 Section 2.4 Complex Numbers

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Section 2.4 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 How to find complex solutions to quadratic equations

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

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

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

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

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

Real Number System Each set is a subset of the 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

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

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

Definition of i The number i is such that Imaginary Unit

Complex Numbers REAL Imaginary Complex

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.

Examples

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.

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

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

Example

Example

Example

Example Cancel the i factor

Add Collect like terms.

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

Multiplication F O I L

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

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

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

Simplify as much as possible.

Use the Quadratic Formula

Homework Section , 83 odd