Real Numbers Class 10 Notes CBSE Maths Chapter 1 (Free PDF Download)

Real Numbers:

• All rational and irrational numbers taken together make the real numbers. On the number line, any real number can be plotted.

Euclid’s Division Lemma: (Not available in the current syllabus)

• A lemma is a verified statement that is used to prove another. Euclid’s Division Lemma states that for any two integers $\text{a}$ and $\text{b}$, there exists a unique pair of integers $\text{q}$ and $\text{r}$ such that a = b x q + r where 0 ≤ r < b.

• The lemma can be simply stated as : Dividend = Divisor x Quotient + Remainder

• For any pair of dividend and divisor, the quotient and remainder obtained are going to be unique.

Euclid’s Division Algorithm:

An algorithm is a set of well-defined steps that describe how to solve a certain problem. The Highest Common Factor (HCF) of two positive integers is computed using Euclid's division algorithm. 

Follow the steps below to find the HCF of two positive integers, say c and d with c > d: 

Step 1: We apply Euclid’s Division Lemma to find two integers q and r such that c = d x q + r where 0 ≤ r < d.

Step 2: If r = 0, the H.C.F is d, else, we apply Euclid’s division Lemma to d (the divisor) and r (the remainder) to get another pair of quotient and remainder.

Step 3: Repeat Steps 1–3 until the remainder is zero. The needed HCF will be the divisor at the last step.

The Fundamental Theorem of Arithmetic:

The process of expressing a natural number as a product of prime numbers is known as prime factorization.

Apart from the sequence in which the prime components occur, the prime factorisation for a given number is unique.

Example: 12=2 x 2 x 3, here 12 is represented as a product of its prime factors 2 and 3.

Finding LCM and HCF:

• HCF is the product of the smallest power of each common prime factor in the given numbers.

• LCM is the product of the greatest power of each prime factor, involved in the given numbers.

• For any two positive integers $\text{a}$ and $\text{b}$, \[\text{HCF }\left( \text{a, b} \right)\text{  }\!\!\times\!\!\text{  LCM }\left( \text{a, b} \right)\text{ = a  }\!\!\times\!\!\text{  b}\]

• L.C.M can be used to find common occurrence sites. For instance, the time when two people running at different speeds meet, or the ringing of bells with various frequencies.

Rational and Irrational Numbers:

• If a number can be expressed in the form $\text{p/q}$ where \[\text{p}\] and \[\text{q}\] are integers and $\text{q }\ne \text{ 0}$, then it is called a rational number.

• If a number cannot be expressed in the form $\text{p/q}$ where \[\text{p}\] and \[\text{q}\] are integers and $\text{q }\ne \text{ 0}$, then it is called an irrational number.

Number Theory:

• If \[\text{p}\] (a prime number) divides \[{{\text{a}}^{\text{2}}}\], then \[\text{p}\] divides $\text{a}$ as well. For example, $3$ divides \[{{6}^{2}}\], resulting in \[36\], implying that $3$ divides $6$.

• The sum or difference of a rational and an irrational number is irrational

• A non-zero rational and irrational number's product and quotient are both irrational.

• \[\sqrt{p}\] is irrational when \[\text{p}\] is a prime number. For example, $7$ is a prime number and $\sqrt{7}$ is irrational. The preceding statement can be proven by the process of “Proof by contradiction”.

Decimal Expansions of Rational Numbers:

• Let \[\text{x = }\frac{\text{p}}{\text{q}}\] be a rational number with the prime factorization \[{{\text{2}}^{\text{n}}}{{\text{5}}^{\text{m}}}\], where $\text{n}$ and $\text{m}$ are non-negative integers. The decimal expansion of \[\text{x}\] then comes to an end. Then \[\text{x}\] has a non-terminating repeating decimal expansion (recurring).

• If $\frac{\text{a}}{\text{b}}$ is a rational number, then its decimal expansion would terminate if both of the following conditions are satisfied :

1. The H.C.F of $\text{a}$ and $\text{b}$ is $1$.

2. \[\text{b}\] can be expressed as a prime factorisation of $2$ and $5$ i.e in the form \[{{\text{2}}^{\text{n}}}{{\text{5}}^{\text{m}}}\] where either $\text{m}$ or $\text{n}$, or both can be zero.

• If the prime factorisation of \[\text{b}\] contains any number other than $2$ or $5$, then the decimal expansion of that number will be recurring

Class 10 notes of real numbers are given here in detail, go through all the concepts given below for better understanding.

