Class 9 – Surds
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Introduction: Consider the following decimals, 0.75, 0.216, 0. , . Each of these decimals can be written in the form of i.e., Rational number equal to a repeating decimal where Finally every terminating or nonterminating repeating decimal can be expressed as a rational number. The set of rational numbers is not adequate to find the solutions of all polynomial equations. 
For example, the equation cannot be solved in set of rational numbers (Q) because the number whose square is 2, written as , is not a rational number. To prove that there is no rational number whose square is 2: If possible, let be the rational number whose square is 2. , where and p is an even integer because the square of an even integer is even. Let p = 2m, where . …..(i) Again, q is an even integer. Let where . ….(ii) From (i) and (ii) p and q have common factor 2. This contradicts the fact that p and q have no common factor. Hence, our assumption was wrong. Hence, there is no rational number whose square is 2, i.e., is not a rational number. 

Irrational numbers: The numbers which cannot be written as form where are called irrational numbers.
The set of irrational numbers are denoted by .
Example:
(i) irrationals.
(ii) An irrational number is a decimal expansion that is neither terminating nor repeating.
A few irrational numbers are:
0.313113111…………………
8.121221222……………………
(iii) The square roots of 2, 3, 5, 6, 7 and 8 are all irrational numbers.
1. For a rational number and irrational number the sum, difference, product and quotient will be an irrational number.
2. For two irrational numbers, the sum, difference product and quotient need not be irrational.
Find irrational numbers between two given numbers:
Example: Write three irrational numbers between 0.3 and 0.4.
Solution: We have to write a pattern which is neither terminating nor repeating decimals.
The three irrational numbers can be:
0.313113111……
0.323223222……..
0.343443444……..
Example: Write an irrational number between:
a) b) 5 and 7
Solution:
a) A real number between
So, an irrational number between
b)
The square root of any integer between 25 and 49, (except 36) is an irrational number.
So, we can take any one of the irrational numbers.
.
Between two irrationals there exist infinite number of irrationals is called a density property.
Finding a rational or irrational number between two irrational numbers:
Example: Find a rational number between and .
Solution: Write a rational number between 5 and 7 which is a perfect square.
For example, 5.76, 5.8081, 5.8564…,6.25,….. are some perfect squares lying between 5 and 7
Now 5 < 5.76 < 7
So, a rational number between
Real Number: The collection of rational numbers and irrational numbers is called set of real numbers. If Q is the set of rational numbers and is the set of irrational numbers then is the set of real numbers.
So , and every real number is either rational or irrational.
The set R of real numbers is the biggest set of numbers that you have learned till now. The section of the set of real numbers are shown below.
Square root by division method: In this section, we will learn to find out the square roots of numbers by division method. This method is also called the dot or comma or bar method.
(i) Finding square root of a perfect square number.
Example: Find the square root of the number 6084.
Example: Find the square root of 15129.
Solution:
(ii) Finding square roots of numbers which are not perfect squares:
Representation of in decimal form: Let us find by the division method. It willbe found that it is nonrepeating and nonterminating decimal.
Hence,
(iii) Finding square roots of decimals:
Example: Find square root of 0.549081
Solution: Process (1) Leaving the first decimal place to the right (you know that the place right to it is tenth’s place start with 100th place put a dot on it, place dot leaving digit after digit, do as was done earlier.
To find the least numberto be added or subtracted to make a given number as a perfect square.
Example: Find the least number to be subtracted from 2921 to make it a perfect square.
Solution:
From the long division method, we notice that remainder is 5.
_ The least number tobe subtracted from 2921 to make it a perfect square is ‘5’.
Example:Find the least number to be added to 15120 to make it a perfect square.
Solution:
From the long division method, we notice that (and )
_ The least number tobe added to make it a perfect square is ‘9’.