Types of Numbers
Natural Numbers
Counting numbers 1, 2, 3, 4, 5,.....are known as natural numbers. The set of all natural
numbers can be represented by : N = {1, 2, 3, 4, 5,....}
Whole Numbers
If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5..... are called
whole numbers. The set of whole numbers can be represented by : W = {0, 1, 2, 3, 4, 5, .... }
Clearly, every natural number is a whole number but 0 is also a whole number which is
not a natural number.
Integers
All counting numbers and their negatives including zero are known as integers. The set of
integers can be represented by: Z or I = {....., –4, –3, –2, –1, 0, 1, 2, 3, 4,......}
Positive integers
+
The set I = {1, 2, 3, 4,....} is the set of all positive integers. Clearly, positive integers and
natural numbers are synonyms.
Negative integers
–
The set I = {....., –3, –2, –1} is the set of all negative integers.
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, 🟋 0 is neither negative nor positive.
Rational numbers
The numbers of the form , where p and q are integers and q≠0, are known as rational
numbers, e.g. etc. The set of all rational numbers is denoted by Q.
i.e.
Since every natural number 'a' can be written as , so 'a' is a rational number. Since 0
can be written as and every non-zero integer 'a' can be written as , so 0 and every
non zero integer is also a rational number.
Every rational number has a peculiar characteristic that when expressed in decimal form
is expressible either in terminating decimals or non-terminating repeating decimals.
For example : =3.1428714287, ...., etc.
The recurring decimals have been given a short notation as
0.333...=
4.1555...=
0.323232...=
Irrational numbers
Those numbers which when expressed in decimal form are neither terminating nor
repeating decimals are known as irrational numbers, e.g. π etc.
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, ⚫ Note, that the exact value of π is not is rational, while π is an irrational
number. is approximate value of π. Similarly, 3.14 is not an exact value of π.
Real numbers
The rational and irrational numbers combined together are called real numbers, e.g.
, etc. are real numbers. The set of real numbers is denoted by R.
⚫ Note that, the sum, difference or non-zero product of a rational and irrational
number is irrational.
e.g. 3 + ,4– , are all irrational.
🟋 Prime numbers
Except 1, each natural number which is divisible by only 1 and itself is called as
prime number e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,.....etc.
⚫ There are total 25 prime numbers upto 100 and 45 upto 200.
⚫ 2 is the only even prime number and the least prime number.
⚫ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
are prime numbers upto 100.
🟋 Co-prime
A pair of two natural numbers having no common factor, other than 1, is called a
pair of co-prime.
For example: (3, 5), (4, 5), (5, 6), (7, 9), (6, 7) etc, are co-primes.
🟋 Twin primes
Prime numbers differing by 2 are called twin primes, e.g. (3, 5), (5, 7), (11, 13) etc,
are called twin primes.
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Natural Numbers
Counting numbers 1, 2, 3, 4, 5,.....are known as natural numbers. The set of all natural
numbers can be represented by : N = {1, 2, 3, 4, 5,....}
Whole Numbers
If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5..... are called
whole numbers. The set of whole numbers can be represented by : W = {0, 1, 2, 3, 4, 5, .... }
Clearly, every natural number is a whole number but 0 is also a whole number which is
not a natural number.
Integers
All counting numbers and their negatives including zero are known as integers. The set of
integers can be represented by: Z or I = {....., –4, –3, –2, –1, 0, 1, 2, 3, 4,......}
Positive integers
+
The set I = {1, 2, 3, 4,....} is the set of all positive integers. Clearly, positive integers and
natural numbers are synonyms.
Negative integers
–
The set I = {....., –3, –2, –1} is the set of all negative integers.
[ 63 ]
, 🟋 0 is neither negative nor positive.
Rational numbers
The numbers of the form , where p and q are integers and q≠0, are known as rational
numbers, e.g. etc. The set of all rational numbers is denoted by Q.
i.e.
Since every natural number 'a' can be written as , so 'a' is a rational number. Since 0
can be written as and every non-zero integer 'a' can be written as , so 0 and every
non zero integer is also a rational number.
Every rational number has a peculiar characteristic that when expressed in decimal form
is expressible either in terminating decimals or non-terminating repeating decimals.
For example : =3.1428714287, ...., etc.
The recurring decimals have been given a short notation as
0.333...=
4.1555...=
0.323232...=
Irrational numbers
Those numbers which when expressed in decimal form are neither terminating nor
repeating decimals are known as irrational numbers, e.g. π etc.
[ 64 ]
, ⚫ Note, that the exact value of π is not is rational, while π is an irrational
number. is approximate value of π. Similarly, 3.14 is not an exact value of π.
Real numbers
The rational and irrational numbers combined together are called real numbers, e.g.
, etc. are real numbers. The set of real numbers is denoted by R.
⚫ Note that, the sum, difference or non-zero product of a rational and irrational
number is irrational.
e.g. 3 + ,4– , are all irrational.
🟋 Prime numbers
Except 1, each natural number which is divisible by only 1 and itself is called as
prime number e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,.....etc.
⚫ There are total 25 prime numbers upto 100 and 45 upto 200.
⚫ 2 is the only even prime number and the least prime number.
⚫ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
are prime numbers upto 100.
🟋 Co-prime
A pair of two natural numbers having no common factor, other than 1, is called a
pair of co-prime.
For example: (3, 5), (4, 5), (5, 6), (7, 9), (6, 7) etc, are co-primes.
🟋 Twin primes
Prime numbers differing by 2 are called twin primes, e.g. (3, 5), (5, 7), (11, 13) etc,
are called twin primes.
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