1. CHAPTER 1 – REAL NUMBERS

2. Learning Objectives:
Describe the set of real numbers, all its subsets and their relationships.
Identify and use the arithmetic properties of subsets of integers, rational, irrational and real numbers, including closure properties for the four basic arithmetic operations where applicable.

3. The Natural Numbers
The numbers 1, 2, 3, 4, 5, … are called the natural numbers (counting numbers).
The group of all natural numbers is denoted by N.
The numbers 1, 3, 5, 7, … are odd natural numbers whereas the numbers 2, 4, 6, 8, … are the even natural numbers.

4. Whole numbers
The collection of all natural numbers along with zero is known as the set of whole numbers.
They are 0, 1, 2, 3, ….
It is denoted by W.

5. Integers
The set of all integers is denoted by Z. The set of integers include the natural numbers, the number zero and the negatives of the natural numbers.
Thus, they are
…, -3, -2, -1, 0, 1, 2, 3, ….

6. Integers 1,2,3 … are called positive integers and
-1, -2, -3,… are called negative integers.
0 (zero) is neither positive nor negative.
There is no largest or smallest integers.

7. Rational Numbers The numbers which can be expressed in the form
where p and q are integers and q≠ 0,
are called rational numbers.
For eg,
-3, 0, 8 etc. are rational numbers.

8. Rational Numbers Any terminating decimal number is a rational number.
Eg: 0.34, , etc.
Any non-terminating recurring (repeating) decimal is a rational number.
Eg: ….
…..

9. Irrational Numbers The numbers which cannot be expressed in the form
where p and q are integers and q≠ 0,
are called irrational numbers.
For eg,
are irrational numbers.

10. Irrational Numbers
Any non-terminating non-recurring (non-repeating) decimal is an irrational number.
Eg: ….
…..

11. Real Numbers
Real numbers consist of both rational and irrational numbers.
The set of all real numbers is denoted by R.

12. Number Line
A real number line is a straight line, each point of which represents each real number, preserving the ordering.

13. Operations on Rational Numbers
1) Addition

14. Operations on Rational Numbers
2) Subtraction

15. Operations on Rational Numbers
3) Multiplication

16. Operations on Rational Numbers
4) Division

17. SUMMARY

18. SUMMARY

19. Classify the following numbers as rational and irrational