Number Theory
Branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). Sometimes called “higher arithmetic,” it is among the
oldest and most natural of mathematical pursuits.
OR
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of
the integers and integer-valued functions.
Example
For instance, they attached significance to perfect numbers—i.e., those that equal the sum of their proper divisors. Examples are 6 (whose
proper divisors 1, 2, and 3 sum to 6) and 28 (1 + 2 + 4 + 7 + 14).
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Natural Numbers (N)
(also called positive integers, counting numbers, or natural numbers); They start from 1 and go to infinity, i.e., 1, 2, 3, 4, 5, 6, and so on. They are
also called positive integers. In the set form, they can be written as: {1, 2, 3, 4, 5, …}. Natural numbers are represented by the symbol N.
Properties of Natural Numbers:
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Addition of natural numbers is closed, associative, and commutative.
Natural Number multiplication is closed, associative, and commutative.
The identity element of a natural number under addition is zero.
The identity element of a natural number under Multiplication is one.
Natural numbers properties are segregated into four main properties which include:
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Closure property
Commutative property
Associative property
Distributive property
Each of these properties is explained below in detail.
Closure Property
Natural numbers are always closed under addition and multiplication. The addition and multiplication of two or more natural numbers will
always yield a natural number. In the case of subtraction and division, natural numbers do not obey closure property, which means
subtracting or dividing two natural numbers might not give a natural number as a result.
 Addition: 1 + 2 = 3, 3 + 4 = 7, etc. In each of these cases, the resulting number is always a natural number.
 Multiplication: 2 × 3 = 6, 5 × 4 = 20, etc. In this case also, the resultant is always a natural number.
 Subtraction: 9 – 5 = 4, 3 – 5 = -2, etc. In this case, the result may or may not be a natural number.
 Division: 10 ÷ 5 = 2, 10 ÷ 3 = 3.33, etc. In this case, also, the resultant number may or may not be a natural number.
Note: Closure property does not hold, if any of the numbers in case of multiplication and division, is not a natural number. But for addition and
subtraction, if the result is a positive number, then only closure property exists.
For example:
 -2 x 3 = -6; Not a natural number
 6/-2 = -3; Not a natural number
Associative Property
The associative property holds true in case of addition and multiplication of natural numbers i.e. a + ( b + c ) = ( a + b ) + c and a × ( b × c
) = ( a × b ) × c. On the other hand, for subtraction and division of natural numbers, the associative property does not hold true. An
example of this is given below.
 Addition: a + ( b + c ) = ( a + b ) + c => 3 + (15 + 1 ) = 19 and (3 + 15 ) + 1 = 19.
 Multiplication: a × ( b × c ) = ( a × b ) × c => 3 × (15 × 1 ) = 45 and ( 3 × 15 ) × 1 = 45.
 Subtraction: a – ( b – c ) ≠ ( a – b ) – c => 2 – (15 – 1 ) = – 12 and ( 2 – 15 ) – 1 = – 14.
 Division: a ÷ ( b ÷ c ) ≠ ( a ÷ b ) ÷ c => 2 ÷( 3 ÷ 6 ) = 4 and ( 2 ÷ 3 ) ÷ 6 = 0.11.
Commutative Property
For commutative property
 Addition and multiplication of natural numbers show the commutative property. For example, x + y = y + x and a × b = b × a
 Subtraction and division of natural numbers do not show the commutative property. For example, x – y ≠ y – x and x ÷ y ≠ y ÷ x
Distributive Property
 Multiplication of natural numbers is always distributive over addition. For example, a × (b + c) = ab + ac
 Multiplication of natural numbers is also distributive over subtraction. For example, a × (b – c) = ab – ac
Operations With Natural Numbers
An overview of algebraic operation with natural numbers i.e. addition, subtraction, multiplication and division, along with their respective
properties are summarized in the table given below.
Properties and Operations on Natural Numbers
Operation
Closure Property
Commutative Property
Associative Property
Addition
Yes
Yes
Yes
Subtraction
No
No
No
Multiplication
Yes
Yes
Yes
Division
No
No
No
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Whole Numbers (W).
This is the set of natural numbers, plus zero, i.e., {0, 1, 2, 3, 4, 5, …}. Whole numbers are represented by the symbol W.
Every natural number is a whole number but 0 is a whole number which is not a natural number
Facts:
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All the natural numbers are whole numbers
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All counting numbers are whole numbers
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All positive integers including zero are whole numbers
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All whole numbers are real numbers
Properties of Whole Numbers:
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Whole numbers are closed under addition and multiplication.
Zero is the additive identity element of the whole numbers.
1 is the multiplicative identity element.
It obeys the commutative and associative property of addition and multiplication.
It satisfies the distributive property of multiplication over addition and vice versa.
Closure Property
They can be closed under addition and multiplication, i.e., if x and y are two whole numbers then x. y or x + y is also a whole number.
Commutative Property of Addition and Multiplication
The sum and product of two whole numbers will be the same whatever the order they are added or multiplied in, i.e., if x and y are two whole
numbers, then x + y = y + x and x . y = y . x
Additive identity
When a whole number is added to 0, its value remains unchanged, i.e., if x is a whole number then x + 0 = 0 + x = x
Multiplicative identity
When a whole number is multiplied by 1, its value remains unchanged, i.e., if x is a whole number then x.1 = x = 1.x
Associative Property
When whole numbers are being added or multiplied as a set, they can be grouped in any order, and the result will be the same, i.e. if x, y and z
are whole numbers then x + (y + z) = (x + y) + z and x. (y.z)=(x.y).z
Distributive Property
If x, y and z are three whole numbers, the distributive property of multiplication over addition is x. (y + z) = (x.y) + (x.z), similarly, the distributive
property of multiplication over subtraction is x. (y – z) = (x.y) – (x.z)
Multiplication by zero
When a whole number is multiplied to 0, the result is always 0, i.e., x.0 = 0.x = 0
Division by zero
Division of a whole number by o is not defined, i.e., if x is a whole number then x/0 is not defined.
Difference between natural and whole numbers
Natural numbers
Whole numbers
Natural numbers are counting numbers beginning with the number 1
All non-zero integers are included in the natural numbers set.
Whole numbers are natural numbers but beginning with the number 0
All positive integers are included in the whole numbers set.
All natural numbers are considered as whole numbers
Represented by the letter 'N'
Natural numbers are closed under addition and multiplication
All whole numbers are not considered natural numbers
Represented by the letter 'W'
Whole numbers are closed under addition, and multiplication.
. Natural number set => N = {1, 2, 3,…,∞}
It contain positive integers.
Whole number set => W = {0, 1, 2, 3,…,∞}
It contain positive integers and 0.
Natural numbers are always greater than 0.
It doesn't contain fractions, decimals, and negative numbers.
Whole numbers are always greater than -1.
It doesn't contain fractions, decimals, and negative numbers.
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Integers (Z)(I)
This is the set of all whole numbers plus all the negatives (or opposites) of the natural numbers, They contain all the numbers which lie between
negative infinity and positive infinity. They can be positive, zero, or negative but cannot be written in decimal or fraction. i.e., {… , ⁻2, ⁻1, 0, 1, 2,
…}. We can say that all whole numbers and natural numbers are integers, but not all integers are natural numbers or whole numbers.. The
symbol Z represents integers.As a set, it can be represented as follows:
Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}
Only whole numbers are integers. Therefore fractions, decimals, percent, and exponents can never be integers.
Example
Example of integers: -100,-12,-1, 0, 2, 1000, 989 etc…
Types of Integers
Integers come in three types:
 Zero (0)
 Positive Integers (Natural numbers)
 Negative Integers (Additive inverse of Natural Numbers)
Zero
Zero is neither a positive nor a negative integer. It is a neutral number i.e. zero has no sign (+ or -).
Positive Integers
The positive integers are the natural numbers or also called counting numbers. These integers are also sometimes denoted by Z+. The positive
integers lie on the right side of 0 on a number line.
Z+ → 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30,….
Negative Integers
The negative integers are the negative of natural numbers. They are denoted by Z–. The negative integers lie on the left side of 0 on a number
line.
Z– → -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22,
-23, -24, -25, -26, -27, -28, -29, -30,…..
Properties of Integers:
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Integers are closed under addition, subtraction, and multiplication.
The commutative property is satisfied for addition and multiplication of integers.
It obeys the associative property of addition and multiplication.
It obeys the distributive property for addition and multiplication.
Additive identity of integers is 0.
Multiplicative identity of integers is 1.
The major Properties of Integers are:
1. Closure Property
2. Associative Property
3. Commutative Property
