Real Numbers Class-10th Board

Real Numbers Class-10th Board: The concept of real numbers is an integral part of mathematics that students must master to build a strong foundation for higher studies. Real numbers encompass various types of numbers such as natural numbers, whole numbers, integers, rational, and irrational numbers, all of which form the real number system. In this post, we will cover the fundamental concepts, properties, theorems, and examples related to real numbers as per the Class 10 curriculum.

Vidoe Tutorial of Real Numbers Class-10th Board:

Class 10th BSEB Board PYQs Solution and Answer Key

Types of Real Numbers Class-10th Board

Natural Numbers:

Natural numbers are the set of positive integers that start from 1 and go on infinitely. They are often referred to as “counting numbers. Generally, It is represented by N.

Set Representation:

N = {1, 2, 3, 4, 5, 6, 7, 8, …}

Properties:

• The smallest natural number is 1.
• They are infinite and continue endlessly.
• Natural numbers do not include zero (0), negative numbers, fractions, or decimals.

Whole Numbers:

Whole numbers are the set of all non-negative integers. This set includes all natural numbers along with zero.

Generally, It is represented by W.

Set Representation:

W = {0, 1, 2, 3, 4, 5, 6, 7, 8, …}

Properties:

• The smallest whole number is 0.
• Whole numbers do not include negative numbers, fractions, or decimals.
• Like natural numbers, the set of whole numbers is infinite and continues endlessly.

Integers:

Integers are the set of all whole numbers and their negatives. This includes positive numbers, zero, and negative numbers without fractions or decimals. Generally, It is denoted by Z.

Set Representation:

Z = {…, -2, -1, 0, 1, 2, …}

Properties:

• Integers include both positive and negative numbers, along with zero.
• They do not include fractions or decimals.
• The set of integers is infinite in both directions (positive and negative).

Rational Numbers:

A rational numbers is any numbers that can be represented in the form of p/q format. where, p and q are integers and q should not be equal to zero. It is generally denoted by Q.

Rational numbers include both positive and negative fractions, whole numbers, and integers, as they can all be expressed as fractions.

Set Representation:

Q = { p/q | p,q ∈ Z and q ≠ 0 }

Irrational Numbers:

A irrational numbers is any numbers that can not be represented in the form of p/q format. where, p and q are integers and q should not be equal to zero. It is generally denoted by I.

Their decimal expansions are non-terminating and non-repeating.

Euclid’s Division Lemma in Real Numbers Class-10th Board

Statement: For any two positive integers a and b, there exist unique integers q and r such that:

a = bq + r, 0 ≤ r < b

Here, is the dividend, b is the divisor, q is the quotient, and r is the remainder.

Application:

Euclid’s division lemma is used to find the Highest Common Factor (HCF) of two numbers.

Example:

Find the HCF of 455 and 42 using Euclid’s division lemma.

1. Divide 455 by 42 => 455 = 42 * 10 + 35
2. Divide 42 by 35 => 42 = 35 * 1 + 7
3. Divide 35 by 7 => 35 = 7 * 5 + 0

Hence, the remainder is 0, and the HCF is 7.

Fundamental Theorem of Arithmetic in Real Numbers Class-10th Board

Statement: Every composite number can be expressed (or factorized) as a product of primes, and this factorization is unique, except for the order of the prime factors.

Application:

This theorem is used for finding the LCM and HCF of two numbers using their prime factorization.

Example:

Factorize 140.

140 = 2² * 5 * 7

HCF and LCM in Real Numbers Class-10th Board

Relation between HCF and LCM:

HCF * LCM = Product of the two numbers

Example:

Step-01: Prime Factorization:

12 = 2² * 3, 18 = 2 * 3²

Step-02: HCF = Product of smallest powers of common prime factors:

HCF = 2¹ * 3¹ = 6

Step-03: LCM = Prodcut of highest powers of all prime factors:

LCM = 2² * 3² = 36

Verification: HCF * LCM = 6 * 36 = 216 and 12 * 18 = 216

Decimal Expansion of Rational and Irrational Numbers in Real Numbers Class-10th Board

Terminating Decimal: A rational number has a terminating decimal expansion if the denominator (in its lowest form) has only powers of 2 or 5.

