CBSE Class 10 Mathematics Real Numbers Topic Real Numbers Topics discussed in this document: Introduction Rational numbers Fundamental theorem of Arithmetic Decimal representation of Rational numbers Terminating decimal Non-terminating repeating decimals Irrational numbers Surd General form of a surd Operations on surds · Addition and subtraction · Multiplication of surds More Topics under Class 10th Real Numbers (CBSE): Real numbers Laws of logarithms Common and natural logarithms Visit Edvie.com for more topics

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REAL
NUMBERS
Class 10 Notes
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,
Once we have decided that the order
will be ascending, then the way the
number is factorized, is unique.
Example:
Example: ​
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.
Example:
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.
If a Rational number, in its
standard form has no other prime

2.
i.e., ​ If we use the
same primes, we will get powers of
primes.
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.
Example:
∴​ ​ = 1.3636……=
Similarly
Observe the following decimals as rationals.
Now, (i)
(ii)
(iii)

3.
(iv)
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 ​ be a rational number whose decimal expansion terminates. Then ​x​ can be
expressed in the form of ​ where ​ and ​ are coprimes and the prime factorization
of ​ is of the form ​ where ​ , ​ are non-negative integers.
​Let ​ be a rational number, such that the prime factorization of ​q​ is of
the form ​ where ​ , ​ are non-negative integers. Then ​ has a decimal
expansion which terminates. This is known as converse of fundamental theorem of
arithmetic.
Observe the following decimals as rationals.
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.
Example:

4.
⇒​ Period = 142857
⇒ ​Periodicity = 6.
From above example, the conclusion is: Let ​ be a rational number, such that the
prime factorization of ​ is not of the form ​ where ​ , ​ are non-negative integers.
Then, ​ 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)
Solution:
(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)
Solution:
(i)
(ii)

5.
(iii)
Irrational numbers:​ ​A number which cannot be written in the form of ​ where ​
and ​ 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 ​ be a prime number. If ​ divides ​ , where ​ is a positive integer. Then
divides ​ .
Proof:​ ​Let the positive integer be ​
Prime factorization of ​ where ​are prime
numbers.
Therefore
Given ​ divides ​ ​⇒​ ​ is one of the prime factors of ​ .
The only prime factor of ​ is
∴​ If ​ divides ​, then ​ is one of
Since ​ is one of ​ ​⇒​ ​ divides ​
Example: ​Verify the statement above for ​ = 2 and ​ = 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.

6.
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 ​ 2 ​
​⇒​ 5 divides ​ ……..(2) (Theorem)
∴​
From (1) , (3)
∴​ 5 divides ​ ⇒ ​5 divides ​ …… (4) (Theorem)
From (2), (4)
5 divides ​ and ​ .
But ​ and ​ 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 ​ are rational then

7.
∴​ 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 ​ and ​ are irrational, but
which is rational.
ii) The product of two irrational numbers need not be irrational.
Example: ​, then both ​ and ​ 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 ​ , where ​ is positive rational
number, ​ 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.,
Example:​ ​i)
ii)
Multiplication of surds:
i) Surds of the same order can be multiplied as

8.
ii) Surds of different order can be multiplied by reducing them to the same
order.
Example:
i)
ii)
Division of surds: ​Apply same procedure as in case of multiplication of surds.
Example:
i)
ii)
iii)
The 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 ‘​ ’ is called radicand and the symbol ​ is called
radical sign.
ii) are not surds as they are not irrational numbers.

9.
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.
Example: ​ 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,

10.
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, ​ where on the number line.
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