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… … FYI…The line above the decimals indicate that number repeats.

Terminating Decimals Terminating decimals are decimals that contain a finite number of digits. Examples:

The Complex Number System All numbers in the world Represented by Complex Number System Real Numbers Imaginary Numbers

The Complex Number System Complex Number System Real Numbers RationalIntegers Whole Numbers Natural Numbers Irrational 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

Real Numbers Real numbers consist of all numbers that can be represented on a number line. Represented by

Complex Number System Real Numbers RationalIrrational 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: – … – … – (pi) –

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 Examples:

Complex Number System Real Numbers RationalIntegers 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

Whole Numbers Whole numbers are the set of numbers that include 0 plus the positive numbers. {0, 1, 2, 3, 4, 5,…} Represented by

Natural Numbers Natural numbers are the set of counting numbers. {1, 2, 3,…} Represented by or

Venn Diagram of the Complex Numbers Irrational Numbers Rational Numbers Complex Numbers Imaginary NumbersReal Numbers Integers Whole Numbers Natural Numbers

Example Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that apply. a.117 b.0 c … 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 Multiplication Commutative a+b = b+a ab = ba Associative(a+b)+c = a+(b+c) (ab)c = a(bc) Identitya + 0 = aa1 = a Inversea + (-a) = 0 a = 1 Opposite Reciprocal Distributive Property a(b + c) = ab + ac

Examples of Properties Name the property displayed: (x – 5) = (-2 + x) – 5 2. (-2) ( -½ ) = (4 – 5) = (4 – 5)2 4. x(y – w) = xy – xw

Order of Operations 1.Parenthesis/Grouping Symbols 2.Exponents 3.Multiplication and Division – left to right 4.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 Operations Evaluate the following: (12 + 8) ÷ (2 + 11) ÷ 5