REAL

NUMBERS
A project submitted to:
The Department of Mathematics
Kathmandu Model College Bagbazar,
Kathmandu

Submitted by:
Anuj Bhandari
Science- XI D3

A Report in Partial fulfilment of the requirements of the Internal

Evaluation for Grade XI 2081-08-26
DECLARATION

I hereby confirm that the study and analysis presented in this project
report are entirely my work, completed under the guidance of Mr. Chuda
Prasad Pokhrel. Throughout this project, I have focused on exploring real
number systems to improve my understanding and skills in this important
area of mathematics.

I declare that this report is a genuine representation of my efforts and has

not been submitted elsewhere for any academic or other purposes. It
reflects the result of many hours of research, study, and problem-solving.

I have ensured that all sources of information used in this project are
properly acknowledged. Any quotes, ideas, or concepts taken from other
sources have been credited to their original authors or organizations,
demonstrating my commitment to honesty and transparency.

By signing this declaration, I affirm my dedication to maintaining high

academic standards and ethical practices.

Signature:
Anuj Bhandari
Science- XI D3
ACKNOWLEDGEMENT

I am deeply grateful to everyone who has supported me in bringing this

project together and transforming my ideas into something meaningful. I
would like to express my sincere thanks to the Department of
Mathematics at Kathmandu Model Secondary School for their
encouragement and guidance throughout this process.

I am especially thankful to my teacher, Mr. Chuda Prasad Pokhrel, for his

valuable advice and inspiration. Lastly, I extend my heartfelt gratitude to
my fellow college mates and family members for their kind support and
helpful suggestions.

Signature:
Name of HOD:
(Department of Mathematics)
Date
TABLE OF CONTENTS
Background ............................................................................................................ 5

Introduction ........................................................................................................... 6

Algebraic property of real number ......................................................................... 9

Order of property ................................................................................................... 11

Geometrical representation ................................................................................... 12

Inequality ............................................................................................................... 14

Intervals ................................................................................................................ 16

The absolute value of function ...............................................................................18

Summary, … ......................................................................................................... 24

Conclusion ........................................................................................................... 24
BACKGROUND OF REAL NUMBER SYSTEM:

Ancient civilizations, such as the Babylonians and Egyptians, had

numerical systems but did not have a formal concept of real numbers.
Euclid’s “Elements” (circa 300 BC) included an axiomatic treatment
of geometry but did not delve into real numbers extensively. Indian
and Arabic mathematicians made significant contributions to number
systems, including the development of decimal notation and zero.
They worked with number systems that included positive and negative
numbers, zero, and fractions, which influenced the understanding of
real numbers. Isaac Newton and Gottfried Wilhelm Leibniz developed
calculus, which heavily relied on real numbers and their properties. In
the 20th Century and beyond, advances in mathematical logic, set
theory, and formal axiomatic systems further refined the understanding
of real numbers.
INTRODUCTION:
We use real numbers to represent various quantities such as time,
distance, area, profit, loss, temperature, speed, etc. So these numbers
are frequently needed in the real world. Moreover, it provides the
foundation for modern mathematics. The real number system evolved
by the expansion of the notion of what we meant by the word
‘number’. In the beginning, ‘number’ meant something one could
count. The concept of these ‘numbers’ started in ancient times to
figure out how many sheep a farmer owns. In this section, we begin
with a discussion of various sets of real number (R) and their
relationships. We then discuss the two major properties of R: namely
algebraic and order properties. Finally, we discuss a well-known set,
called the interval and a function known as absolute value:
1. Set of natural numbers:
The type of numbers first to come into existence were the counting
numbers 1, 2, and 3. These are called natural numbers. The set of natural
numbers is denoted by N.
Thus, N= {1,2,3,4, .............. }

2. Whole numbers:
At some point, the idea of ‘zero’ came to be considered as a number. This
made it possible to answer the situation. “If the farmer does not have any
sheep, then how many sheep does that farmer own?” The set of natural
numbers together with zero is called whole numbers. So the set of whole
number is {0, 1, 2, 3, …}
 3. Set of integers:
After the most valuable creation of the number ‘0’ (zero), the negative
natural numbers appeared very late in history. The set consisting of whole
numbers 0, 1, 2, 3, and the negative natural numbers 1, -2, -3...... form a
set of integers which is denoted by I or Z
Z= {...-3, -2, -1,0,1,2, 3, ............ }

4. Set of rational numbers:

Rational numbers are numbers that can be expressed as fractions where

both the numerator and denominator are integers, and the denominator is
not zero. In other words, a rational number can be written as p/q, where p
and q are integers and q ≠ 0.
Examples of rational numbers:

* Fractions: ½, ¾, -5/7
* Integers: 2, -3, 0 (can be written as 2/1, -3/1, 0/1)
* Terminating decimals: 0.5, 0.75, 1.25
* Repeating decimals: 0.333..., 0.666..., 0.142857...
 5. Set of irrational numbers:
Irrational numbers are numbers that cannot be expressed as simple
fractions. They have decimal expansions that neither terminate nor
repeat. Some common examples include:

* The square root of 2 (√2)

* Pi (π)
* The golden ratio (φ)
* Euler’s number (e)
These numbers play a crucial role in various fields of mathematics and
science.

