Learning Outcomes: Define natural numbers, whole number, integers, prime numbers, rational numbers and irrational numbers. (a) (b) Represent rational and irrational numbers in decimal form. Represent the relationship of number sets in a real number system diagrammatically showing and . (c) At the end of the lesson, students should be able to Represent open, closed and half-open intervals and their representations on the number line. (d) Find union and intersection of two or more intervals with the aids of number line. (e) 1.1 Real Numbers SM015 - C 01 Page 1
• Positive Numbers used for counting. • Natural Numbers, Example 1 (a) Circle the Natural Numbers from the following: • Zero and Natural numbers • Whole Numbers, Example 1 (b) Circle the Whole Numbers from the following: • Zero, Natural numbers & their negatives. • • Integers, Example 1 (c) Circle the Integers from the following: Classification of Real Numbers - I SM015 - C 01 Page 2
Numbers that can be represented in the form of where and . • (a) terminating decimals E.g. • The decimal representation can be either (a) non-terminating & repeating decimals E.g. Rational Numbers, Example 1 (d) Circle the Rational Numbers from the following: • Numbers that are NOT rational numbers. The decimal representation is non-terminating & non-repeating decimals. • Irrational Numbers, E.g. Example 1 (e) Circle the Irrational Numbers from the following: Classification of Real Numbers - II SM015 - C 01 Page 3
• The union of rational numbers and irrational numbers Real Numbers, Example 1 (f) Circle the Real Numbers from the following: • Natural numbers that have exactly 2 distinct natural number divisors Prime Numbers Example 1 (g) Circle the Prime Numbers from the following: • Integers that are divisible by 2. Even Numbers Example 1 (h) Circle the Even Numbers from the following: • Integers that are NOT even numbers. Odd Numbers Example 1 (i) Circle the Odd Numbers from the following: Classification of Real Numbers - III SM015 - C 01 Page 4
Real numbers can be represented geometrically by points on a straight line, called as the real number line. The real numbers on the number line are ordered in increasing magnitude from the left to the right. All sets of real numbers can be written in the form of either finite intervals [open , closed, & half-open] or infinite intervals: Inequality Interval Notation Set Notation Number Line Closed Interval Open Interval Halfopen Interval Infinite Interval Real Number Line SM015 - C 01 Page 5
Intersection The intersection of two sets and denoted by , is the set of all elements which belongs to both and . E.g. Given and . Union The union of two sets and denoted by , is the set of all elements which belongs to either or . E.g. Given and . Intersection & Union of Intervals SM015 - C 01 Page 6
By using number lines, find each of the following: (a) Solutions: (b) Solutions: (c) Solutions: Example 2 [PowerPoint Notes Lecture 1 Example 3] SM015 - C 01 Page 7