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Section 2.4 Complex Numbers

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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 find complex solutions to quadratic equations

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Real Number System Natural {1, 2, 3, 4,…} How many natural numbers are there?

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Real Number System Whole Natural {0, 1, 2, 3, 4,…} How many whole numbers are there?

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Real Number System Integers Whole Natural {...-3, -2, -1, 0, 1, 2, 3, …} How many integers numbers are there?

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Real Number System Rational Integers Whole Natural Fractions How many rational numbers are there?

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

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Real Number System Rational Integers Whole 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. Irrational Real Numbers

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

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

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Definition of i The number i is such that Imaginary Unit

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Imaginary REAL Complex Complex Numbers

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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.

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Examples

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

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

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

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Example

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Cancel the i factor

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Add Collect like terms.

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

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Multiplication FOIL

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Simplify each expression. Express your answer in form. Combine like terms. Recall i 2 =-1 F-O-I-L Combine like terms.

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Write in the form Multiply by the conjugate factor. 2

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

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Simplify as much as possible.

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Use the Quadratic Formula

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Homework Section , 83 odd

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