Operations on Real Numbers
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The sum, difference and the product of two rational numbers is always a rational number. The quotient of a division of one rational number by a non-zero rational number is a rational number. Rational numbers satisfy the closure law under addition, subtraction, multiplication and division. They also satisfy the commutative law and associative law under addition and multiplication.

The sum, difference, multiplication and division of irrational numbers are not always irrational. Irrational numbers do not satisfy the closure property under addition, subtraction, multiplication and division.
Real numbers satisfy the commutative, associative and distributive laws. These can be stated as:

Commutative law of addition: a + b = b + a

Commutative law of multiplication: a × b = b × a

Associative law of addition: a + (b + c) = (a + b) + c

Associative law of multiplication: a × (b × c) = (a × b) × c

Distributive law: a × (b + c) = (a × b) + (a × c) or (a + b) × c = (a × c) + (b × c)

Real numbers can be represented on the number line. The square root of any positive real number exists and that also can be represented on number line.

The sum or difference of a rational number and an irrational number is an irrational number.
The product or division of a rational number with an irrational number is an irrational number.

Some of the basic identities involving square roots are:
If a, b, c and d are positive real numbers,

ab = a b 

a b  = a b

( a  + b )( a  b ) = a – b

(a + b )(a – b ) = a2 – b

( a + b )( c d ) = ac ad + bc + bd

( a + b )2 = a + 2 ab + b

If the product of two irrational numbers is a rational number, then each of the irrational numbers is called the rationalising factor of the other. The process of multiplying an irrational number with its rationalising factor to get the product as a rational number is called rationalisation of the given irrational number.

Summary

The sum, difference and the product of two rational numbers is always a rational number. The quotient of a division of one rational number by a non-zero rational number is a rational number. Rational numbers satisfy the closure law under addition, subtraction, multiplication and division. They also satisfy the commutative law and associative law under addition and multiplication.

The sum, difference, multiplication and division of irrational numbers are not always irrational. Irrational numbers do not satisfy the closure property under addition, subtraction, multiplication and division.
Real numbers satisfy the commutative, associative and distributive laws. These can be stated as:

Commutative law of addition: a + b = b + a

Commutative law of multiplication: a × b = b × a

Associative law of addition: a + (b + c) = (a + b) + c

Associative law of multiplication: a × (b × c) = (a × b) × c

Distributive law: a × (b + c) = (a × b) + (a × c) or (a + b) × c = (a × c) + (b × c)

Real numbers can be represented on the number line. The square root of any positive real number exists and that also can be represented on number line.

The sum or difference of a rational number and an irrational number is an irrational number.
The product or division of a rational number with an irrational number is an irrational number.

Some of the basic identities involving square roots are:
If a, b, c and d are positive real numbers,

ab = a b 

a b  = a b

( a  + b )( a  b ) = a – b

(a + b )(a – b ) = a2 – b

( a + b )( c d ) = ac ad + bc + bd

( a + b )2 = a + 2 ab + b

If the product of two irrational numbers is a rational number, then each of the irrational numbers is called the rationalising factor of the other. The process of multiplying an irrational number with its rationalising factor to get the product as a rational number is called rationalisation of the given irrational number.

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