugates to write the quotient of two complex numbers in standard form • How to use the Quadratic Formula to find complex solutions to quadratic equations.

  3. Real Number System Natural How many irrational numbers are there? Whole Integers Irrational Rational

  4. Real Number System Natural Each set is a subset of the Real Number System. The union of all these sets forms the real number system. The number line is our model for the real number system. Whole Integers Irrational Rational Real Numbers

  5. Definition of Square Root If a2 = n then a is a square root of n. 42 = (4)(4) = 16  4 is a square root of 16 (-4)2 = (-4)(-4) = 16  -4 is a square root of 16

  6. What is the square root of -16? Whatever it is it is not on the real number line.

  7. Definition of i The number i is such that Imaginary Unit

  8. Complex Numbers REAL Imaginary Complex

  9. 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 then the number a + bi = a is a real number. • If b≠ 0, then the number a + bi is called an imaginary number. • A number of the form bi, where b ≠ 0 is called a pure imaginary number.

  10. Examples

  11. If you square a radical you get the radicand 2 2 Whenever you have i2 the next turn you will have -1 and no i.

  12. Equality of Complex numbers If a + bi = c + di, then a = c and b = d.

  13. Is a negative times a negative always positive? Trick question. This is not a negative times a negative.

  14. Example

  15. Example

  16. Example

  17. Example Cancel the i factor

  18. Collect like terms. Add

  19. First distribute the negative sign. Now collect like terms. Subtract

  20. Multiplication F O I L

  21. Simplify each expression. Express your answer in a + bi form. F-O-I-L Recall i2=-1 Combine like terms. Combine like terms.

  22. Write in the form 2 Multiply by the conjugate factor.

  23. Powers of i Anything other than 0 raised to the 0 is 1. Anything raised to the 1 is itself.

  24. Simplify as much as possible.

  25. Homework Section 1.5 Complex Numbers Page 129 1-4, 17-39 odd, 49-55 odd, 63, 65, 73, 88

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