How many properties in rational numbers in 8th class? explain with examples.
Answers
Step-by-step explanation:
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:
Closure Property
Commutative Property
Associative Property
Distributive Property
Identity Property
Inverse Property
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
Commutative Property
For rational numbers, addition and multiplication are commutative.
Commutative law of addition: a+b = b+a
Commutative law of multiplication: a×b = b×a
For example:
Commutative law example
Subtraction is not commutative
Associative Property
Rational numbers follow the associative property for addition and multiplication.
Suppose x, y and z are rational, then for addition: x+(y+z)=(x+y)+z
For multiplication: x(yz)=(xy)z.
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
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
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]
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
Answer:
Properties of Rational Numbers
By: DHRUVAN
Step-by-step explanation:
Closure Property
For two rational numbers x, y the addition, subtraction, multiplication results always yield a rational number. The Closure Property isn’t applicable for the division as division by zero isn’t defined. In other words, we can say that closure property is applicable for division too other than zero.
4/7 + 2/3 =26/21
4/3 – 2/4 = 6/12
3/5. 2/3 = 6/15
Commutative Property
Considering two rational numbers x, y the addition and multiplication are always commutative. Subtraction doesn’t obey commutative property. You can get a clear idea of this property by having a look at the solved examples.
Commutative Law of Addition: x+y = y+x Ex: 1/3+2/3 = 3/3
Commutative Law of Multiplication: x.y = y.x Ex: 1/2.2/3 =2/3.1/2 =2/6
Subtraction x-y≠y-x Ex: 4/3-1/3 = 3/3 whereas 1/3-4/3=-3/3
Division isn’t commutative x/y ≠y/x Ex: 3/9÷1/2=6/9 whereas 1/2 ÷3/9 =9/6
Associative Property
Rational Numbers obey the Associative Property for Addition and Multiplication. Let us assume x, y, z to be three rational numbers then for Addition, x+(y+z)=(x+y)+z
whereas for Multiplication x(yz)=(xy)z
Ex: 1/3 + (1/4 + 3/3) = (1/3+ 1/4) + 3/3
⇒19/12 =19/12
Distributive Property
Let us consider three rational numbers x, y, z then x . (y+z) = (x . y) + (x . z). We will prove the property by considering an example.
Ex: 1/3.(1/4+2/5) =(1/3.1/4)+(1/3.2/5)
1/3.(17/20)= 1/12+2/10
17/60 =17/60
Thus, L.H.S = R.H.S
Identity and Inverse Properties of Rational Numbers
Identity Property: We know 0 is called Additive Identity and 1 is called Multiplicative Identity of Rational Numbers.
Ex: 1/4+0 = 1/4(Additive Identity)
5/3.1 = 5/3(Multiplicative Identity)
Inverse Property: For a Rational Number x/y additive inverse is -x/y and multiplicative inverse is y/x.
Ex: Additive Inverse of 2/3 is -2/3
Multiplicative Inverse of 4/5 is 5/4
There are few other properties that you need to be aware of Rational Numbers and they are explained below.