4. Distributive Property
5. Additive Inverse Property
6. Multiplicative Inverse Property
7. Identity Property
Closure Property
According to the closure property of integers, when two integers are added or multiplied together, it results in an integer only. If a and b are
integers, then:
 a + b = integer
 a x b = integer
Examples:
2 + 5 = 7 (is an integer)
2 x 5 = 10 (is an integer)
Commutative Property
According to the commutative property of integers, if a and b are two integers, then:
 a+b=b+a
 axb=bxa
Examples:
3 + 8 = 8 + 3 = 11
3 x 8 = 8 x 3 = 24
But for subtraction and division, commutative property does not obey.
Associative Property
As per the associative property , if a, b and c are integers, then:
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a+(b+c) = (a+b)+c
ax(bxc) = (axb)xc
Examples:
2+(3+4) = (2+3)+4 = 9
2x(3×4) = (2×3)x4 = 24
This property is applicable for addition and multiplication operations only.
Distributive property
According to the distributive property of integers, if a, b and c are integers, then:
a x (b + c) = a x b + a x c
Example: Prove that: 3 x (5 + 1) = 3 x 5 + 3 x 1
LHS = 3 x (5 + 1) = 3 x 6 = 18
RHS = 3 x 5 + 3 x 1 = 15 + 3 = 18
Since, LHS = RHS
Hence, proved.
Additive Inverse Property
If a is an integer, then as per additive inverse property of integers,
a + (-a) = 0
Hence, -a is the additive inverse of integer a.
Multiplicative inverse Property
If a is an integer, then as per multiplicative inverse property of integers,
a x (1/a) = 1
Hence, 1/a is the multiplicative inverse of integer a.
Identity Property of Integers
The identity elements of integers are:
a+0 = a
ax1=a
Example: -100,-12,-1, 0, 2, 1000, 989 etc…
As a set, it can be represented as follows:
Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}
Rules of Integers
Rules defined for integers are:
 Sum of two positive integers is an integer
 Sum of two negative integers is an integer
 Product of two positive integers is an integer
 Product of two negative integers is an integer
 Sum of an integer and its inverse is equal to zero
 Product of an integer and its reciprocal is equal to 1
Addition of Signed Integer Numbers
While adding the two integers with the same sign, add the absolute values, and write down the sum with the sign provided with the numbers.
For example,
(+4) + (+7) = +11
(-6) + (-4) = -10
While adding two integers with different signs, subtract the absolute values, and write down the difference with the sign of the number which
has the largest absolute value.
For example,
(-4) + (+2) = -2
(+6) + (-4) = +2.
Subtraction of Signed Integer Numbers
While subtracting two integers, change the sign of the second number which is being subtracted, and follow the rules of addition.
For example,
(-7) – (+4) = (-7) + (-4) = -11
(+8) – (+3) = (+8) + (-3) = +5
Multiplication and Division of Signed Integer Numbers
While multiplying and dividing two integer numbers, the rule is simple.
If both the integers have the same sign, then the result is positive.
If the integers have different signs, then the result is negative.
For example,
(+2) x (+3) = +6
(+3) x (-4) = – 12
Similarly
(+6) ÷ (+2) = +3
(-16) ÷ (+4) = -4
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Rational numbers (Q).
The word “rational” is derived from the word “ratio, ” as rational numbers are the two integers’ ratios. For example, 0.7 is a rational number
because it can be written as 7/10. Other examples of rational numbers are -1/3, 2/5, 99/100, 1.57, etc
OR
This is all the fractions where the top and bottom numbers are integers; e.g., 1/2, 3/4, 7/2, ⁻4/3, 4/1 [Note: The denominator cannot be 0, but the
numerator can be].
OR
Consider a rational number p/q, where p and q are two integers. Here, the numerator p can be any integer (positive or negative), but the
denominator q can never be 0, as the fraction is undefined. Also, if q = 1, then the fraction is an integer.
The symbol Q represents rational numbers
Example
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1/9 – Both numerator and denominator are integers.
7 – Can be expressed as 7/1, wherein 7 is the quotient of integers 7 and 1.
√16 – As the square root can be simplified to 4, which is the quotient of fraction 4/1
0.5 – Can be written as 5/10 or 1/2 and all terminating decimals are rational.
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0.3333333333 – All recurring decimals are rational.
Number 5 can be written as 5/1 where both 5 and 1 are integers.
0.5 can be written as ½, 5/10, 25/50 or 10/20 and in the form of all terminating decimals.
√81 is a rational number, as it can be simplified to 9 and can be expressed as 9/1.
0.8888888 is recurring decimals and is a rational number
Properties of Rational Numbers:
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Rational numbers are closed under addition, subtraction, multiplication, and division.
It satisfies commutative and associative property under addition and multiplication.
It obeys distributive property for addition and subtraction.
The results are always a rational number if we multiply, add, or subtract any two rational numbers.
A rational number remains the same if we divide or multiply both the numerator and denominator with the same factor.
If we add zero to a rational number then we will get the same number itself.
Rational numbers are closed under addition, subtraction, and multiplication.
Arithmetic Operations on Rational Numbers
In Maths, arithmetic operations are the basic operations we perform on integers. Let us discuss here how we can perform these operations on
rational numbers, say p/q and s/t.
Addition: When we add p/q and s/t, we need to make the denominator the same. Hence, we get (pt+qs)/qt.
Example: 1/2 + 3/4 = (2+3)/4 = 5/4
Subtraction: Similarly, if we subtract p/q and s/t, then also, we need to make the denominator same, first, and then do the subtraction.
Example: 1/2 – 3/4 = (2-3)/4 = -1/4
Multiplication: In case of multiplication, while multiplying two rational numbers, the numerator and denominators of the rational numbers are
multiplied, respectively. If p/q is multiplied by s/t, then we get (p×s)/(q×t).
Example: 1/2 × 3/4 = (1×3)/(2×4) = 3/8
Division: If p/q is divided by s/t, then it is represented as:
(p/q)÷(s/t) = pt/qs
Example: 1/2 ÷ 3/4 = (1×4)/(2×3) = 4/6 = 2/3
Multiplicative Inverse of Rational Numbers
As the rational number is represented in the form p/q, which is a fraction, then the multiplicative inverse of the rational number is the reciprocal of
the given fraction.
For example, 4/7 is a rational number, then the multiplicative inverse of the rational number 4/7 is 7/4, such that (4/7)x(7/4) = 1
How to identify rational numbers?
To identify if a number is rational or not, check the below conditions.
 It is represented in the form of p/q, where q≠0.
 The ratio p/q can be further simplified and represented in decimal form.
The set of rational numerals:
1. Include positive, negative numbers, and zero
2. Can be expressed as a fraction
Standard Form of Rational Numbers
The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore
the divisor is positive.
For example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend is only one. So we can
say that rational number ⅓ is in standard form.
Positive and Negative Rational Numbers
As we know that the rational number is in the form of p/q, where p and q are integers. Also, q should be a non-zero integer. The rational number
can be either positive or negative. If the rational number is positive, both p and q are positive integers. If the rational number takes the form (p/q), then either p or q takes the negative value. It means that
-(p/q) = (-p)/q = p/(-q).
Now, let’s discuss some of the examples of positive and negative rational numbers.
Positive Rational Numbers
Negative Rational Numbers
If both the numerator and denominator are of
the same signs.
If numerator and denominator are of opposite
signs.
All are greater than 0
All are less than 0
Examples of positive rational numbers: 12/17,
9/11 and 3/5
Examples of negative rational numbers: -2/17,
9/-11 and -1/5.
How to Find the Rational Numbers between Two Rational Numbers?
There are “n” numbers of rational numbers between two rational numbers. The rational numbers between two rational numbers can be found
easily using two different methods. Now, let us have a look at the two different methods.
Method 1:
Find out the equivalent fraction for the given rational numbers and find out the rational numbers in between them. Those numbers should be the
required rational numbers.
Method 2:
Find out the mean value for the two given rational numbers. The mean value should be the required rational number. In order to find more
rational numbers, repeat the same process with the old and the newly obtained rational numbers.
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Irrational Numbers(Q’)
Irrational numbers cannot be written in fraction form, i.e., they cannot be written as the ratio of the two integers. The symbol Q’ represents
irrational numbers
OR
Those numbers which when expressed in decimal form are neither terminating nor repeating decimals are known as irrational numbers
Any number that is not rational is called an irrational number. Irrational number cannot be expressed in the form of fraction
Example
A few examples of irrational numbers are √2, √5, 0.353535…, π, and so on. You can see that the digits in irrational numbers continue for infinity
with no repeating pattern.
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√2 – √2 cannot be simplified and so, it is irrational.
√7/5 – The given number is a fraction, but it is not the only criteria to be called as the rational number. Both numerator and denominator need
to integers and √7 is not an integer. Hence, the given number is irrational.
3/0 – Fraction with denominator zero, is irrational.
