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CBSE Maths Chapter 1 Real Numbers Formulas for Class 10

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Last updated date: 23rd Jul 2024
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Real Numbers Formulas for Class 10 CBSE Maths

The world of numbers is vast and fascinating, and real numbers are the foundation upon which much of mathematics is built. Real numbers are a mix of rational and irrational numbers. Real Numbers Formulas for Class 10 play a crucial role in various important concepts. Clear understanding of real numbers is essential before diving into these formulas. This article presents a short list of formulas with examples to aid students in learning and memorizing them.


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CBSE Maths Chapter 1 List of Real Numbers Formulas for Class 10

CBSE Class 10 Maths Chapter 1 all formulas on Real Numbers cover fundamental concepts in number theory. The chapter includes important formulas related to prime numbers, Euclid's division lemma, and the Fundamental Theorem of Arithmetic. Students will explore these key mathematical principles to build a strong foundation in the understanding of real numbers.


S No

Type of Real Numbers

Description

1

Natural Numbers

N = {1,2,3,4,5  >

It is the counting numbers

2

Whole Numbers

W= {0,1,2,3,4,5>

It is the counting numbers + zero

3

Integers

All whole numbers, including negative numbers + Positive numbers like -4,-3,-2,-1,0,1,2,3,4,5… so on.


Like whole numbers, integers don’t include fractions or decimals.

4

Positive Integers

Z+ = 1,2,3,4,5, ……

5

Negative Integers

Z– = -1,-2,-3,-4,-5, ……

6

Rational Numbers

A number is called rational if it can be expressed in the form of p/q where p and q are integers (q> 0).

For example, P/q, ⅘.

7

Irrational Number

A number is called irrational if it cannot be expressed in the form of p/q where p and q are integers (q> 0).

For example, √3.

8

Prime Numbers Rule

If 'x' is a prime number and 'x' divides $a^2$, then 'x' also divides 'a', where 'x' is a positive integer.

9

Real Numbers

A real number is a number that can be found on the number line. Real Numbers are the numbers that we normally use and apply in real-world applications.

Real numbers include Natural Numbers, Integers, Whole Numbers, Rational Numbers, Fractions, and Irrational Numbers.



Properties of Real Number Formula for Class 10:


S No

Properties of Real Numbers

Description

1

Closure

Addition, subtraction, multiplication, and division of real numbers (except division by zero) yield real numbers.

2

Commutative

Addition and multiplication are commutative:

$a + b = b + a$

$a \times b = b \times a$

3

Associative

Addition and multiplication are associative:

$(a + b) + c = a + (b + c)$

$(a \times b) \times c = a \times (b \times c)$

4

Distributive

Multiplication over addition is distributive 

$a \times (b + c) = a \times b + a \times c$

5

Identity

Addition has an identity element (0: a + 0 = a) and multiplication has an identity element (1: a $\times$ 1 = a)

6

Inverse

Every non-zero real number has an additive inverse (-a) and a multiplicative inverse (1/a)



Other Important Real Number Formula for Class 10:


S.No

Other Formula of Real Numbers

Description

1

Euclid’s Division Lemma

If you have two positive numbers, say a and b, there are whole numbers m and n such that 

$a=bm+n, 0 \leq n < b$

2

Euclid’s Division Algorithm

To find the Highest Common Factor (HCF) of two positive numbers 'a' and 'b', where a is greater than b:

  1. Use Euclid’s Division Lemma to find m and n such that $a=bm+n, 0 \leq n <b$

  2. If n = 0, then the HCF is 'b'. If $n \neq 0$, apply Euclid’s Lemma to 'b' and 'n'.

  3. Repeat this process until the remainder is 0. The divisor at this point is the HCF of (a, b), and the HCF of (a, b) is also the HCF of (b, n).

3

HCF (Highest Common Factor)

The largest integer that is a factor of both a and b. It can be found using Euclid's algorithm or prime factorization.

Formula: HCF(a, b) = Product of smallest powers of common factors in prime factorizations of a and b.

4

LCM (Least Common Multiple)

The smallest positive integer that is a multiple of both a and b.

Formula: LCM(a, b) = Product of highest powers of each prime factor involved in a and b.

