# Properties Of Rational Numbers

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```					           Properties Of Rational Numbers
Properties Of Rational Numbers

We will discuss different properties of rational numbers in this session. Rational
numbers are the numbers which can be expressed in the form of p/q, where p and q
are the integers and q in not equal to zero.

Here we will take the properties of rational numbers:

1. Closure property: We mean by closure property that if there are two rational
numbers, then

Closure property of addition holds true, which means that the sum of two rational
numbers is also a rational number.

Closure property of subtraction holds true, which means that if there exist two rational
numbers, then the difference of the two rational numbers is also a rational number.

Closure property of multiplication holds true, which means that if there exist two
Know More About Left And Right Hand Limits

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rational numbers, then the product of the two rational numbers is also a rational
number.

Closure property of division holds true, which means that if there exist two rational
numbers, then the quotient of the two rational numbers is also a rational number.

2. Commutative property of rational number: Commutative property of rational
numbers holds true for addition and multiplication but does not hold true for
subtraction and division.

It means that if p1/q1 and p2/q2 are any two rational numbers, then according to
commutative property of rational numbers, we mean that :

P1/q1 + p2/q2 = p2/q2 + p1/q1

P1/q1 * p2/q2 = p2/q2 * p1/q1

P1/q1 - p2/q2 <> p2/q2 - p1/q1

P1/q1 ÷ p2/q2 <> p2/q2 ÷ p1/q1

3. Additive Identity of Rational numbers: According to additive identity property, If we
have a rational number p/q, then there exist a number zero (0),

such that if we add the number zero to any number, the result remains unchanged.
So we write it as : p/q + 0 = p/q

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4. Multiplicative identity of Rational numbers: According to multiplicative identity
property of rational numbers, If we have a rational number p/q, then there exist a
number one (1), such that if we multiply the number one to any number, the result
remains unchanged. So we write it as : p/q * 1 = p/q

5. Power of zero: By the property Power of zero, we mean that there exists a number
zero, such that if we multiply zero to any rational number, then the product id zero
itself. So if we have p/q as a rational number, then we say: p/q * 0 = 0 .

6. Associative property of Rational numbers: Associative property of rational numbers
holds true for addition and multiplication but does not hold true for subtraction and
division. It means that if p1/q1 , p2/q2 and p3/q3 are any three rational numbers, then
according to associative property of rational numbers, we mean that :

(P1/q1 + p2/q2) + p3/q3 = P1/q1 + (p2/q2 + p3/q3 )

(P1/q1 * p2/q2) * p3/q3 = P1/q1 * (p2/q2 * p3/q3 )

(P1/q1 - p2/q2) - p3/q3 <> P1/q1 - (p2/q2 - p3/q3 )

(P1/q1 ÷ p2/q2) ÷ p3/q3 <> P1/q1 ÷ (p2/q2 ÷ p3/q3 )

7.    Distributive property of multiplication over addition and subtraction of rational
numbers holds true, which states:

P1/q1 * ( p2/q2 + p3/q3) = (p1/q1 * p2/q2) + (p1/q1 * p3/q3)

P1/q1 * ( p2/q2 - p3/q3) = (p1/q1 * p2/q2) - (p1/q1 * p3/q3)

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Introduction to Rational numbers

Today, I will tell you a story. Once there was a family of Natural numbers where all counting
numbers used to live. One day a guest named zero visited the house and requested for a
permission to stay there.

All were happy; they requested the eldest member of the family Mr. infinite (∞) to grant the
permission for 0. The permission was granted and the name of the house was changed to
house of Whole numbers.

Now, after some time negative numbers also visited the house and requested for the
permission to be the part of the family. They were permitted and now the family became the
family of Integers i.e. -∞ . ……..-3,-2,-1,0,1,2,3,…….∞. On seeing the family living together,
some numbers which were in form of p/q, where p and q are natural numbers also asked for a
permission to stay there. They were called fractions.

Some fractions are 4/7, 2/5 …etc. The family of fractions also told that if you all see the
denominator with you, which is not usually visible, then you will also become the part of
fractions family. All the numbers started trying it and realized that they all are the part of
fractions. But this was not true for negative integers.

The meeting was held, in which it was decided that a name Rational number will be given to
the family. A family of Rational Numbers consists of all the numbers which can be expressed
in form of p/q, where p and q are integers, but q≠ 0.

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