π – As the decimal value of π is never-ending, never-repeating and never shows any pattern. Therefore, the value of pi is not exactly equal to
any fraction. The number 22/7 is just and approximation.
0.3131131113 – The decimals are neither terminating nor recurring. So it cannot be expressed as a quotient of a fraction.
5/0 is an irrational number, with the denominator as zero.
π is an irrational number which has value 3.142…and is a never-ending and non-repeating number.
√2 is an irrational number, as it cannot be simplified.
0.212112111…is a rational number as it is non-recurring and non-terminating.
Pi, π
3.14159265358979…
Euler’s Number, e
2.71828182845904…
Golden ratio, φ
1.61803398874989….
1.5=3/2 (ratio)
Rational
pi= 3.14159………..=?/?(no ratio)
irrational
Properties of Irrational Numbers:
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Irrational numbers do not satisfy the closure property.
It obeys commutative and associative property under addition and multiplication.
Irrational Numbers are distributive under addi tion and subtraction.
The addition of an irrational number and a rational number gives an irrational number. For example, let us assume that x is an irrational
number, y is a rational number and the addition of both the numbers x +y gives a rational number z.
Multiplication of any irrational number with any nonzero rational number results in an irrational number. Let us assume that if xy=z is rational,
then x =z/y is rational, contradicting the assumption that x is irrational. Thus, the product xy must be irrational.
The least common multiple (LCM) of any two irrational numbers may or may not exist.
The addition or the multiplication of two irrational numbers may be rational; for example, √2. √2 = 2. Here, √2 is an irrational number. If it is
multiplied twice, then the final product obtained is a rational number. (i.e) 2.
The set of irrational numbers is not closed under the multiplication process, unlike the set of rational numbers.
Are Irrational Numbers Real Numbers?
In Mathematics, all the irrational numbers are considered as real numbers, which should not be rational numbers. It means that irrational
numbers cannot be expressed as the ratio of two numbers. The irrational numbers can be expressed in the form of non-terminating fractions and
in different ways. For example, the square roots which are not perfect squares will always result in an irrational number.
Sum and Product of Two Irrational Numbers
Now, let us discuss the sum and the product of the irrational numbers.
Product of Two Irrational Numbers
Statement: The product of two irrational numbers is sometimes rational or irrational
For example, √2 is an irrational number, but when √2 is multiplied by √2, we get the result 2, which is a rational number.
(i.e.,) √2 x √2 = 2
We know that π is also an irrational number, but if π is multiplied by π, the result is π 2, which is also an irrational number.
(i.e..) π x π = π2
It should be noted that while multiplying the two irrational numbers, it may result in an irrational number or a rational number.
Sum of Two Irrational Numbers
Statement: The sum of two irrational numbers is sometimes rational or irrational.
Like the product of two irrational numbers, the sum of two irrational numbers will also result in a rational or irrational number.
For example, if we add two irrational numbers, say 3√2+ 4√3, a sum is an irrational number.
But, let us consider another example, (3+4√2) + (-4√2 ), the sum is 3, which is a rational number.
So, we should be very careful while adding and multiplying two irrational numbers, because it might result in an irrational number or a rational
number.
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N=22/7 looks like rational as expressed in p/q form but 22/7 = 3.142857……… ,So it is irrational
‘0’ or ‘Zero’ is a rational number. Its reason is that ‘0’ or ‘Zero’ belongs to the set of integers and we have discussed that all
the integers are rational numbers.
Difference between rational and irrational numbers
Rational
Numbers that can be expressed as a ratio of two number (p/q form) are
termed as a rational number.
Rational Number includes numbers, which are finite or are recurring in
nature.
Rational Numbers includes perfect squares such as 4, 9, 16, 25, and
so on
Both the numerator and denominator are whole numbers, in which the
denominator is not equal to zero.
On the basis of square root
Irrational
Numbers that cannot be expressed as a ratio of two numbers are
termed as an irrational number.
These consist of numbers, which are non-terminating and nonrepeating in nature.
Irrational Numbers includes surds such as √2, √3, √5, √7 and so on.
Solvable square roots are rational
insolvable square roots are irrational
e.g √9/4=3/2
Q’ = √2 = 1.41421356……….
Perfect squares are rational numbers
Surds are irrational numbers
e.g 4, √ 49, √ 324, √ 1089 and √ 1369.
e.g The examples of the surds are √2, √3 and √7. After taking the
square roots of these surds, we get 1.41, 1.73 and 2.64 respectively.
1.41, 1.73 and 2.64 are not integers.
On the basis of decimals
o Non terminating (e.g 3. 14159……….= 22/7) irrational
o Non terminating and non recurring e.g
0.1312879……(irrational )
On the basis of decimals
o Terminating (e.g 0.15=15/100(p/q)) so rational
o Non terminating and recuuring e.g
0.13131313…………(rational)
Irrational numbers cannot be written in fractional form.
On the basis of square root
Similarities between Rational and Irrational Numbers
1. Both belong to real numbers
2. There exist rational numbers between any two rational numbers similarly there exist irrational numbers between any two irrational
numbers.
3. The sum of two rational numbers is a rational number and the sum of two irrational numbers is an irrational number.
4. The difference between two rational numbers is a rational number and the difference of two irrational numbers is an irrational number.
Arithmetic Rules For Rational And Irrational Numbers
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The sum of two irrational numbers may be an irrational number or a rational number, as an example (√2+ 4), (π + 2) are irrational
numbers and √2 + (-√2) = 0
The product of an irrational number to a rational number is an irrational number, as an example, 2√5,2π are irrational number.
The product of two irrational number may be a rational or irrational number, as an example √2×–√2=-2,√2×√3 = √6
The product of two identical irrational numbers may be rational or irrational, as an example √2×√2 = 2,
The division of two irrational numbers can be rational or irrational, as an example 2√2/3√2= 2/3 , 2√2/√3 etc.
Find Irrational Numbers Between Two Rational Numbers
It is easy for us to find irrational numbers between two rational numbers. We try to learn this concept with the help of an example. Find irrational
numbers between 3 and 4. We can find the irrational numbers between these two rational numbers by following these steps;
 First of all, we should find squares of the given numbers. In this case, the squares of 3 and 4 are 9 and 16 respectively.
 Secondly, you should find the prime numbers between their squares. The prime numbers between 9 and 16 are 11 and 13.
 By taking the square root of these prime numbers, we get the required irrational numbers. The square roots of 11 and 13 are 3.316624… and
3.6055512… respectively. As 3.316624… and 3.6055512… are non-repeating decimals. That’s why these are irrational numbers.
Key Point
The numbers which are written without denominators are rational numbers. The examples of these kinds of numbers are 8 and 9. These
numbers are written in the form of p/q as 8/1 and 9/1. The numbers whose denominators are 0 are called irrational numbers like 8/0 and 9/0.
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Real numbers (R)
(also called measuring numbers or measurement numbers). Real numbers are the set of all rational and irrational numbers. This includes all
numbers that can be written as a decimal. This includes fractions written in decimal form e.g., 0.5, 0.75 2.35, ⁻0.073, 0.3333, or 2.142857. It also
includes all the irrational numbers such as π, √2 etc. Every real number corresponds to a point on the number line. Real numbers can also be
positive, negative or zero.
All integers are real numbers, but not all real numbers are integers. Real numbers include all the integers, whole numbers, fractions, repeating
decimals, terminating decimals, and so on.
The symbol R represents real numbers.
At the same time, the imaginary numbers are the un-real numbers, which cannot be expressed in the number line and is commonly used to
represent a complex number.
Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by
the symbol “R”. All the natural numbers, decimals and fractions come under this category. See the figure, given below, which shows the
classification of real numerals.
Set of Real Numbers
The set of real numbers consist of different categories, such as natural and whole numbers, integers, rational and irrational numbers. In the table
given below, all these numbers are defined with examples.
Category
Natural
Numbers
Definition
Example
Contain all counting numbers which start
from 1.
All numbers such as 1, 2, 3, 4,5,6,…..…
N = {1,2,3,4,……}
Whole
Numbers
Collection of zero and natural number.
W = {0,1,2,3,…..}
All numbers including 0 such as 0, 1, 2,
3, 4,5,6,…..…
Integers
The collective result of whole numbers
and negative of all natural numbers.
Includes: -infinity (-∞),……..-4, -3, -2, 1, 0, 1, 2, 3, 4, ……+infinity (+∞)
Rational
Numbers
Numbers that can be written in the form
of p/q, where q≠0.
Examples of rational numbers are ½,
5/4 and 12/6 etc.
Irrational
Numbers
All the numbers which are not rational
and cannot be written in the form of p/q.
Irrational numbers are nonterminating and non-repeating in
nature like √2
Properties of Real Numbers:



Real Numbers are commutative, associate, and distributive under addition and multiplication.
Real numbers obey the inverse property.
Additive and multiplicative identity elements of real numbers are 0 and 1, respectively.
There are four main properties which include commutative property, associative property, distributive property and identity property. Consider “m,
n and r” are three real numbers. Then the above properties can be described using m, n, and r as shown below:
Commutative Property
If m and n are the numbers, then the general form will be m + n = n + m for addition and m.n = n.m for multiplication.
 Addition: m + n = n + m. For example, 5 + 3 = 3 + 5, 2 + 4 = 4 + 2
 Multiplication: m × n = n × m. For example, 5 × 3 = 3 × 5, 2 × 4 = 4 × 2
Associative Property
If m, n and r are the numbers. The general form will be m + (n + r) = (m + n) + r for addition(mn) r = m (nr) for multiplication.
 Addition: The general form will be m + (n + r) = (m + n) + r. An example of additive associative property is 10 + (3 + 2) = (10 + 3) + 2.
 Multiplication: (mn) r = m (nr). An example of a multiplicative associative property is (2 × 3) 4 = 2 (3 × 4).
Distributive Property
For three numbers m, n, and r, which are real in nature, the distributive property is represented as:
m (n + r) = mn + mr and (m + n) r = mr + nr.
 Example of distributive property is: 5(2 + 3) = 5 × 2 + 5 × 3. Here, both sides will yield 25.
Identity Property
There are additive and multiplicative identities.
 For addition: m + 0 = m. (0 is the additive identity)
 For multiplication: m × 1 = 1 × m = m. (1 is the multiplicative identity)