5

Important relation of HCF and LCM

HCF(a, b) $\times$ LCM(a, b) = a $\times$ b (product of two numbers equals their HCF times their LCM)



Applications of Real Numbers Formulas for Class 10

Class 10 Maths chapter 1 all formulas on Real numbers play a crucial role in various aspects of everyday life, not just academics. Here are some key applications of real number formulas learned in Class 10:


Mathematics:

  • Arithmetic: Formulas for HCF (Highest Common Factor) and LCM (Least Common Multiple) help solve problems involving fractions, ratios, and simplifying expressions.

  • Algebra: Understanding properties of real numbers like commutativity, associativity, and distributivity aids in simplifying equations, manipulating expressions, and solving complex problems.

  • Geometry: Formulas for areas, volumes, distances, and slopes involve real numbers and their operations, essential for various geometric calculations.

  • Trigonometry: Understanding sine, cosine, tangent, and other trigonometric functions relies heavily on real numbers and their properties.


Real-world Applications:

  • Physics: Real numbers are the foundation for representing physical quantities like temperature, mass, velocity, acceleration, and energy. Formulas are used in calculations related to motion, forces, work, power, and other physical phenomena.

  • Chemistry: Chemical formulas, concentrations, reaction rates, and stoichiometry calculations all involve real numbers and their manipulation.

  • Economics and Finance: Real numbers are vital for calculations involving percentages, interest rates, ratios, profits, losses, and other financial metrics.

  • Engineering and Technology: Engineers rely on real numbers for design calculations, measurements, simulations, and data analysis in various fields like civil, mechanical, electrical, and software engineering.


Everyday Life:

Real numbers formulas for class 10 help us in everyday situations. Here are some ways these formulas are used:


  • Cooking: Recipes utilize fractions and proportions, which are rooted in real numbers, for accurate ingredient measurements and scaling recipes.

  • Travel and Navigation: Distances, speeds, times, and directions involve real number calculations for planning trips and routes.

  • Shopping and Budgeting: Calculating expenses, discounts, taxes, and managing finances all rely on real number computations.


These are just a few examples, and the applications of real number formulas are practically endless! As you learn and practice these concepts, remember that they have far-reaching significance beyond the classroom, impacting various aspects of the world around you.


Tips to Memorize Real Numbers Formulas for Class 10:

Memorizing formulas can be challenging, but there are several techniques and tips that can help you retain the class 10 Maths chapter 1 formula of real numbers. Here are some suggestions:


  • Before memorizing formulas, make sure you understand the underlying concepts. Knowing how and why a formula works will make it easier to remember.

  • Write the formulas on one side of an index card and the explanations or examples on the other side. Quiz yourself regularly using these flashcards.

  • Repeat the formulas regularly. The more you repeat them, the more likely they are to stick in your memory. Reviewing them periodically reinforces your memory.

  • Create acronyms or mnemonic devices to remember the formulas. Associate each letter or word with a specific formula or concept. This can make it easier to recall the information.

  • Group similar formulas together. For example, group formulas related to arithmetic operations, algebraic expressions, or geometric shapes. This can help you remember them in clusters, making it easier to recall.

  • Apply the formulas by solving problems. The more you practice using the formulas in real-world scenarios, the better you'll remember them.

  • Prepare a formula sheet that you can refer to regularly. Having a condensed reference with all the formulas in one place can be handy for quick reviews.

  • Use the formulas in practical situations. When you encounter problems or situations that require the use of these formulas, apply them. Real-world application enhances memory.

  • Break down complex formulas into smaller parts. Learn each part individually and then combine them to understand the complete formula.


Examples on Real Numbers Formulas for Class 10 

Example 1: Check if √2 is an irrational number.

Solution: √2 is indeed an irrational number. This means it cannot be expressed as a fraction of two integers, and its decimal representation never ends or repeats. There are various ways to verify this, here are two common approaches:


First Approach:

Assume that √2 is rational. This means it can be written as a fraction a/b, where a and b are integers with no common factors (coprime).

Squaring both sides of the equation, we get $2 = \dfrac{a^2}{b^2}$.

Multiplying both sides by $b^2$, we get $2b^2 = a^2$.

This implies that $a^2$ is even (since it's equal to 2 times another integer).

Since the square of an odd number is always odd, this means a itself must be even (the square of an even number is also even).

Let a = 2c, where c is another integer.

Substitute this back into the equation $2b^2 = a^2$, we get $2b^2 = (2c)^2 = 4c^2$.

Dividing both sides by 2, we get $b^2 = 2c^2$.

This implies that $b^2$ is also even, meaning b itself must be even.

We have now shown that both a and b are even

Therefore,  √2 is rational must be false, and it follows that √2 is irrational.