6. Set of real numbers:

Real numbers encompass all numbers that can be represented on a
number line. They include both rational numbers (like integers and
fractions) and irrational. Numbers (like √2 and π). Essentially, any
number you can think of, except for complex numbers, is a real number.
ALGEBRAIC PROPERTIES OF REAL NUMBER:
The algebraic properties of real numbers are based on two operations,
namely addition and multiplication. All the algebraic properties of real
numbers can be derived from these two operations. The binary
operations addition and multiplication on the set of real numbers satisfy
the following properties:

1. CLOSURE PROPERTY: if a,b. R then a + b∈ R (the sum

of real numbers is a real number) a ∙ b∈ R (the product of real
numbers is a real number)

2. COMMUTATIVE PROPERTY: if a,b R then,

a+b=b+a
a·b=b·a

3. ASSOCIATIVE PROPERTY: if a,b ∈ R then,

a + (b + c) = (a + b) + c , a · (b · c) = (a · b) · c
4. EXISTENCE OF THE IDENTITY ELEMENT
There exists 0 ∈ R : 0 + a = a + 0 = a, for all a ∈ R
There exists 1 ∈ R : 1∙a = a = a ∙ 1 for all a ∈ R, a≠ 0

Here, 0 and 1 are called additive identity and multiplicative identity,

respectively.
5. EXISTENCE OF INVERSE ELEMENT

For each a ∈ R, there exists -a ∈ R, such that.

a + (-a) = 0 = a + (-a)
For each a≠0, a ∈ R, there exists a-1 ∈ R such that
a ∙ a-1 =1= a-1 ∙ a

6. DISTRIBUTIVE PROPERTY: if a, b, c ∈ R then

a · (b + c) = a ∙ b + a · c (left distributive property) (b + c) a = b ·

a + c · a (right distributive property)
ORDER PROPERTY OF REAL NUMBERS
The order property of real numbers states that any two real numbers can
be compared using the inequality symbols < (less than), > (greater than),
or = (equal to). In other words, real numbers can be arranged in a linear
order.
Here are the key properties of the order property of real numbers:
Trichotomy Law: For any two real numbers a and b, exactly one of the
following is true:

* a<b
* a=b
* a>b
1. Transitive Law: If a < b and b < c, then a < c.
2. Ordering is Compatible with Addition: If a < b, then a + c < b
+ c for any real number c.
3. Ordering is Compatible with Multiplication: If a < b and c > 0,
then ac < bc.
4. Ordering is Compatible with Multiplication (Negative Case):
If a < b and c < 0, then ac > bc.
These properties allow us to compare and order real numbers, which is
fundamental to many mathematical concepts and applications.
GEOMETRICAL REPRESENTATION OF REAL
NUMBERS
The real number line is a visual representation of the setof all real
numbers. It's a horizontal line that extends infinitely in both directions.
Each point on this line corresponds to exactly one real number, and each
real

number corresponds to exactly one point on the line. The above is the
geometrical representation of realnumbers. The key components of the
real number line include origin, positive numbers, negative numbers and
scale.
The origin, labelled as 0, serves as the reference point,dividing the line
into positive and negative sides.
Positive numbers are located to the right of the origin, while negative
numbers are to the left.
The distance between consecutive integers on the line is uniform,
representing the scale. Every point on the line corresponds to a unique real
number, and vice versa.
This includes rational numbers, which can be expressed as fractions or
decimals, and irrational numbers, which cannot be expressed as simple
fractions.

Locating Irrational Numbers in a Number Line

To locate irrationals in a real number line, we take √2 as an example. For

this consider the following figure:

In the figure, ABCD is a square of unit length. Then, the length of BD=√12
+ 12=√2. Now we draw an arc with a center at D and a radius equal to
length BD = √2 that meets the line at point P. Clearly,

BD = DT =√2. Thus, T represents the irrational number.

√2.
Similarly, considering a rectangle of length √2 and width 1 unit, we can
locate √3. In this way, we can represent all real numbers on the real
number line.
The position of the real numbers on the number line establishes an order
for the real numbers. The numbers go on increasing from left to right.
There is no greatest real number and least real number in the system of
real numbers.