Prime Numbers (P)
A prime number is a positive integer having exactly two factors. If p is a prime, then it’s only factors are necessarily 1 and p itself. Any number
which does not follow this is termed as composite numbers, which means that they can be factored into other positive integers.
OR
Prime number is a whole number greater than 1 which are divisible by 1 and itself. Prime number is denoted by P
P={2,3,5,7,11,13,17,19,23,29,…………………….}
Even Prime Number
As we know, the prime numbers are the numbers that have only two factors and the numbers that are evenly divisible by 2 are even numbers.
Therefore, 2 is the only prime number that is even and the rest of the prime numbers are odd numbers, hence called odd prime numbers.
Twin Prime Numbers
The prime numbers that have only one composite number between them are called twin prime numbers or twin primes. The other definition of
twin prime numbers is the pair of prime numbers that differ by 2 only. For example, 3 and 5 are twin primes because 5 – 3 = 2.
The other examples of twin prime numbers are:
 (5, 7)
[7 – 5 = 2]
 (11, 13) [13 – 11 = 2]
 (17, 19) [19 – 17 = 2]
 (29, 31) [31 – 29 = 2]
 (41, 43) [43 – 41 = 2]
 (59, 61) [61 – 59 = 2]
 (71, 73) [73 – 71 = 2]
Coprime Numbers
The pair of numbers that have only one factor in common between them, are called coprime numbers. Prime factors and coprime numbers are
not the same. For example, 6 and 13 are coprime because the common factor between them is 1 only.
Properties of Prime Numbers
Some of the properties of prime numbers are:




