Revision Notes: Rational and Irrational Numbers | Mathematics Class 9 ICSE PDF Download

Rational Numbers 

• A rational number is a number that can be expressed in the form a/b where  
• a and b are integers, and b ≠ 0. The set of rational number is denoted by Q. Thus 
  Q = {a/b = a, b ∈ Z, b ≠ 0}
• Between two rational numbers, there always exists infinite rational numbers.
• Every integer (positive, negative or Zero) and every decimal number is a rational number. 
• All terminating and non-terminating but repeating (or periodic or recurring) decimals are rational numbers.  
• Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2ⁿ5^m, where n, m are non-negative integers. Then, x has a decimal expansion which terminates. 
• Every rational number can be expressed either as a terminating decimal or as a recurring decimal.

Irrational Numbers 

• The real numbers, which are not rational, are called irrational numbers.
• The square roots, cube roots etc. of natural numbers are irrational numbers; if their exact value can not be obtained. 
• All non-terminating non-repeating decimals are irrational numbers. 
• π is an irrational number. 
• If m is not a perfect square, the √m is irrational.
• If a and b are two positive numbers such that ab is not a perfect square, then 
  (a) A rational number between a and b is (a+b)/2
  (b)  An irrational number between a and b is √ab

More about Irrational Numbers 

• For any two positive rational numbers x and y,
  
  If √x and √y are irrational numbers, then,
  If √x and √y are irrational numbers, then,
•  If a + b√x = c + d√x ⇒ a = c and b = d 
• The negative of an irrational number is always irrational. 
• The sum of a rational and an irrational number is always irrational.
• The product of a non-zero rational number and an irrational number is always irrational. 
• The sum, difference, product and quotient of two irrational numbers may not be an irrational number. 

Real Numbers 

• The set of the union of rational and irrational numbers forms a set of real numbers. Thus a real number can be associated with a point on the number line. 
  
  

Surds 

• If x is a positive rational number and n is a positive integer such that is irrational, then x^1/n is called a surd or a radical.
• Every surd is an irrational number, but every irrational number is not a surd. π is an irrational number but it is not a surd. 
• When two irrational numbers are multiplied, their product is called a rational number. This process of removing the radical sign from the denominator is called rationalization and the two irrational numbers are called rationalizing factors of each other. 
• The irrational numbers  are called conjugates of each other.
• A number of the form  is called a surd of the order n where a is a positive rational number which is not equal to the n^th power of a rational number and n is a natural number greater than or equal to 2.