Second Approach

While not a formal proof, we can also consider the decimal representation of √2. It starts with 1.414..., and continues with an infinite string of non-repeating digits. If √2 were rational, its decimal representation would eventually either terminate (becoming a repeating decimal) or start repeating a specific pattern. Since this doesn't happen, it's another strong indication that √2 is irrational.

These are just two ways to approach the problem. There are other proofs and methods available depending on your level of mathematical knowledge.


Therefore, the square root of 2 (√2) is an irrational number.


Example 2: Mohan is organizing chocolates in his shop, with 400 dark chocolates and 220 milk chocolates. How many chocolates should be in each stack if Mohan wants an equal number in each stack?

Solution: If Mohan has 400 dark chocolates and 220 milk chocolates, and he wants to organize them into stacks with an equal number of chocolates in each stack, we need to find the greatest common divisor (GCD) of the total number of chocolates and the desired number of stacks.

The GCD is the largest number that is a factor of both the total number of chocolates (620) and the desired number of stacks (4). In this case, the GCD is 4.

Therefore, Mohan can create 4 stacks with 155 chocolates in each stack 

$\dfrac{620\text{ chocolates }}{4\text{ stacks }} $ = 155 chocolates

So the answer is 155


How Chapter 1 Maths Class 10 Formulas on Real Numbers Will Help Students?

  • The formulas help students to simplify calculations easily.

  • Knowing the  CBSE Class 10 Maths Chapter 1all Formulas on Real Numbers will help students to prepare more effectively for the exams.

  • It helps students to save valuable time in solving questions in the exam.

  • Silly mistakes can be avoided while solving problems related to real numbers.

To help students get good scores in Class 10th board exams,  we have put together some of the important CBSE Real Numbers formulas for Class 10 that will further help them prepare well for the exams. Candidates can download the formula page PDF from the link given on this page.


Conclusion

In Class 10 Maths Chapter 1 formula, we dove into the world of Real Numbers, unraveling essential formulas to grasp their essence. These foundational formulas equip us to understand and manipulate real numbers effectively. From Euclid's Division Lemma to the Fundamental Theorem of Arithmetic, each formula unveils a unique aspect of real numbers. These concepts, like the HCF and LCM, lay the groundwork for more advanced mathematical explorations. By mastering the Chapter 1 Maths Class 10 formulas, we not only comprehend the intricacies of real numbers but also pave the way for a solid mathematical journey ahead. Explore, practice, and let these formulas be your guiding lights in the realm of numbers.

FAQs on CBSE Maths Chapter 1 Real Numbers Formulas for Class 10

1. What are the important Real Numbers formulas for Class 10?

1. Euclid’s Division Lemma: For positive integers a and b, there are whole numbers m and n such that a = bm + n, where 0 ≤ n < b.


2. Euclid’s Division Algorithm: Based on Euclid’s division lemma, it finds the Highest Common Factor (HCF) of positive integers a and b.


  1. Apply division lemma to find m and n: a = bm + n, where 0 ≤ n < b.

  2. If n = 0, HCF is b. If n ≠ 0, apply Euclid’s lemma to b and n.

  3. Repeat until remainder is zero; the divisor is HCF of (a, b). HCF(a, b) = HCF(b, n).


3. Prime Divisibility: If 'x' is a prime number and 'x' divides a², then 'x' divides 'a', where 'x' is a positive integer.

2. Why is it Important to Solve Problems Related to Real Number Formulas for Class 10?

Solving these problems helps understand essential concepts with applications in arithmetic.


Examples include proving irrationality using the Fundamental Theorem of Arithmetic and determining termination of decimal expansion in rational numbers.

3. What are Important Concepts Covered under class 10 maths chapter 1 formula on Real Numbers?

  • Properties of real numbers.

  • Equation for Euclid's Lemma and its proof.

  • Fundamental Theorem of Arithmetic.

  • Finding HCF and LCM via Prime Factorization Method.

4. How Many Formulas are There in Real Numbers Formulas for Class 10?

There are three primary formulas covering properties and theorems of real numbers, applicable in various mathematical levels.

5. How can I Memorize chapter 1 maths class 10 formula on Real Numbers?

  • Read notes in the textbook to understand the formula concept.

  • Write down the formula multiple times with explanations for better understanding.

  • Save formula images on devices for quick revision.

  • By mastering these formulas, students not only solve class 10 questions effectively but also build a foundation for advanced mathematics