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Non-negative counting numbers excluding zero are called Natural Numbers.
N = 1, 2, 3, 4, 5, ……….
All natural numbers including zero are called Whole Numbers.
W = 0, 1, 2, 3, 4, 5, ……………. (Scroll down to continue …)
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All natural numbers, negative numbers and 0, together are called Integers.
Z = – 3, – 2, – 1, 0, 1, 2, 3, 4, …………..
The number ‘a’ is called Rational if it can be written in the form of r/s where ‘r’ and ‘s’ are integers and s ≠ 0,
Q = 2/3, 3/5, etc. all are rational numbers.
To find the rational number between two given numbers ‘a’ and ‘b’.
Example:
Find 2 rational numbers between 4 and 5.
Solution:
To find the rational number between 4 and 5
To find another number we will follow the same process again.
Hence the two rational numbers between 4 and 5 are 9/2 and 17/4.
Remark: There could be unlimited rational numbers between any two rational numbers.
The number ‘a’ which can’t be written in the form of p/q is called irrational. Here, p and q are integers and q ≠ 0. You can say that the numbers which are not rational are called Irrational Numbers.
Example – √7, √11 etc.
All numbers including both rational and irrational numbers are called Real Numbers.
R = – 2, – (2/3), 0, 3 and √2
1. Rational Numbers
If the rational number is in the form of a/b, then we can get two situations by dividing a by b.
a. If the remainder becomes zero
While dividing if we get zero as the remainder after some steps then the decimal expansion of such a number is called terminating.
Example:
7/8 = 0.875
b. If the remainder does not become zero
While dividing if the decimal expansion continues and not becomes zero then it is called non-terminating or repeating expansion.
Example:
1/3 = 0.3333….
Hence, the decimal expansion of rational numbers could be terminating or non-terminating recurring and vice-versa.
2. Irrational Numbers
If we do the decimal expansion of an irrational number then it would be non –terminating non-recurring and vice-versa. i. e. the remainder does not become zero and also not repeated.
Example:
π = 3.141592653589793238……
To represent the real numbers on the number line, we use the process of successive magnification. We visualise the numbers through a magnifying glass on the number line.
Example:
Step 1: The number lies between 4 and 5, so we divide it into 10 equal parts. Now for the first decimal place, we will mark the number between 4.2 and 4.3.
Step 2: Now we will divide it into 10 equal parts again. The second decimal place will be between 4.26 and 4.27.
Step 3: Now we will again divide it into 10 equal parts. The third decimal place will be between 4.262 and 4.263.
Step 4: By doing the same process again we will mark the point at 4.2626.
1. The sum, difference, product and quotient of two rational numbers will be rational.
Example:
2. If we add or subtract a rational number with an irrational number then the outcome will be irrational.
Example:
If 5 is a rational number. √7 is an irrational number. Then, 5 + √7 and 5 – √7 are irrational numbers.
3. If we multiply a non-zero rational number with an irrational number, the outcome will be irrational. If we divide a non-zero rational number with an irrational number, the outcome will also be irrational.
Example:
If 7 is a rational number and √5 is an irrational number then 7√7 and 7/√5 are irrational numbers.
4. The sum, difference, product and quotient of two irrational numbers could be rational or irrational.
Example:
To find √x geometrically
1. First, mark the distance x unit from point A on the line. This ensures that AB equals x unit.
2. From B mark a point C with the distance of 1 unit, so that BC = 1 unit.
3. Take the midpoint of AC and mark it as O. Then take OC as the radius and draw a semicircle.
4. From the point B draw a perpendicular BD which intersects the semicircle at point D.
The length of BD = √x.
To mark the position of √x on the number line, we will take AC as the number line. B will be zero. So C is point 1 on the number line.
Now we will take B as the centre and BD as the radius. We will draw the arc on the number line at point E.
Now E is √x on the number line.
If p and q are two positive real numbers
Examples:
1. Simplify
We will use the identity
2. Simplify
We will use the identity
Rationalising the denominator means to convert the denominator containing a square root term into a rational number. This is done by finding the equivalent fraction of the given fraction.
For which we can use the identities of the real numbers.