Introduction to Real Numbers

Real Numbers

Real numbers consist of the union of all the rational and irrational numbers, and any of the real numbers can be plotted on the number line for representation.

CBSE Class 10 Math Revision Notes

• Revision Notes For CBSE Class 10 Math Chapter 1 Real Numbers

• Revision Notes For CBSE Class 10 Math Chapter 2 Polynomials

• Revision Notes For CBSE Class 10 Math Chapter 3 Pair of Linear Equations In Two Variables

• Revision Notes For CBSE Class 10 Math Chapter 4 Quadratic Equations

• Revision Notes  For CBSE Class 10 Math Chapter 5 Arithmetic Progression

• Revision Notes For CBSE Class 10 Math Chapter 6 Triangles

• Revision Notes For CBSE Class 10 Math Chapter 7 Coordinate Geometry

• Revision Notes For CBSE Class 10 Math Chapter 8 Introduction To Trigonometry

• Revision Notes For CBSE Class 10 Math Chapter 9 Some Application of Trigonometry

• Revision Notes For CBSE Class 10 Math Chapter 10 Circles

• Revision Notes For CBSE Class 10 Math Chapter 11 Constructions

• Revision Notes For CBSE Class 10 Math Chapter 12 Area Related To Circles

• Revision Notes For CBSE Class 10 Math Chapter 13 Surface Area And Volume

• Revision Notes For CBSE Class 10 Math Chapter 14 Statistics

• Revision Notes For CBSE Class 10 Math Chapter 15 Probability

Euclid’s Division Lemma (Not available in the current syllabus)

• According to Euclid’s Division Lemma statement, the given two integers a and b, there exists a unique pair of integers q and r, such that a = b × q + r and 0 ≤ r < b.

• The above lemma is equivalent to dividend = divisor × quotient + remainder.

• For all the given pair of dividend and divisor, the obtained quotient and remainder is always going to be unique.

Euclid’s Division Algorithm

• This method is very important to find the HCF of two numbers, consider two integers a and b where a > b.

• We apply Euclid’s Division Lemma to find two integers q and r such that a = b × q + r and 0 ≤ r < b.

• If the remainder is equal to 0, the H.C.F is b, else, we can apply Euclid’s Division Lemma to b (the divisor) and r (the remainder) to get a different pair of quotient and remainder.

• This method is repeated until a remainder of zero is obtained and the divisor in that step is the H.C.F of the given set of numbers.

Method of Finding The Fundamental Theorem of Arithmetic

Prime Factorisation

• This method is used to express a natural number as a product of prime numbers.

• Example: 36 = 2 × 2 × 3 × 3 is the prime factorisation of 36.

Fundamental Theorem of Arithmetic

• This theorem states, the prime factorisation of a given number is always unique if the arrangement of the prime factors is ignored.

• Example: 36 = 2 × 2 × 3 × 3 or, 36 = 2 × 3 × 2 × 3.

• In the above example, 36 is represented as a product of prime factors (2s and 3s) in which the arrangement of the factor doesn’t matter.

Method of Finding LCM

Example: To find the LCM of 36 and 56,

• 36 = 2 × 2 × 3 × 3
  56 = 2 × 2 × 2 × 7

• The common prime factors are 2 × 2.

• The uncommon prime factors are 3 × 3 for 36 and 2 × 7 for 56.

• LCM of 36 and 56 = 2 × 2 × 3 × 3 × 2 × 7 which is 504.

Method of Finding HCF

There are two methods to find the HCF of the number, which are Prime factorisation and Euclid’s division algorithm.

Prime Factorisation:

• The two numbers are given, where both of them can be expressed as products of their respective prime factors, later, we can select the prime factors that are common to both the numbers

• Ex – To find the H.C.F of the two numbers 20 and 24.
  20 = 2 × 2 × 5 and 24 = 2 × 2 × 2 × 3.

• The factor common to 20 and 24 is 2 × 2, which is 4, which in turn is the H.C.F of 20 and 24.

Euclid’s Division Algorithm: (Not available in the current syllabus)

• It is the repeated use of Euclid’s division lemma to find the H.C.F of two numbers.

Product of Two Numbers = HCF × LCM of the Two Numbers.

• For any two positive integers a and b,

a × b = H.C.F × L.C.M.

