Engr. Mexieca M. Fidel

THE REAL NUMBER SYSTEM

WRITE SETS USING SET NOTATION

A set is a collection of objects called the elements or members of the set. Set braces { } are usually used to enclose the elements. In Algebra, the elements of a set are usually numbers.

• Example 1: 3 is an element of the set {1,2,3} Note: This is referred to as a Finite Set since we can count the elements of the set.

• Example 2: N= {1,2,3,4,…} is referred to as a Natural Numbers or Counting Numbers Set.

• Example 3: W= {0,1,2,3,4,…} is referred to as a Whole Number Set.

A set is a collection of objects called the elements or members of the set. Set braces { } are usually used to enclose the elements.

• Example 4: A set containing no numbers is shown as { } Note: This is referred to as the Null Set or Empty Set. Caution: Do not write the {0} set as the null set. This set contains one element, the number 0.

• Example 5: To show that 3 “is a element of” the set {1,2,3}, use the notation: 3 {1,2,3}. Note: This is also true: 3 N

• Example 6: 0 N where is read as “is not an element of”

WRITE SETS USING SET NOTATION

Two sets are equal if they contain exactly the same elements. (Order doesn’t matter)

• Example 1: {1,12} = {12,1} • Example 2: {0,1,3} {0,2,3}

WRITE SETS USING SET NOTATION

In Algebra, letters called variables are often used to represent numbers or to define sets of numbers. (x or y). The notation {x|x has property P}is an example of “Set Builder Notation” and is read as:

{x x has property P} 

the set of all elements x such that x has a property P

• Example 1: {x|x is a whole number less than 6} Solution: {0,1,2,3,4,5}

• Example 2: {x|x is a natural number greater than 12} Solution: {13,14,15,…}

1-1 Using a number line

One way to visualize a set a numbers is to use a “Number Line”.

• Example 1: The set of numbers shown above includes positive numbers, negative numbers and 0. This set is part of the set of “Integers” and is written:

I = {…, -2, -1, 0, 1, 2, …}

-2 -1 0 1 2 3 4 5

1-1 Using a number line

Each number on a number line is called the coordinate of the point that it labels, while the point is the graph of the number.

• Example 1: The fractions shown above are examples of rational numbers. A rational number is one than can be expressed as the quotient of two integers, with the denominator not 0.

-2 -1 0 1 2 3 4 5

coordinate

Graph of -1

o12

114

o o

1-1 Using a number line

Decimal numbers that neither terminate nor repeat are called “irrational numbers”.

• Example 1: Many square roots are irrational numbers, however some square roots are rational.

• Irrational: Rational:

2 7 4 16

-2 -1 0 1 2 3 4 5

coordinate

Graph of -1

o12

114

o o2

7

4 16 o o oo o

Circumferencediameter

OPERATIONS INVOLVING SETS

REAL NUMBERS (R)Definition:

REAL NUMBERS (R)- Set of all rational and irrational numbers.

SUBSETS of RDefinition:

RATIONAL NUMBERS (Q)- numbers that can be expressed as a quotient a/b, where a and b are integers.- terminating or repeating decimals- Ex: {1/2, 55/230, -205/39}

SUBSETS of RDefinition:

INTEGERS (Z)- numbers that consist of positive integers, negative integers, and zero,- {…, -2, -1, 0, 1, 2 ,…}

SUBSETS of RDefinition:

NATURAL NUMBERS (N)- counting numbers- positive integers- {1, 2, 3, 4, ….}

SUBSETS of RDefinition:

WHOLE NUMBERS (W)- nonnegative integers- { 0 } {1, 2, 3, 4, ….}- {0, 1, 2, 3, 4, …}

SUBSETS of RDefinition:

IRRATIONAL NUMBERS (Q´)- non-terminating and non-repeating decimals- transcendental numbers- Ex: {pi, sqrt 2, -1.436512…..}

The Set of Real Numbers

Q

Q‘(Irrational Numbers)Q(Rational Numbers)

Z(Integers)

W(whole numbers)

N(Natural numbers)

PROPERTIES of RDefinition:

CLOSURE PROPERTYGiven real numbers a and b,

Then, a + b is a real number (+), or a x b is a real number (x).

PROPERTIES of RExample 1:

12 + 3 is a real number. Therefore, the set of reals is CLOSED with respect to addition.

PROPERTIES of RExample 2:

12 x 4.2 is a real number. Therefore, the set of reals is CLOSED with respect to multiplication.

PROPERTIES of RDefinition:

COMMUTATIVE PROPERTYGiven real numbers a and b,Addition: a + b = b + aMultiplication: ab = ba

PROPERTIES of RExample 3:Addition:

2.3 + 1.2 = 1.2 + 2.3Multiplication:

(2)(3.5) = (3.5)(2)

PROPERTIES of RDefinition:

ASSOCIATIVE PROPERTYGiven real numbers a, b and c,

Addition: (a + b) + c = a + (b + c)Multiplication: (ab)c = a(bc)

PROPERTIES of RExample 4:Addition:

(6 + 0.5) + ¼ = 6 + (0.5 + ¼) Multiplication:

(9 x 3) x 4 = 9 x (3 x 4)

PROPERTIES of RDefinition:

DISTRIBUTIVE PROPERTY of MULTIPLICATION OVER ADDITIONGiven real numbers a, b and c,

a (b + c) = ab + ac

PROPERTIES of RExample 5:4.3 (0.11 + 3.02) = (4.3)(0.11) + (4.3)(3.02)

Example 6:2x (3x – b) = (2x)(3x) + (2x)(-b)

PROPERTIES of RDefinition:

IDENTITY PROPERTYGiven a real number a,Addition: 0 + a = a Multiplication: 1 x a = a

PROPERTIES of RExample 7:Addition:

0 + (-1.342) = -1.342 Multiplication:

(1)(0.1234) = 0.1234

PROPERTIES of RDefinition:

INVERSE PROPERTYGiven a real number a,Addition: a + (-a) = 0 Multiplication: a x (1/a) = 1

PROPERTIES of RExample 8:Addition:

1.342 + (-1.342) = 0 Multiplication:

(0.1234)(1/0.1234) = 1

EXERCISESTell which of the properties of real numbers justifies each of the following statements.1. (2)(3) + (2)(5) = 2 (3 + 5)2. (10 + 5) + 3 = 10 + (5 + 3)3. (2)(10) + (3)(10) = (2 + 3)(10)4. (10)(4)(10) = (4)(10)(10)5. 10 + (4 + 10) = 10 + (10 + 4)6. 10[(4)(10)] = [(4)(10)]107. [(4)(10)]10 = 4[(10)(10)]8. 3 + 0.33 is a real number

TRUE OR FALSE 1. The set of WHOLE numbers is closed with respect to multiplication.

TRUE OR FALSE2. The set of NATURAL numbers is closed with respect to multiplication.

TRUE OR FALSE3. The product of any two REAL numbers is a REAL number.

TRUE OR FALSE4. The quotient of any two REAL numbers is a REAL number.

TRUE OR FALSE5. Except for 0, the set of RATIONAL numbers is closed under division.

TRUE OR FALSE6. Except for 0, the set of RATIONAL numbers contains

the multiplicative inverse for each of its members.

TRUE OR FALSE7. The set of RATIONAL numbers is associative under multiplication.

TRUE OR FALSE8. The set of RATIONAL numbers contains the additive inverse for each of its members.

TRUE OR FALSE9. The set of INTEGERS is commutative under subtraction.

TRUE OR FALSE10. The set of INTEGERS is closed with respect to division.