Every number greater than 1 can be divided by at least one prime number.
Every even positive integer greater than 2 can be expressed as the sum of two primes.
Except 2, all other prime numbers are odd. In other words, we can say that 2 is the only even prime number.
Two prime numbers are always coprime to each other.

Each composite number can be factored into prime factors and individually all of these are unique in nature.
How to Find Prime Numbers?
The following two methods will help you to find whether the given number is a prime or not.
Method 1:
We know that 2 is the only even prime number. And only two consecutive natural numbers which are prime are 2 and 3. Apart from those, every
prime number can be written in the form of 6n + 1 or 6n – 1 (except the multiples of prime numbers, i.e. 2, 3, 5, 7, 11), where n is a natural
number.
For example:
6(1) – 1 = 5
6(1) + 1 = 7
6(2) – 1 = 11
6(2) + 1 = 13
6(3) – 1 = 17
6(3) + 1 = 19
6(4) – 1 = 23
6(4) + 1 = 25 (multiple of 5)
…
Method 2:
To know the prime numbers greater than 40, the below formula can be used.
n2 + n + 41, where n = 0, 1, 2, ….., 39
For example:
(0)2 + 0 + 0 = 41
(1)2 + 1 + 41 = 43
(2)2 + 2 + 41 = 47
Is 1 a Prime Number?
Conferring to the definition of the prime number, which states that a number should have exactly two factors for it to be considered a prime
number. But, number 1 has one and only one factor which is 1 itself. Thus, 1 is not considered a Prime number.
Examples: 2, 3, 5, 7, 11, etc
In all the positive integers given above, all are either divisible by 1 or itself, i.e. precisely two positive integers.
Smallest Prime Number
The smallest prime number defined by modern mathematicians is 2. To be prime, a number must be divisible only by 1 and the number itself
which is fulfilled by the number 2.
Largest Prime Number
As of January 2020, the largest known prime number is 2^(82,589,933) – 1 a number which has 24,862,048 digits. It was found by the Great
Internet Mersenne Prime Search (GIMPS) in 2018.

