The Number System

The Complex Number System and Operations with Numbers

Repeating Decimals

• Repeating decimals are decimals that contain a infinite number of digits.

• Examples: 0.333… 7.689689…

FYI…The line above the decimals indicate that numberrepeats.

9.1

Terminating Decimals

• Terminating decimals are decimals that contain a finite number of digits.

• Examples:36.80.1254.5

The Complex Number System

• All numbers in the world• Represented by ℂ

Complex Number System

Real Numbers

Imaginary Numbers

Imaginary Numbers

Imaginary numbers are all the numbers that deal with the square root of a negative number and contain the letter i in it.

Example:

You will learn more about these numbers in Algebra 2

i525

Real Numbers

• Real numbers consist of all numbers that can be represented on a number line.

• Represented by ℝ

Complex Number System

Real Numbers

Rational Irrational

Imaginary Numbers

Irrational Numbers

• Irrational numbers are any numbers that cannot be

expressed as .

• They are expressed as non-terminating, non-repeating

decimals; decimals that go on forever without

repeating a pattern.

• Examples of irrational numbers:

– 0.34334333433334…– 45.86745893…– (pi)–

b

a

2

Rational Numbers

• Rational numbers are any numbers that can be expressed in the form of , where a and b are integers, and b ≠ 0.

• They can always be expressed by using terminating decimals or repeating decimals.

• Represented by ℚ• Examples:

b

a

3.0,125.0,5

2

Complex Number System

Real Numbers

Rational

Integers

Whole Numbers

Natural Numbers

Irrational

Imaginary Numbers

Integers

• Integers are the set of whole numbers and their opposites.

{…,-3, -2, -1, 0, 1, 2, 3,…}

• Represented by ℤ

Whole Numbers

• Whole numbers are the set of numbers that include 0 plus the positive numbers.

{0, 1, 2, 3, 4, 5,…}

• Represented by 𝕎

Natural Numbers

• Natural numbers are the set of counting numbers.

{1, 2, 3,…}

• Represented by ℕ or ℙ

Venn Diagram of the Complex Numbers

Irrational Numbers

Rational Numbers

Complex Numbers

Imaginary NumbersReal Numbers

Integers

Whole Numbers

NaturalNumbers

Example

• Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that apply.

a. 117

b. 0

c. -12.64039…

d. -½

e. 6.36

f. -3

FYI…For Your Information

• When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non-repeating. Therefore, those numbers are always irrational.

Properties of Real Numbers

Property Addition MultiplicationCommutative a+b = b+a ab = ba

Associative (a+b)+c = a+(b+c) (ab)c = a(bc)

Identity a + 0 = a a•1 = a

Inverse a + (-a) = 0 a = 1Opposite Reciprocal

Distributive Property

a(b + c) = ab + ac

a

1

Examples of PropertiesName the property displayed:

1. -2 + (x – 5) = (-2 + x) – 5

2. (-2) ( -½ ) = 1

3. 2(4 – 5) = (4 – 5)2

4. x(y – w) = xy – xw

Order of Operations

1. Parenthesis/Grouping Symbols

2. Exponents3. Multiplication and Division

– left to right4. Addition and/or

Subtraction – left to right

Grouping Symbols

Grouping symbols include parenthesis, braces, brackets, numerators and denominators of fractions and underneath a radical or inside absolute value symbols.

Examples – Using Order of

OperationsEvaluate the following:

1. 22(12 + 8) 5

2. 52 ÷ (2 + 11)

3. 7 • 12 + 30 ÷ 5