Example:
Rationalise the denominator of 7/(7- √3).
Solution:
We will use the identityhere.
If we have a and b as the base and m and n as the exponents, then
1. am × an =am+n
2. (am)n = amn
4. am bm = (ab)m
5. a0 = 1
6. a1 = a
7. 1/an = a-n
Example:
Simplify the expression (2x3y4) (3xy5)2.
Solution:
Here we will use the law of exponents
am × an =am+n and (am)n = amn
(2x3y4)(3xy5)2
(2x3y4)(3 2 x 2 y10)
18. x3. x2. y4. y10
18. x3+2. y4+10
18x5y14
Here’s a simple outline for an eBook on Real Numbers:
Title: Understanding Real Numbers: A Comprehensive Guide
Table of Contents
What are Numbers?
Introduction to Real Numbers
Why Are Real Numbers Important?
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Closure Property
Commutative Property
Associative Property
Distributive Property
Identity and Inverse Elements
Concept of Number Line
Plotting Real Numbers on the Number Line
Understanding Density of Real Numbers
Definition of Rational Numbers
Properties of Rational Numbers
Definition of Irrational Numbers
Examples of Irrational Numbers (like √2, π, e)
Proving √2 is Irrational
Finite and Infinite Decimals
Terminating and Non-Terminating Decimals
Relationship between Decimals and Fractions
Addition and Subtraction
Multiplication and Division
Operations with Decimals
Operations with Irrational Numbers
Definition of Absolute Value
Geometric Representation on the Number Line
Properties of Absolute Value
Understanding Infinite Sets
Limits and Real Numbers
Approaching Infinity on the Number Line
In Geometry (Pythagorean Theorem)
In Calculus (Limits, Derivatives, and Integrals)
In Daily Life (Measurements, Finance, etc.)
Real Numbers in Algebra
Real Numbers and Functions
Real Numbers and Continuity
Summary of Key Concepts
Importance of Mastering Real Numbers
How Real Numbers Apply to Higher Mathematics
Chapter 1: Introduction to Real Numbers
What Are Numbers?
Numbers are abstract symbols used to represent quantities. Throughout history, different types of numbers have been developed to address various mathematical problems.
Introduction to Real Numbers
Real numbers are all the numbers that can be found on the number line. This includes rational numbers (such as 5, -3, and 0.75) and irrational numbers (such as √2 and π). Together, they form the building blocks of modern mathematics.
Real numbers are used to measure continuous quantities like distance, time, and weight. They are integral to the concepts of calculus, physics, engineering, and many other fields.
Why Are Real Numbers Important?
Real numbers play a critical role in mathematics. They allow us to describe the size of objects, calculate areas and volumes, and express very large or very small values. Without real numbers, much of modern science and technology would not exist.
Chapter 2: Classification of Numbers
Natural Numbers
The set of natural numbers consists of counting numbers, such as 1, 2, 3, and so on. These are the simplest type of numbers and do not include zero.
Whole Numbers
Whole numbers are like natural numbers but also include zero. Thus, the set is {0, 1, 2, 3,…}.
Integers
Integers expand on whole numbers by including negative numbers. The set of integers is {…, -3, -2, -1, 0, 1, 2, 3,…}.
Rational Numbers
Rational numbers are numbers that can be expressed as a fraction of two integers (a/b), where b ≠ 0. Examples of rational numbers include 1/2, -4, and 0.75.
Irrational Numbers
Irrational numbers cannot be expressed as a fraction of two integers. Examples include √2, π, and e. These numbers have non-repeating, non-terminating decimal expansions.
Chapter 3: Properties of Real Numbers
Closure Property
The set of real numbers is closed under addition, subtraction, multiplication, and division (except division by zero). This means that the result of any of these operations on two real numbers will always yield another real number.
Commutative Property
For any two real numbers a and b:
Addition: a + b = b + a
Multiplication: a × b = b × a
Associative Property
For any three real numbers a, b, and c:
Addition: (a + b) + c = a + (b + c)
Multiplication: (a × b) × c = a × (b × c)
Distributive Property
The distributive property connects addition and multiplication:
a × (b + c) = (a × b) + (a × c)
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