Composite Numbers(C)
A composite number has more than two factors, which means apart from getting divided by number 1 and itself, it can also be divided by
at least one integer or number. We don’t consider ‘1’ as a composite number.
OR
Natural numbers greater than 1 which are not prime are composite numbers. Composite numbers denoted by C
Example
4, 6, 8, 9, 12 are composite numbers
12 is a composite number because it can be divided by 1,2,3,4,6 and 12. So, the number ‘12’ has 6 factors.
12/1 = 12
12/2 =6
12/3 =4
12/4 =3
12/6 =2
12/12 =1
Smallest Composite Number
As you have understood already about the composite number, now let us know the smallest number which is composite in nature. If we see the
list of composite numbers, it starts from 4, 6, 8, 9, 10, 12, 14, 15, and so on. So you can see here 4 is the smallest number, which has factors
apart from 1 and itself, such as;
Factors of 4 are 1, 2, and 4.
Prime Numbers and Composite Numbers
Prime Numbers
Composite Numbers
A prime number has two factors
only.
A composite number has more than two factors.
It can be divided by 1 and the
number itself.
For example, 2 is divisible by 1 and
2.
Examples: 2, 3, 7, 11, 109, 113,
181, 191, etc.

Imaginary Numbers(i)
It can be divided by all its factors. For example, 6 is
divisible by 2,3 and 6.
Examples: 4, 8, 10, 15, 85, 114, 184, etc.
Numbers other than real numbers are imaginary or complex numbers. When we square an imaginary number, it gives a negative result, which
means it is a square root of a negative number, for example, √-2 and √-5. When we square these numbers, the results are -2 and -5. The square
root of negative one is represented by the letter i, i.e.
i = √-1
Example 1
What is the square root of -16? Write your answer in terms of the imaginary number i.
Solution

Step 1: Write the square root form.
√(-16)

Step 2: Separate -1.
√(16 × -1)

Step 3: Separate square roots.
√(16) × √(-1)

Step 4: Solve the square root.
4 × √(-1)

Step 5: Write in the form of i.
4i
Sometimes you get an imaginary solution to the equations.
Example 2
Solve the equation,
x2 + 2 = 0
Solution

Step 1: Take the constant term on other side of the equation.
x2 = -2

Step 2: Take the square root on both sides.
√x2 = +√-2 or -√-2

Step 3: Solve.
x = √(2) × √(-1)
x = +√2i or -√2i

Step 4: Verify the answers by plugging values in the original equation and see if we get 0.
x2 + 2
(+√2i)2 + 2 = -2 + 2 = 0
(as i = √-1 and square of i is -1)
(-√2i)2 + 2 = -2 + 2 = 0
(as i = √-1 and square of i is -1)
Just because their name is “imaginary” does not mean they are useless. They have many applications. One of the greatest applications of
imaginary numbers is their use in electric circuits. The calculations of current and voltage are done in terms of imaginary numbers. These
numbers are also used in complex calculus computations. In some places, the imaginary number is also represented by the letter j.
Properties of Imaginary Numbers:
Imaginary Numbers has an interesting property. It cycles through 4 different values each time when it is under multiplication operation.





1×i=i
i × i = -1
-1 × i = -i
-i × i = 1
So, we can write the imaginary numbers as:
i = √1





i2 = -1
i3 = -i
i4 = +1
i4n = 1
i4n-1= -i
Importance of imaginary numbers














Mathematically iota(i) or √-1 may appear in solving the equations like
X2+1=0
X2 =-1
X2=+-√-1
X=+-√-1
X=+-i
Physically iota is used in quantum world (in which particles like electron, proton and their motion is discussed ).specially the motion of electron
around the nucleus in an atom is described by equation involving in “i” (iota)
Like imaginary numbers, complex numbers are also not useless. They are used in many applications like Signals and Systems and Fourier
Transform.
Complex Numbers
An imaginary number is combined with a real number to obtain a complex number. It is represented as a + bi, where the real part and b are the
complex part of the complex number. Real numbers lie on a number line, while complex numbers lie on a two-dimensional flat plane.
Complex numbers represented by C
C= sum of real numbers and imaginary numbers
C=R+ imaginary
Normally expressed as C= a+ib
Where a= real part
b=imaginary part
(iota) I =√-1
Properties of Complex Numbers:
The following properties hold for the complex numbers:
Associative property of addition and multiplication.
Commutative property of addition and multiplication.
Distributive property of multiplication over addition.
Transcendental Numbers
The numbers which can never be the zero (or root) of a polynomial equation with rational coefficients are called transcendental numbers. Not all
irrational numbers are transcendental numbers, but all transcendental numbers are irrational numbers.
Constructing numbers
The numbers we meet at school are generally represented by using combinations of ten number symbols (also called numerals or digits) plus the
symbols ".", "+", and "–" (e.g., 5, 27, 35.8, ⁻4)The ten number symbols we use are:
1 2 3 4 5 6 7 8 9 as well as 0.

Even numbers
Even numbers are those numbers which are divisible by 2 and odd numbers which are not divisible by two”.
OR
“Even numbers are those which when divided by 2 leaves no remainder or as 0 and Odd numbers are those numbers which when divided by 2
leaves a remainder of 1”
Example
Some of the examples of even numbers are:
20, 46, 68, 100, 112, 446, and so on.
Properties of Even
The following are the properties of even and odd numbers:







