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Real Numbers
  • Irrational decimal expansion
  • Properties of rational numbers
  • Converting non-terminating recurring decimals to rational numbers
  • Real numbers
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Real Numbers
  • Laws of logarithms
  • Common and natural logarithms
  • Polynomials
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Introduction:All the numbers including whole numbers, integers, fractions and decimals can be written in the form of


Rational numbers: A rational number is a number which can be written in the form of  where both p and q are integers and .

They are a bigger collection than integers as there can be many rational numbers between two integers. All rational numbers can be written either in the form of terminating decimals or non-terminating repeating decimals.


Fundamental theorem of Arithmetic: Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.


1.     In general, given a composite number x, written in product of primes as  where  are primes and written in ascending order,
 If we use the same primes, we will get powers of primes.


Once we have decided that the order will be ascending, then the way the number is factorized, is unique.




Decimal representation of Rational numbers: Every rational number when written in decimal form is either a terminating decimal or a non-terminating repeating decimal.

Terminating decimal: A terminating decimal is a decimal that contains finite number of digits.


In the above examples, we can observe that the denominators of the rational numbers don’t have any other prime factors except 2 or 5 or both. Hence at some stage a division of numerator by 2 or 5 the remainder is zero and we get a terminating decimal.

Description: If a Rational number, in its standard form has no other prime factor except 2 or 5 it can be expressed as a terminating decimal.

Non-terminating repeating decimals: If a rational number in its standard form has prime factors other than 2 or 5 or in addition to 2 and 5, the division does not end. During the process of division we get a digit or a group of digits recurring in the same order. Such decimals are called non terminating repeating decimals.

We draw a line segment over the recurring part to indicate the non-terminating nature.



    = 1.3636……=


Observe the following decimals as rationals.

Now,    (i)




It is observed that the decimal expression expressed in simplest rational form, the denominator is having only powers of 2 or 5, or both 2 and 5.

From above examples, the conclusion is:

Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form of  where p and q are coprimes and the prime factorization of q is of the form  where n, m are non-negative integers.

Description: Let  be a rational number, such that the prime factorization of q is of the form  where n, m are non-negative integers. Then x has a decimal expansion which terminates. This is known as converse of fundamental theorem of arithmetic.

Observe the following decimals as rationals.

Description: i) The recurring part of the Non – terminating decimal is called period.

Example: Period = 3

ii) Number of digits in the period is called periodicity.


Period = 142857

Periodicity = 6.

From above example, the conclusion is: Let  be a rational number, such that the prime factorization of q is not of the form  where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).

Example: Without actual division, state whether the following rational numbers are terminating or nonterminating repeating decimals.

(i)                         (ii)                        (iii)


(i)  is terminating decimal.

(ii)  is non-terminating, repeating decimal as the denominator is not of the form

(iii)  is non-terminating, repeating decimal as the denominator is not of the form

Example:Write the decimal expansion of the following rational numbers without acutal division.

(i)                            (ii)                           (iii)





Irrational numbers:A number which cannot be written in the form of  where p and q are integers and  (or) a decimal number which is neither terminating nor repeating is called irrational number.

Example: etc.

Before proving … etc.. are irrational using fundamental theorem of arithmetic. There is a need to learn another theorem which is used in the proof.

Theorem: Let p be a prime number. If p divides a2, where a is a positive integer. Then p divides a.

Proof:Let the positive integer be a.

Prime factorization of  where are prime numbers.


Given p divides a2 p is one of the prime factors of a2.

The only prime factor of a2is

  If p divides , then p is one of    

Since p is one of   p divides a

Example: Verify the statement above for p = 2 and a2 = 64

Clearly 64 is even number which is divisible by 2

8 is also divisible by 2 which is even number.

  The theorem above is verified.

Now, by using the above theorem it is easy to prove  are irrational. For this contradiction technique is used.

Example:Prove that  is irrational.

Solution:Let  is not an irrational

5 divides a2   5 divides a ……..(2) (Theorem)

From (1) , (3)

5 divides b25 divides b …… (4) (Theorem)

From (2), (4)

5 divides a and b.

But a and b are co – primes (by assumption)

It is contradiction to our assumption.

 is not a rational

is an irrational. 

Example: Show that  is irrational.

Solution: Let  is not an irrational

Here a, b are rational then

a rational is not equal to irrational

our assumption is wrong.

 is an irrational.


i)               The sum of the two irrational numbers need not be irrational.
Example: If
, then both x and y are irrational, but  which is rational.

ii)             The product of two irrational numbers need not be irrational.
, then both x and y are irrational, but  which is rational.

Surd:An irrational root of a rational number is called a surd.

General form of a surd:  is called a surd of order n, where a is positive rational number, n is a positive integer greater than 1 and  is not a rational number.

Example: i)  are surds of order 2.

               ii) , are surds of order 3.

Operations on surds:

Addition and subtraction:Similar surds can be added or subtracted. Addition and subtraction can be done using distributive law. i.e.,



Multiplication of surds:

i)               Surds of the same order can be multiplied as

ii)             Surds of different order can be multiplied by reducing them to the same order.




Division of surds: Apply same procedure as in case of multiplication of surds.





Description: order of radicals  are 4, 6 and 6 respectively we note that  and the order of is 2. Thus the order of surd is not a property of the surd itself, but of the way in which it is expressed.


i)              If  is a surd, the ‘a’ is called radicand and the symbol  is called radical sign.

ii)             are not surds as they are not irrational numbers.

iii)          In a surd the radicand should always be a rational number. So  and  are roots of an irrational number, hence cannot called surds.

iv)            All surds are irrational numbers, whereas all irrational numbers are not surds.
 is irrational but not a surd.

Real numbers:A number whose square is non-negative, is called a real number.

In fact, all rational and  irrational numbers form the collection of real numbers.

Every real number is either rational or irrational.

Consider a real number.

i)               If it is an integer or it has a terminating or repeating decimal representation then it is rational.

ii)             If it has a either-terminating nor-repeating decimal representation then it is irrational.

 Rational and irrational number together form the collection of all real numbers


In the given number line, 4/10 is to the left of 5/10, 6/10,7/10 etc. Thus.

The above implies that 5/10 lines between 4/10 and 6/10. Consider two rational numbers 5/10 and 6/10. We can find a rational number between them. For example,

Again, between 5/10 and 11/20, we can find another rational number. For example,

and the process continues. Thus, we can find a rational number between any two rational numbers however close they may be end, hence, infinite rational numbers lies between two rational numbers. This property of rational numbers explains that rational numbers and present ever, y where on the number line.