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Real numbers are a fundamental part of mathematics, forming the basis for most mathematical operations. Class 10 Real Numbers Notes introduce key concepts that help solve basic mathematical problems and lay the foundation for advanced topics such as algebra, geometry, and calculus. Understanding real numbers is essential as they have practical applications in everyday life—for instance, measuring distances or handling financial transactions.
Subject-wise material for Class 10, Chapter 1 Real Numbers, is curated with the revised CBSE pattern in mind. The chapter comprises 6 marks in the final exam. At Educart, we provide a detailed explanation of real numbers. Let’s now explore the important formulas and questions covered in Class 10 Real Numbers Notes.
Class 10 math Chapter 1 notes cover all the main concepts like Euclid's Division Lemma and Arithmetic Fundamental Theorem. Real numbers include every number other than the complex numbers and the downloadable notes PDFs provided below are detailed and in easy-to-understand language.
Real numbers include all numbers that can be plotted on a number line. They consist of both rational and irrational numbers, which can appear as fractions, decimals, and square roots. Real numbers cover a vast range of values, including positive and negative numbers, as well as zero.
On a number line:
Real numbers are further classified into different categories, as explained below:
Rational numbers can be expressed as the ratio of two integers, where the denominator is not zero. In other words, a number is rational if it can be written in the form p/q, where p and q are integers, and q ≠ 0.
Examples: ½, -¾, 5, 0.25
Irrational numbers cannot be expressed as a fraction of two integers. Their decimal representation is non-terminating and non-repeating.
Examples: π (pi), √2, e
Integers include all whole numbers, both positive and negative, along with zero. While all integers are rational numbers, not all rational numbers are integers.
Examples: -3, 0, 5
Whole numbers consist of non-negative integers, including zero.
Examples: 0, 1, 2, 3, 4
Natural numbers are also known as counting numbers and do not include zero.
Examples: 1, 2, 3, 4, …
A number line visually represents numbers in order. On this line:
Rational numbers fall on specific points, whereas irrational numbers are placed between them without precise locations.
Real numbers follow important arithmetic and algebraic properties, which help simplify calculations and solve mathematical problems efficiently.
This property states that the order of numbers does not affect the result.
This property states that the way numbers are grouped does not change the result.
This property states that multiplication distributes over addition or subtraction.
a×(b+c)=a×b+a×c
Identity elements do not change a number’s value when applied.
Every real number has an inverse that helps in balancing equations.
This property states that real numbers remain within the set when performing operations.
Real numbers follow fundamental arithmetic operations, including addition, subtraction, multiplication, and division.
When adding or subtracting real numbers, we follow these rules:
Multiplication: Multiply the absolute values of the numbers and apply the sign rules:
Example: (-3) × (+4) = -12
Division
Divide the absolute values of the numbers and determine the sign based on division rules:
Example: (+6) ÷ (-3) = -2
Real numbers can be represented in decimal form. Their decimal expansion can be of three types:
Terminating Decimals:
These decimals have a finite number of digits after the decimal point.
Examples: 0.5, 1.75, -3.25
Non-Terminating, Repeating Decimals:
These decimals continue indefinitely but follow a cyclic pattern.
Examples: 0.333... (1/3), 0.142857... (1/7)
Non-Terminating, Non-Repeating Decimals:
These decimals expand infinitely without a repeating sequence. They are irrational numbers.
Examples: √2, π, e
When a fraction contains a square root in the denominator, we can rationalise it. This means eliminating the square root from the denominator by multiplying both the numerator and denominator by an appropriate value to form a perfect square.
To rationalise 1/√2, multiply both the numerator and denominator by √2:
1/√2 x √2/√2
= √2/2
This simplifies the fraction while ensuring the denominator is a rational number.
Real numbers can be expressed in different forms, depending on their properties:
Decimal Form: As discussed earlier, real numbers can be represented as decimals, which may be terminating or non-terminating.
Fractional Form: Rational numbers are commonly written as fractions, where the numerator and denominator are integers.
Radical Form: Irrational numbers, especially those involving square roots, are often represented using radical symbols (e.g., √2, √5).
Exponential Form: Numbers like e (Euler’s number) or powers of 10 are expressed in exponential notation, such as 10³ or e².
Some important questions related to the Real Numbers.
a) They consist exclusively of positive integers, excluding fractions and other representations.
b) They represent precise fractions and terminating decimals that can be plotted on a number line.
c) They include only integers, excluding decimals, fractions, and irrational numbers.
d) They include all values that appear on the number line, covering both rational and irrational numbers.
Answer: d) They include all values that appear on the number line, covering both rational and irrational numbers.
a) The distributive property explains how multiplication over addition correctly combines values.
b) The commutative property states that changing the order of multiplication does not affect the result.
c) The associative property implies that regrouping numbers in multiplication affects the outcome.
d) The identity property ensures that multiplying by one retains the original value.
Answer: b) The commutative property states that changing the order of multiplication does not affect the result.
a) Multiply both the numerator and denominator by the root to remove the radical from the denominator.
b) Multiply only the numerator by the square root, changing its form without matching the denominator.
c) Divide the numerator and denominator by square roots wherever possible for better proportions.
d) Remove the square root manually, altering the numerator and denominator separately.
Answer: a) Multiply both the numerator and denominator by the root to remove the radical from the denominator.
a) Real numbers are limited to positive integers and do not include decimals or fractions.
b) Real numbers are plotted with negatives on the left, positives on the right, and zero at the centre.
c) Real numbers are mainly shown as fractions, with limited use of number lines.
d) Real numbers are confined to a flat surface, excluding negative numbers and large values.
Answer: b) Real numbers are plotted with negatives on the left, positives on the right, and zero at the centre.
Mastering real numbers is essential for building a strong foundation in mathematics. Here are some effective study tips to help you understand the chapter better:
While studying real numbers, students often make these common mistakes. Here’s how to avoid them:
Mistake: Assuming that any non-repeating decimal is irrational.
Tip: Remember, decimals that terminate (e.g., 0.5) or repeat (e.g., 0.333...) are rational, whereas numbers like π or √2 are irrational.
Mistake: Incorrectly applying commutative, associative, or distributive properties when simplifying expressions.
Tip: Carefully follow the correct order of operations and apply properties correctly.
Mistake: Miscalculating signs, especially when multiplying or dividing negative numbers.
Tip: Remember:
Mistake: Forgetting that | -5 | = 5, not -5.
Tip: Absolute value represents the distance from zero, meaning it is always positive or zero.
Taking notes is an effective way to reinforce learning, especially for complex topics like real numbers. Here's why:
Organises Information – Summarising concepts in notes helps you structure your learning in an easy-to-understand way.
Quick Review Tool – Notes act as a personal study guide, making revision easier before exams.
Improves Focus – Writing down important concepts keeps you engaged and enhances understanding.
Useful for Future Reference – Well-organised notes can be revisited later when studying more advanced topics in mathematics.
Real numbers play a crucial role in mathematics, forming the basis for topics such as algebra, geometry, and calculus. Understanding their properties, the number line, operations, and decimal representations is essential for success in Class 10 mathematics.
Mastering real numbers will not only help in solving equations but will also prepare students for advanced studies in mathematics and science. Whether working with algebra or geometric calculations, real numbers will always be an integral part of mathematical problem-solving.
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