The properties of rational numbers, Square and square roots, cube and cube roots- A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.Since q may be equal to 1, every integer is a rational number.

1. MATHEMATICS
PRESENTED BY,
AANYA. R
VIII ‘A’
PSSEMR School,
Tholahunse,
Davangere
Submitted to:Girish sir

2. Properties of
Rational numbers

3.  The word rational has evolved from the word
ratio.
 In general, rational numbers are those
numbers that can be expressed in the form of
p/q, in which both p and q are integers and
q≠0.
 The properties of rational numbers are:

4.  Closure Property
 Commutative Property
 Associative Property
 Distributive Property
 Identity Property
 Inverse Property

5. Closure property
For two rational numbers say x and y the results of addition,
subtraction and multiplication operations give a rational number.
We can say that rational numbers are closed under addition,
subtraction and multiplication.
For example:
•(7/6)+(2/5) = 47/30
•(5/6) – (1/3) = 1/2
•(2/5) (3/7) = 6/35

6. Commutative Property
For rational numbers, addition and multiplication are
commutative.
 Commutative law of addition: a+b = b+a
For example:
2+3=3+2=5
 Commutative law of multiplication: a×b = b×a
For example:

7. Associative Property
Rational numbers follow the associative property for
addition and multiplication.
For example: 1/2 + (1/4 + 2/3) = (1/2 + 1/4) + 2/3
⇒ 17/12 = 17/12
And in case of multiplication;
1/2 x (1/4 x 2/3) = (1/2 x 1/4) x 2/3
⇒ 2/24 = 2/24
⇒1/12 = 1/12

8. Distributive Property
The distributive property states, if a, b and c are three
rational numbers, then;
a x (b+c) = (a x b) + (a x c)
Example:
1/2 x (1/2 + 1/4) = (1/2 x 1/2) + (1/2 x 1/4)
LHS = 1/2 x (1/2 + 1/4) = 3/8
RHS = (1/2 x 1/2) + (1/2 x 1/4) = 3/8
Hence, proved

9. Identity Property
0 is an additive identity and 1 is a multiplicative
identity for rational numbers.
Examples:
•1/2 + 0 = 1/2 [Additive Identity]
•1/2 x 1 = 1/2 [Multiplicative Identity]

10. Inverse Property
For a rational number x/y, the additive inverse is -x/y and y/x is
the multiplicative inverse.
Examples:
 The additive inverse of 1/3 is -1/3.
Hence, 1/3 + (-1/3) = 0
 The multiplicative inverse of 1/3 is 3.
Hence, 1/3 x 3 = 1

11. Properties of
Square and square roots

12.  Squares of even numbers are even and squares of odd
numbers are odd.
Example 1 : Even numbers
22² = 484
86² = 7396
Example 2 : Odd numbers
81² = 6561
1001² = 1002001

13.  The numbers that have 0, 1, 4, 5, 6 or 9 in their units
place maybe perfect squares where as the numbers that
have 2, 3, 7 or 8 in their units place are never perfect
squares.
For example:
12²=144 18²=324
13²=169 19²=361
14²=196
15²=225
16²=256
17²=289

14. If a number ends with odd number of zero’s ,it is not
a perfect square
For example:10²=10x10=100
20²=20x20=400
 These examples shows the number of zero’s in the
squares of numbers is always even.

15.  Square of a natural number can be
expressed as a sum of consecutive odd
number
For example:2²=4=1+3
3²=9=1+3+5
4²=16=1+3+5+7 (sum of
consecutive odd number

16. The difference between the square of two
consecutive natural numbers is equal to the sum
of the two number
For example:
2²-1²= 4-1=3 1+2=3
3²-2²= 9-4=5 2+3+5
4²-3²=16-9=7 3+4=7

17.  The square of an odd numbers can be expressed as the sum of two consecutive natural numbers

18. Properties of
Cube and cube roots

19. The cube of an even number will
always be an even number.
Example : 83 = 512, 123 = 1728, etc.
The cube of odd number will always be
an odd number.
Example : 73 = 343, 193 = 6589, etc.