• For example, consider 36 and 56, the H.C.F is 4 and the L.C.M is 504

36 × 56 = 2016

4 × 504 = 2016

Thus, solving both you get the same answer 36 × 56 = 4 × 504.

• This relationship is not true for 3 or more numbers.

Applications of HCF & LCM in Real-World Problems

L.C.M is mainly used to find the point of the common occurrence of two or more numbers.

For example, the ringing of bells that ring with different frequencies and to find the time where two persons are running at different speeds, meet, and so on.

Revisiting Irrational Numbers

Irrational Numbers

Any number which cannot be expressed in the form of $ \dfrac{p}{q} $ (where p and q are integers and q ≠ 0.) is an irrational number, some of the examples: √2, π, e and so on.

Number Theory: Interesting Results

• If a number p (a prime number) divides a2, then p divides a, example for this type of theory is: 3 divides 62 i.e 36, which implies that 3 divides 6.

• The sum or difference between a rational and an irrational number is irrational.

• The product and quotient of a non-zero rational number and irrational number always result in irrational numbers.

• √p is irrational when ‘p’ is a prime, for example, you can consider, 7 is a prime number and √7 is irrational, this statement can be proven by the method of “Proof by contradiction”.

Proof by Contradiction

In the method of contradiction, check whether the given statement is true. 

(i)  First we have to assume that the given statement is TRUE.

(ii) Then solving, we arrive at some point that contradicts our assumption, thereby proving the contrary.

Eg: Prove that $ \sqrt{7} $ is irrational.

Assumption: $ \sqrt{7} $ is rational.

Since it is rational $ \sqrt{7} $ can be expressed as:

$ \sqrt{7} = \dfrac{a}{b} $, in this a and b are coprime integers, such that $b \neq 0$.

On squaring, $ \dfrac{a^2}{b^2} = 7 $

Hence, 7 divides a, and then, there exists a number c such that $a = 7c$. Then, $a^2 = 49 c^2$. Hence, $7 b^2 = 49 c^2$ or $b^2 = 7 c^2$.

Hence 7 divides b, since 7 is a common factor for both a and b, it contradicts our assumption that a and b are coprime integers.

Hence, our initial assumption is wrong that $ \sqrt{7} $ is a rational number and therefore, $ \sqrt{7} $ is an irrational number.

Revisiting Rational Numbers and Their Decimal Expansions (Not available in the current syllabus)

Rational Numbers

The real numbers can be written in the form of p/q, where p and q are integers and q is not equal to zero, example of a rational number is  -1/2, 4/5, 1, 0, -3, and so on.

Terminating and Nonterminating Decimals

• Terminating decimals are kind of decimals having a certain point of end, for example, 0.2, 2.56, and so on.

• Non-terminating decimals are a kind of decimals where the decimals don’t have any end so it is known as non-terminating decimals, for example, 0.333333….., 0.13135235343…

Non-terminating decimals can be:

a) Recurring –  same part of the decimals keep on repeating  (0.142857142857….)

b) Non-Recurring – No repetition of part of decimal takes place, an example is = 3.1415926535…

How to check if the given rational number is terminating or not?

Consider, if a/b is a rational number, then its decimal expansion would terminate if the below-given conditions are satisfied:

a) The H.C.F of two integers a and b is 1.

b) b can be expressed as a prime factorisation of 2 and 5 i.e b = 2m × 5n where either m or n, or both can = 0.

Consider, if the prime factorisation of ‘b’ has any other number rather than 2 or 5, then the decimal expansion of that number will be recurring.

Example: 

• 1/40 = 0.025 is a terminating decimal, as the Highest Common Factor of two number 1 and 40 is 1, and the denominator (40) can be expressed in the form of  23 × 51.

• 3/7 = 0.428571 is a recurring decimal as the Highest Common Factor of 3 and 7 is 1 and the denominator of (7) is equal to71.

Benefits of Notes of Real Numbers Class 1

General Tips

• While revising, go through each and every point and concept related to the chapter carefully in notes of Class 10 maths chapter because consider it as an easy topic some of the important points will be missed.

• Do revise the Class 10 Maths Ch 1 Notes Introduction and the important topics because some of the basic concepts of this chapter might repeat in further Classes.

Conclusion

Real Numbers Class 10 Notes is one of the best study materials for the preparation of the final examination. Students can make use of different Notes of Class 10 Maths Chapter 1 to make their learning more convenient and saves their time. Vedantu is one of the best platforms for the students, which has different notes on real number Class 10  available free of cost for their academics.

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