DIGITAL LIBRARY OF GLT SARAWATI BAL MANDIR, NEHRU NAGAR, BY HITESH MEENA CLASS 9th - A

1. Introduction to Real NumbersIntroduction to Real Numbers
andand
Their PropertiesTheir Properties

2. Two Kinds of Real Numbers
• Rational Numbers
• Irrational Numbers

3. Rational Numbers
• A rational number is a real
number that can be written
as a ratio of two integers.
• A rational number written in
decimal form is terminating
or repeating.

4. Examples of Rational
Numbers
•16
•1/2
•3.56
•-8
•1.3333…
•- 3/4

5. Irrational Numbers
• An irrational number is a
number that cannot be
written as a ratio of two
integers.
• Irrational numbers written as
decimals are non-terminating
and non-repeating.

6. Examples of Irrational
Numbers
• Square roots of
non-perfect
“squares”
• Pi
17

7. What are integers?
• Integers are the
whole numbers and
their opposites.
• Examples of integers
are
6
-12
0
186
-934

8. Using Exponents
 If “a” is a real number and “n” is a natural number, then an
=
a•a•a•••a•a (n factors of a).
where n is the exponent, a is the base, and an
is an
exponential expression. Exponents are also called powers.
To find the value of a whole number exponent:
100
= 1, 20
= 1, 80
= 1, #0
= 1
101
= 10, 21
= 2, 81
= 8, #1
= #
102
= 10 x 10 = 100, 22
= 2 x 2 = 4, 82
= 8 x 8 = 64
103
= 10 x 10 x 10 = 1000, 23
= 2 x 2 x 2 = 8
104
= 10 x 10 x 10 x 10 = 10,000 24
= 2 x 2 x 2 x 2 = 16
(-10)3
= (-10)(-10)(-10) (12).5
=

9. Using the Identity Properties
“additive identity”
 Zero is the only number that can be added to
any number to get that number.
0 is called the “identity element for addition”
a + 0 = a Example 1: 4 + 0 = 4
“multiplicative identity”
 One is the only number that can be multiplied
by any number to get that number.
1 is called the “identity element for
multiplication”
a • 1 = a Example 2: 4 • 1 = 4

10. The Real Number SystemThe Real Number SystemReal Numbers
Rational Numbers Irrational Numbers
3
1/2
-2
15%
2/3
1.456
-
0.7
0
√3 2
π
−√5
2
3π
4

11. The Real NumberThe Real Number
SystemSystem Real Numbers
Rational Numbers Irrational Numbers
31/2 -2
15
%
2/3
1.45
6
-
0.7
0
√3 2
π
−√5
2
3π
4
Integers

12. The Real Number System
Real Numbers
Rational Numbers Irrational Numbers
31/2
-2
15
%
2/3
1.45
6
-
0.7
0
√3 2
π
−√5
2
3π
4
Integers
Whole

13. The Real Number System
Real Numbers
Rational Numbers Irrational Numbers
3
1/2
-2
15
%
2/3
1.45
6
-
0.7
0
√3 2
π
−√5
2
3π
4
Integers
Whole
Natural

14. Finding Additive inversesFinding Additive inverses
 For any real number x, the number –x is theFor any real number x, the number –x is the
additive inverse of x.additive inverse of x.
Example 1:Example 1:
Number
Inverse
Additive
6 - 6
- 4 4
- 8.7 8.7
0 0
2
3
2
3
−

15. Symbol Meaning Example
= is equal to 4 = 4
≠ is not equal to 4 ≠ 5
< is less than 4 < 5
≤ is less than or equal -4 ≤ -3
> is greater than -4 > -5
≥ is greater than or equal -8 ≥ - 10

16. Number Reciprocal or Inverse Additive Inverse
−6 6
0.05 20 -0.05
0 none 0
2
5
−
5
2
−
2
5
1
6
−
11
7
7
11
7
11
−