The sum of two even numbers is an even number
The sum of two odd numbers is an even number
The sum of even and an odd number is an odd number
Even number is divisible by 2, and leaves the remainder 0
An odd number is not completely divisible by 2, and leaves the remainder 1.
An even number ends with 0, 2, 4, 6, and 8
An odd number ends with 1, 3, 5, 7, and 9
Property
Property Name
Operation
Operation Description
Example
Property
1
Property of
Addition
Even + Even =
Even
Adding even and even number will
always result in an even number
14 + 6 =
20
Property
2
Property of
Subtraction
Even – Even =
Even
Subtracting even from even number
will result in an even number.
16 – 6 =
10
Property
3
Property of
Multiplication
Even × Even =
Even
Multiplying even and even number
will always result in an even number.
6 × 4 = 24
Property of Addition
 Adding even and odd (or vice-versa), the resulting number is always odd.
Ex: 8 + 5 = 13,
5 + 18 = 23
 Adding even and even, the resulting number is always even.
Ex: 12+8 = 20
 Adding odd and odd, the resulting number is always even.
Ex: 13 + 9 = 22
Property of Subtraction
 Subtracting even from odd (or vice-versa), the resulting number is always odd.
Ex: 7 – 4 = 3,
10 – 5 = 5
 Subtracting even from even, the resulting number is always even.
Ex: 16 – 6 = 10
 Subtracting odd from odd, the resulting number is always even.
Ex: 21 – 13 = 8
Property of Multiplication
 Multiplying even and even will always result in an even number.
Ex: 6 × 4 = 24,
12 × 4 = 48
 Multiplying even and odd numbers will result in an even number.
Ex: 4 × 5 = 20,
6 × 3 = 18
 Multiplying odd and odd numbers will always give an odd number.
Ex: 3 × 5 = 15,
5 × 9 = 45
How to Check a Number is an Even or Odd Number?
As we know now that “Even numbers those numbers which end with 0,2,4,6,8 and odd numbers are those numbers which end with 1,3,5,7,9.”
So, first, look at the number in the one’s place. This single number will tell whether the entire number is odd or even.
Let us take a number 1131. To check whether the given number is even or odd, first, divide the number by 2.
While dividing the number 1131 by 2, we get the remainder 1. So, the given number 1131 is an odd number.
How to Know If a Number is Even or Odd?
To find out whether the given number is odd or even, you need to check the number in one’s (or unit’s) place. That particular number in one’s
place will tell you whether the number is odd or even.

Even numbers end with 0, 2, 4, 6, 8
 Odd Numbers end with 1, 3, 5, 7, 9
Think about the number 3, 845, 917 which ends with an odd number i.e. 3, 5 and 7. Therefore, the given numbers 3, 845, 917 is an odd number.
Thus the number is not an even number. In the same way, 8, 322 is an even number as it ends with 8 and 2.
What are Even and Odd Decimals?
Decimals are not even or odd numbers because they are not whole numbers. For example, you can’t say that the fraction 1/3 is odd by the fact
that a denominator is an odd number or 12.34 as an even as its last digit is even. Only integers can be even, or odd, meaning decimals and
fractions cannot be even or odd. Zero, however, is an integer and is divisible by two, so it is even. Odd numbers are those numbers not evenly
divisible by two.
Why Zero is an Even Number?
Zero is an even number because it is an integer multiple of 2, specifically 0 × 2.

Odd numbers
The numbers that cannot be divided evenly into groups of two. Odd numbers always end with a digit of 1, 3, 5, 7, or 9
Some of the examples of odd numbers are:
21, 49, 67, 89, 111, 555, 999 and so on.
Properties of odd numbers
Adding Two Odd Numbers
Any odd number added to another odd number always gives an even number. This statement is also proved below.
Odd + Odd = Even
Proof:
Let two odd numbers be a and b.
These numbers can be written in the form where
a = 2k1 + 1
and b = 2k2 + 1 where k1, k2 ∈ Z
Adding a + b we have,
(2k1 + 1) + (2k2 + 1) = 2k1 + 2k2 + 2 = 2(k1 + k2 + 1) which is surely divisible by 2.
Subtracting Two Odd Numbers
When an odd number is subtracted from an odd number, the resultant number will always be an even number. This is similar to adding two odd
numbers where it was proved that the resultant was always an even number.
Odd – Odd = Even
Multiplication of Two Odd Numbers
If an odd number is multiplied by another odd number, the resulting number will always be an odd number. A proof of this is also given below.
Odd × Odd = Odd
Let two odd number be a and b. These numbers can be written in the form where
a = 2k1 + 1 and b = 2k2 + 1 where k1 , k2 ∈ Z
Now, a × b = (2k1 + 1)(2k2 + 1)
So, a × b = 4k1 k2 + 2k1 + 2k2 + 1
The above equation can be re-written as:
a × b = 2(2k1 k2 + k1 + k2) + 1 = 2(x) + 1
Thus, the multiplication of two odd number results is an odd number.
Division of Two Odd Numbers
Division of two odd numbers always results in Odd number if and only if the denominator is a factor of the numerator, or else the number result in
decimal point number.
Odd ⁄ Odd = Odd
In short:
Operation
Result
ODD + ODD
EVEN
ODD – ODD
EVEN
ODD x ODD
ODD
ODD / ODD
*denominator is a factor of the numerator
ODD
Types of Odd Numbers
There are 2 main types of odd numbers which are consecutive odd numbers and composite odd numbers.
Consecutive Odd Numbers
If ‘a’ is an odd number, then ‘a’ and ‘a + 2’ are called consecutive odd numbers. A few examples of consecutive odd numbers can be
 15 and 17
 29 and 31
 3 and 5
 19 and 21 etc.
Even for negative odd numbers, consecutive ones will be:
 -5 and -3
 -13 and -11, etc.
Composite Odd Number
A composite odd number is a positive odd integer which is formed by multiplying two smaller positive integers or multiplying the number with one.
The composite odd numbers up to 100 are: 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99.