REAL NUMBER SYSTEM

Number Systems Real Rational (fraction) Irrational Integer Whole Natural

Real Number System

Real Numbers Real numbers consist of all the rational and irrational numbers. The real number system has many subsets: – Natural Numbers – Whole Numbers – Integers

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

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

Integers Integers are the set of whole numbers and their opposites. {…,-3, -2, -1, 0, 1, 2, 3,…}

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.

Terminating Decimals Terminating decimals are decimals that contain a finite number of digits. Examples:  36.8   4.5

Repeating Decimals Repeating decimals are decimals that contain a infinite number of digits. Examples:  0.333…   … The line above the decimals indicate that number repeats.

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) –

Set of Real Numbers Rational Numbers –the set of all numbers that can be expressed as a quotient of integers ie. In p/q form where q is not equal to zero Irrational Numbers – the set of all numbers that correspond to points on the number line but that are not rational numbers. That is, an irrational number is a number that cannot be expressed as a quotient of integers. Real Numbers – is the set of all numbers each of which correspond to a point on the number line.

Other Vocabulary Associated with the Real Number System …(ellipsis)—continues without end { } (set)—a collection of objects or numbers. Sets are notated by using braces { }. Finite—having bounds; limited Infinite—having no boundaries or limits Venn diagram—a diagram consisting of circles or squares to show relationships of a set of data.

Venn Diagram of the Real Number System Irrational NumbersRational 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. g.-3

To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn diagram. Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers …

Solution 117 is a natural number, a whole number, an integer, and a rational number. is a rational number. 0 is a whole number, an integer, and a rational number … is an irrational number. -3 is an integer and a rational number is a rational number. is an irrational number. is a rational number.

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.

Example 2-1b Name all of the sets of numbers to which each real number belongs. a. 31 b. c d. e. Answer:natural number, whole number, integer, rational number Answer:integer, rational number Answer:rational number Answer:natural number, whole number, integer, rational number Answer:irrational number The Real Number System (rational and irrational)

Commutative Properties Changing the order of the numbers in addition or multiplication will not change the result. Commutative Property of Addition states: = or a + b = b + a. Commutative Property of Multiplication states: 4 5 = 5 4 or ab = ba.

Associative Properties Changing the grouping of the numbers in addition or multiplication will not change the result. Associative Property of Addition states: 3 + (4 + 5)= (3 + 4)+ 5 or a + (b + c)= (a + b)+ c Associative Property of Multiplication states: (2 3) 4 = 2 (3 4) or (ab)c = a(bc)

Distributive Property Multiplication distributes over addition.

Additive Identity Property There exists a unique number 0 such that zero preserves identities under addition. a + 0 = a and 0 + a = a In other words adding zero to a number does not change its value.

Multiplicative Identity Property There exists a unique number 1 such that the number 1 preserves identities under multiplication. a ∙ 1 = a and 1 ∙ a = a In other words multiplying a number by 1 does not change the value of the number.

Additive Inverse Property For each real number a there exists a unique real number –a such that their sum is zero. a + (-a) = 0 In other words opposites add to zero.

Multiplicative Inverse Property For each real number a there exists a unique real number 1/a such that their product is 1.

State the property or properties that justify the following = Commutative Property

State the property or properties that justify the following. 10(1/10) = 1 Multiplicative Inverse Property

State the property or properties that justify the following. 3(x – 10) = 3x – 30 Distributive Property

State the property or properties that justify the following. 3 + (4 + 5) = (3 + 4) + 5 Associative Property

State the property or properties that justify the following. (5 + 2) + 9 = (2 + 5) + 9 Commutative Property

3 + 7 = Commutative Property of Addition

8 + 0 = 8 Identity Property of Addition

6 4 = 4 6 Commutative Property of Multiplication

17 + (-17) = 0 Inverse Property of Addition

(2 + 1) + 4 = 2 + (1 + 4) Associative Property of Addition

even + even = even Closure Property

5 1 / = 5 1 / 7 Identity Property of Addition

3 / 4 – 6 / 7 = – 6 / / 4 Commutative Property of Addition

1 2 / 5 1 = 1 2 / 5 Identity Property of Multiplication

(fraction)(fraction) = fraction Closure Property

[(- 2 / 3 )(5)]9 = - 2 / 3 [(5)(9)] Associative Property of Multiplication

6(3 – 2n) = 18 – 12n Distributive Property

2x + 3 = 3 + 2x Commutative Property of Addition

ab = ba Commutative Property of Multiplication

a + 0 = a Identity Property of Addition

a(bc) = (ab)c Associative Property of Multiplication

a1 = a Identity Property of Multiplication

a +b = b + a Commutative Property of Addition

a(b + c) = ab + ac Distributive Property

a + (b + c) = (a +b) + c Associative Property of Addition

a + (-a) = 0 Inverse Property of Addition