INEQUALITY

The properties of inequalities with addition and multiplication stem from

basic rules in the real number system. If we know that one number is
less than another, certain operations like addition or multiplication will
either maintain or reverse the inequality. For example, if a < b, adding
the same number to both sides doesn’t change the inequality, but
multiplying by a negative number does. These properties are
foundational and are formally proven to ensure their reliability in all
applicable cases.

We present three key properties of inequalities and their proofs:

1. Addition Property: a < b → a + c < b + c

2. Multiplication by a Positive Number: a < b → ca < cb

if c is positive

3. Multiplication by a Negative Number: a < b → ca > cb

1. Statement: If a < b, then adding c to both sides maintain the
inequality, so a + c < b + c.
Proof:

•Since a < b, we know that b – a > 0, meaning b is greater than a by a

positive amount.

•Now, add c to both a and b to get a + c and b + c.

•Observe that (b + c) – (a + c) = b – a.
Since b – a > 0, it follows that a + c < b + c.
Conclusion: We have proven that if a < b, then a + c < b +c

2. Proof of a < b → ca < cb if c is positive.

1. Statement: If a < b and c is a positive real number, then multiplying
both sides by c maintains the inequality, so ca < cb.
Proof:
•Given a < b, we know that b – a > 0.
•Since c is a positive real number, multiplying b – a by c will keep the
product positive, so c (b – a) > 0.
•Expanding this, we get cb – ca > 0, which implies ca < cb
Conclusion: We have proven that if a<b and c is positive, then ca
< cb
3. Proof of a < b → ca > cb if c is the negative statement: If a < b and c is
a negative real number, then multiplying both sides by c reverses the
inequality, so ca > cb. Proof:
•Given a < b, we have b – a > 0.
•Since c is negative, multiplying b – a by c will reverse the sign, resulting
in c(b – a) < 0.

INTERVALS:
Intervals can define subsets of the real numbers. We use intervals in
calculus and analysis as well as countless other areas of mathematics to
explore problems and describe quantities. Types of Intervals:
1. Open Intervals (a, b):
Definition: Includes all real numbers x such that a< x < b.
Example: If a=2a = 2 and b=5b = 5, then the interval is (2,5)(2, 5), which
includes all numbers between 2 and 5, but not 2 and 5 themselves.

2. Closed Intervals [a, b]:

Definition: Includes all real numbers xx such that
Example: If a=2a = 2 and b=5b = 5, then the interval is [2,5][2, 5], which
includes the numbers 2, 5, and all numbers in between.
3. Half-open (or Half-Closed) Intervals
a. Left Open (a, b]: Includes all real numbers xx such that a<x_<b
Example: If a=2a = 2 and b=5b = 5, then the interval is (2,5] (2, 5],
including numbers between 2 and 5, but not 2, and including 5.
b. Right Open [a, b): Includes all real numbers xx such that a≤x<b
Example: If a=2a = 2 and b=5b = 5, then the interval is [2,5)[2,
5), including numbers between 2 and 5, including 2 but not 5.
4. Unbounded Intervals:
Definition: These intervals extend infinitely in one or both
directions.

Types:

(a,∞)(a, \infinity): All real numbers greater than a.

[a,∞)[a, \infinity): All real numbers greater than or equal to a.
(−∞,b)(-\infinity, b): All real numbers less than b.
(−∞,b](-\infinity, b]: All real numbers less than or equal to b.
(−∞,∞)(-\infinity, \infinity): Represents all real numbers.
THE ABSOLUTE VALUE FUNCTION
The absolute value function is denoted by f(x) = |x|. It takes any real
number x as input and outputs its absolutevalue. This function has a
simple definition:

• If x is greater than or equal to 0, then |x| = x.

• If x is less than 0, then |x| = -x.

For example, |3| = 3 and |-3| = 3, illustrating the

function’s behavior.
Graphing absolute value
The graph of the absolute value function f(x) = |x| is a V-shaped curve. It
is symmetric about the y-axis, with the vertex at the origin (0, 0). The left
side of the graph is a straight line with a slope of -1, while the right side
has a slope of 1.

Absolute value in equalities and inequalities

Absolute value is frequently used in equations and inequalities.

When solving equations involving absolute value, we consider
two cases: one where the expression inside the absolute value is
positive or zero, and another where it is negative. For
inequalities, we need to consider both positive and negative
cases as well, but the solution may involve two intervals.

For example, to solve the equation |x – 2| = 5, we solve the following two

equations: x – 2 = 5 and –(x – 2) = 5.

This yields the solutions x = 7 and x = -3.

To solve the inequality |x + 1| < 3, we solve the inequalities x + 1 < 3 and
–(x + 1) < 3. This gives us the solution -4 < x < 2.
Applications of absolute value
Absolute value finds applications in various fields, including:
• Physics: Calculating distance or displacement without considering
directions
• Computer Science: Measuring errors and deviations.
• Statistics: Determining the spread of data.
• Finance: Calculating the difference between expected and actual
values.
For example, in physics, if an object moves 5 meters to the right and
then 3 meters to the left, the total distance travelled is 8 meters,
calculated using absolute values. The net displacement, however, is 2
meters to the right, indicating the object’s final position relative to its
starting point.