Non-Terminating, Repeating Decimal: A rational number has a non-terminating, repeating decimal expansion if the denominator (in its lowest form) has prime factors other than 2 or 5.

Irrational Numbers: Have non-terminating, non-repeating decimal expansions.

Example:

1/8 = 0.125 (terminating decimal).

1/3 = 0.3333… (non-terminating repeating)

π = 3.1415926535. . . (non-terminating, non-repeating)

Revisiting Rational Numbers and Their Decimal Expansions in Real Numbers Class-10th Board

Rational Numbers: Can be expressed in the form p/q, where p and q are integers, and q not equal to 0.

Irrational Numbers: Cannot be expressed as p/q, and their decimal expansion does not terminte or repeat.

Important Formulas in Real Numbers Class-10th Board

• Euclid’s Division Lemma: a = bq + r
• HCF * LCM = Product of two numbers

Prove that root(5) is irrational

Solution:

Step-01: Assume root(5) is a rational number.

So, root(5) = p/q, where p and q are co-prime integers, and q not equal to 0.

step-02: Squaring both sides:

5 = p² / q²

p² = 5q²

This means p² is divisble by 5, so p must be divisible by 5. Let p = 5k for some interger k.

step-03: Substituting p = 5k into the equation:

(5k)² = 5q²

25 k² = 5q²

q² = 5k²

This implies q² is divisible by 5, so q is also divisible by 5.

step-04: Since both p and q are divisible by 5, they have a common factor of 5, which contradicts the assumption that p and q are co-prime.

Hence, root(5) is irrational.

Use Euclid’s division algorithm to find the HCF of 56 and 72.

Solution: Using Euclid’s division algorithm:

step-01: Divide 72 by 56:

72 = 56 * 1 + 16

Here, the remainder is 16

step-02: Now, divide 56 by 16:

56 = 16 * 3 + 8

The remainder is 8

step-03: Next, divide 16 by 8:

16 = 8 * 2 + 0

The remainder is 0, so the division process stops here.

Hence, the HCF of 56 and 72 is 8.

Find the LCM and HCF of 120 and 144 by prime factorization.

Solution:

Step-01: Prime factorization of 120:

120 = 2³ * 3 * 5

step-02: Prime factorization of 144:

144 = 2⁴ * 3²

step-03: Finding the HCF:

The HCF is the product of the smallest powers of all common prime factors.

HCF = 2³ * 3 = 24

step-04: Finding the LCM:

The LCM is the product of the highest power of all prime factors.

LCM = 2⁴ * 3² * 5 = 720

Thus, the HCF of 120 and 144 is 24, and the LCM of 120 and 144 is 720.

Explain why 13/25 has a terminating decimal expansion

Solution:

A rational number has a terminating decimal expansion if its denominator (in its lowest form) has only the prime factors 2 and/or 5.

step-01: The given number is 13/25.

step-02: The prime factorization of the denomenator 25 is:

25 = 5²

since the denomenator contains only the prime factor 5, the decimal expansion of 13/25 will be terminating.

thus, 13/25 = 0.52 is a terminating decimal.

Show that 7/22 has a non-terminating repeating decimal expansion

Solution:

A rational number has a non-terminating repeating decimal expansion if its denominator (in its lowest form) has prime factors other than 2 or 5.

step-01: The given number is 7/22.

step-02: The prime factorization of 22 is:

22 = 2 * 11

since the denomintor has a prime factor 11 (which is not 2 or 5), the decimal expansion of 7/22 will be non-terminating and repeating.

step-03: Dividing 7 by 22:

7/22 = 0.318181831818183181818. . .

the repeating block is 3181818.

Thus, 7/22 has a non-terminating repeating decimal expansion.

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