# Operations on Real Numbers

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In this section we will discuss operations on real numbers.

The rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication Moreover, if we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number (that is, rational numbers are ‘closed’ with respect to addition, subtraction, multiplication and division). It turns out that irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication. However, the sum, difference, quotients and products of irrational numbers are not always irrational.
√5 + (- √5) = 0
√3 - √3= 0
(√2)(√2)= 2
(√7)/( √7)= 1
All are rational numbers.

When we add and multiply a rational number with an irrational number.

For example, √6 is an irrational number so when we add or subtract any rational number to an irrational number the result will be irrational number only.

Note : The numbers with same radicals are called like terms. Only like terms can be added or subtracted. In like terms only numbers before the radicals is added or subtracted and the radical will remain as it is.

Example :
3√2 and 5√2 ---------> Like terms.
-√7 and 3√7 -------------> Like terms.
2√3 and √5 --------------> Unlike terms.
2√11 and 5√6 ------------> Unlike terms.

Operations on real numbers

1) Add the 2√2 + 5√3 and √2 - 3√3

Solution :
2√2 + 5√3 + √2 + (- 3√3)
= 2√2 + 5√3 + √2- 3√3
= 3 √2 + 2√3
-------------------------------------------------------------
2) Add the -6√3 + 3√2 and -2√2 – 4 √3

Solution :
-6√3 + 3√2 + (-2√2 – 4 √3)
= - 6√3 + 3 √2 - 2√2 - 4√3
= -10√3 + 1√2
-------------------------------------------------------------
3) Multiply (2√3) and (-3√5)

Solution :
2 √3 x (-3√5)
= (2 x -3) √3 x √5
= - 6 √(3x5)
= -6 √15
-------------------------------------------------------------
4) Multiply 3√2 ( 2 + 4√3)

Solution :
3√2 ( 2 + 4√3)
= 3√2 x 2 + 3√2 x 4√3 [ use a distributive law]
= (3x2)√2 + (3x4)( √2 x √3)
= 6√2 + 12√6
-------------------------------------------------------------
5) Divide √15 by √3

Solution :
√15 / √3
= (√3 x √5) / √3
= √5

Real-Numbers

Real Numbers
Representation of real-numbers on number line
Operations on Real Numbers
Rationalization of denominator

From real- number to number system