Title: 1-2 complex and imaginary numbers
1 Complex Numbers
- N-CN.1 Know there is a complex number i such that
i2 1, and every complex has the form a bi
with a and b real. - N-CN.2 Use the relation i2 1 and the
commutative, associative, and distributive
properties to add, subtract, and multiply complex
numbers.
2First a little review
3Real Number System
- 1, 2, 3, 4,
- How many natural numbers are there?
Natural
Infinite
4Real Number System
- 0, 1, 2, 3, 4,
- How many whole numbers are there?
Natural
Whole
Infinite
5Real Number System
Natural
- ...-3, -2, -1, 0, 1, 2, 3,
- How many integers numbers are there?
Whole
Integers
Infinite
6- Fractions
- How many rational numbers are there?
Natural
Whole
Infinite
Integers
Rational
7Real Number System
Natural
- How many irrational numbers are there?
Whole
Integers
Infinite
Rational
Irrational
8Real 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
9Definition 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
The square root of a number is a value that, when
multiplied by itself, gives the number.
10What square root of -16?
- Whatever it is,
- it is not on the real number line.
11- You may remember being told that you can't take
the square root of a negative number, because no
numbers when squared resulted in a negative
product. - Â Squaring a negative number always gives you a
positive. Â
12- however, if we had a number for the square root
of negative one, we could take the square root
of a negative numbers. - So mathematicians came up
with the number i
13Definition of i
Imaginary Unit
The number i is such that
14Definition of pure imaginary numbers
i is not a variable it is a symbol for a
specific number, the square root of negative one.
15Imaginary Numbers
A number that when squared gives a negative
result.
16Complex Numbers
- A complex number is a combination of a
- real number and an imaginary number.
- Standard form is
Real part
Imaginary part
Example 54i
17Complex Numbers
Imaginary
REAL
Complex
18If you square a radical you get the radicand
2
2
Whenever you have i2 , it can be simplified to
-1 because i2 -1
19Examples
20Is a negative times a negative always positive?
Simplify negative radicals before multiplying
21Example
22Example
23Example
24Example
Cancel the i factor
25Add
Collect like terms.
26Subtract
First distribute the negative sign.
Now collect like terms.
27Multiplication
F
O
I
L
28Simplify each expression. Express your answer in
form.
F-O-I-L
Recall i2-1
Combine like terms.
Combine like terms.
29Equality of Complex numbers
- If a bi c di, then a c and bi di.