Properties of absolute value

Absolute value possesses some useful properties:
• |x| ≥ 0 for all x. The absolute value of any number is always non-
negative.
• |x| = |-x| for all x. The absolute value of a number is the same as the
absolute value of its negative counterpart.
• |x| = 0 if and only if x = 0. The absolute value of a number is 0 only
if the number itself is 0.
• |x y| = |x| |y| for all x and y. The absolute value of a product is equal
Proof of some property of Absolute value:

Case I: Suppose x+y≥0. Then, by definition,

1. For any real number x and y, |x +y| = x+y |x| + |y|

As x ≤ |x| and y ≤ |y|
|x+y|≤ |x| +|y|..................(1)
Case II: Suppose x+y<0. Then, |x+y| =−(x+y)=(−x)+(−y)≤ |x|
+|y|
As −x≤ |x| and −y≤|y|
|x|+ |y| |x| + |y|……..(2)

Therefore, from (1) and (2) it follows that x+y ≤ x + y.

Aliter
|x+y|2 =(x+y)2
=x2+2xy+y2= x 2+2xy+ y 2

≤ x 2+2 x y+y 2
= (|x| + |y|)2

Taking only the positive square root, we get x+y ≤ x +y

2.For any real numbers x and y x−y≤ x +y Proof

x−y
= x+(−y)
= |x + (-y)| ≤ |x| + |-y|
=|x| + |y|
(by triangle inequality) x−y≤ x +y

3.For any real numbers x and y x− y ≤x+y Proof

We have x = (x+y)−y ≤x+y +y, by triangle inequality. Thus, x
− y≤x+y .

4.For any real numbers x and y x− y ≤ x−y

Proof

x = (x−y)+y ≤x−y +y, by triangle inequality. Thus, x − y≤x−y

.

5.For any real numbers x, y, and z x−z

≤ x−y +y−z
Proof

∣x−z∣=∣(x−y)+(y−z)∣≤∣x−y∣+∣y−z∣

∣x−z∣≤∣x−y∣+∣y−z∣
. If a be any positive real number and x ∈ R, prove that
|x| ≤ a ⇔ -a ≤ x ≤ a.
Proof:
For all x ∈ R, we have x ≤ |x|. Given |x| ≤ a ⇒ x ≤ |x| ≤ a
⇒ x ≤ a Again, for all x ∈ R, -x ≤ |x|.
Given |x| ≤ a ⇒ -x ≤ |x| ≤ a ⇒ -x ≤ a ⇒ x ≥ -a ⇒ -a ≤ x Combining
(1) and (2), -a ≤ x ≤ a.
Conversely, let -a ≤ x ≤ a.
At first, x ≤ a. If x ≥ 0, then |x| = x. ⇒ |x| = x ≤ a Again, -a
≤ x. ⇒ a ≥ -x ⇒ -x ≤ a.
But for x < 0, |x| = -x. ⇒ |x| = -x ≤ a
Hence, from (3) and (4), for all x ∈ R, |x| ≤ a.
Remark: We can also prove the triangle inequality using the Property 7.
For this we have -|x| ≤ x ≤ |x| and -|y| ≤ y ≤ |y|.
Adding these inequalities, we get
-|x| - |y| ≤ x + y ≤ |x| + |y|
⇒ -(|x| + |y|) ≤ x + y ≤ (|x| + |y|)
Using property 7, we get |x + y| ≤ |x| + |y|. Proved.
SUMMARY
Real numbers encompass all numbers, both rational and irrational.
Rational numbers can be expressed as fractions or
terminating/repeating decimals (e.g., ½, 0.333…). Irrational numbers
cannot be expressed as fractions and have non-repeating decimals (e.g.,
√2, π). Real numbers can be positive, negative, or zero. They form the
foundation of mathematics and are used to measure various quantities
in the real world.

CONCLUSION
In conclusion, the real number system serves as the foundation for
various mathematical operations and problem-solving. It
encompasses all numbers, both rational and irrational, and can be
visualized on a number line. Understanding the properties and
classifications of real numbers is crucial for various mathematical
disciplines and real-world application
BIBLIOGRAPHY: -

• https://study.com/academy/lesson/real-analysis- completeness-of-the-real-

numbers.html

• https://byjus.com/maths/real-numbers-for-class-10/

• https://www.geeksforgeeks.org/what-is-a-real-number- system-in-

mathematics/

• https://flexbooks.ck12.org/cbook/ck-12-cbse-math-class-
10/section/1.1/primary/lesson/real_numbers/

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