Types of Numbers
In mathematics , A number is a value which is used to represent the quantity of an object. We are using numbers in our day to day life.
Types of numbers:
Classified according to the properties and the representation in the number line. there are 8 types of numbers, such as,
Natural numbers
whole numbers
Integers
real numbers
rational numbers
irrational numbers
imaginary numbers
complex numbers
Natural numbers:
Counting numbers are called natural numbers. These numbers consist of positive integers from 1 to infinity. The natural numbers are represented by the letter “N”.
Examples:758 , 234, 45, 890……..
Whole numbers
Whole numbers are also called Natural numbers which start with 0 . whole numbers are positive numbers it does not contain decimal or fraction fractional parts. the whole number is represented by the letter “W”
Examples : 0,345, 566,2929…..
Integers:
Integers are defined as the set of all positive and negative numbers including zero. integers are represented by the letter “Z”.
Examples: -23 , 0 ,3, -45, 67….
Integers - in mathematics, integers are whole numbers and negative numbers. integers does not include fraction, integers and numbers that can be positive, negative or zero, but it cannot be a fraction. The word integer originated from the Latin word “ Integer” which means ( untouched, unchanged, whole) . The symbol of integer is “Z”.
for example; 1,-4, 3, 0,-7
types of integers:
Zero - the number zero is neither positive nor negative
Positive integers - Positive integers are natural numbers, they are also called counting numbers. These numbers usually lie on the right side of the number line.
Negative integers - Negative integers are negative of natural numbers. These numbers usually lie on the left side of the number line.
Number line:
The integer 0 is marked at the centre of the number line.
All the positive integers are marked towards the right of zero.
All the negative integers are marked towards the left of zeo.
Real numbers:
Any numberStatus positive integers, negative integers, decimal numbers or fractional numbers are called real numbers. Real numbers are represented by the letter “R”.
Examples: 43 , 45 , 7.89 , 70, -23…..
Rational numbers:
Any number which is written in a fractional form is known as a rational number.
Rational numbers are represented by the letter “Q”.
Examples: 65, 79 , 44……
Irrational numbers:
Irrational numbers cannot be expressed in the form of fraction. It means a number that cannot be written as the ratio of one over the other. It is represented by the letter “P”.
Examples: 3 ,
Imaginary numbers
Numbers other than real are imaginary or complex numbers. when we square and imaginary number it gives a negative result which mean it is a square root of a negative number.
Examples: -3, -5
Complex numbers:
An imaginary number is combined with the real number to obtain a complex number. It is represented as a+bi, Where the real part and we are the complex part of the complex number. Real numbers lie on the number line while complex numbers lie on a two dimensional flat plane.
Addition and subtraction of integers
rules for addition and subtraction of integers
Adding two positive integer result in positive integers where is adding two negative integer will result in negative integer ( same sign have to be added and for the same sign) for example: +2+3=5 ; -2-3=-5
Addition of two different sign integers will result in subtraction only and the sign of the result will be the same as the larger number. ( different sign have to be subtracted and put the greater number sign) For example: -4+6=+2 ; +4-6=-2
Properties of addition of integers:
Closure property: the sum of any two integers always results in an integer. For example: -15 + 8 = -7 (7 is an integer) ; 18+(-9) = 9 (9 is also an integer)
commutative property: even if we change the order of the integer the total of any two integers remains the same. For example: -23+ 14 = 14+ (-23) = -9
Associative property: when three or more integers are added, the grouping of integers can be interchanged. for example -12 + (11+9) = (-12 + 11) +9 = 8
Additive identity: when an integer is added with zero the result is the integer itself. The additive Identity of an integer is zero. for example: 0+12 = 12
additive inverse: Two Converse integers are the term additive inverse of one another. For example 5+(-5) = 0
properties of subtraction of integers:
Closure property: the difference of any two integers always results in an integer. For example: -12- 9 = -21 (-21 is an integer) ; 5 -(-9) = 14 (14 is an integer)
Commutative property: the difference between any two integers is not equal when its order is changed. for example 7-4 =3 but 4-7 = -3 ; so 7-4 ≠ 4-7.
Associative property: when 3 or more integers are subtracted The result changes if the grouping of the integers are changed. for example: (12-4) -2 = 6 , but 12-(4-2) = 10: so (12-4)-2 ≠ 12-(4-2).
Multiplication of integers :
Multiplication is nothing but a repeated addition but the rules of multiplication of integers are different from the rules of addition.
Possibilities of multiplication of integers:
Multiplication of integers can be done between two positive integers. when we multiply two positive integers the resultant answer is always positive integer. For example: +2 x +3 = +6 : +4 x +3 =+12.
Multiplication of integers can be done between two negative integers. when we multiply two negative integers the resultant answer will always be a positive integer. For example: -2 x -3 = +6; -4 x -3 = +12.
Multiplication of integers can be done between one positive and one negative integer, the resultant answer will always be negative. for example -2 x +3 = -6 ; 4 x -3 = -12.
properties of multiplication of integers:
closure property : when two integers are multiplied the resultant answer would always be an integer. A x b ∈ Z for example; -2 x -3 = -6 .
Commutative property: even if we change the order of the in feature the product remains the same. A x b =b x a. For examples -2 x 3 = 3 x -2 = 6
Associative property: when three or more integers are multiplied the groups of the integers can be interchanged. a x (b x c ) = (a x b ) x c , For example : -3 x ( 4 x-2 ) = (-3 x 4 ) x -2 = 24.
Division of integers: Division of integer means grouping an integer, in a specific number of groups. For example : -6 ÷ 2 = 3.
Possibilities of division of integers:
Division can be done between two positive integers. The resultant answer will always be positive . 14 ÷7 = 2
Division can be done between two negative integers. the resultant answer will always be positive. -14 ÷-7 = 2
Division can be done between a positive integer and a negative integer. the resultant will always be negative. -14 ÷ 7 = -2.
Properties of division of integers:
Closure property: If a and b are two integers then a. a ÷ b it is not always an integer. for example: 2 ÷ 3 = 2/3 ∉ Z
Commutative property: If a and b are two integers, then a ÷ b ≠ b ÷ a . for example ⅔ ≠ 3/2.
Associative property: if a b and c are integers then a ÷( b ÷ c) ≠ ( a ÷ b) ÷ c.
Division by 1: if a is an integer then a ÷ 1 = a, a Is an integer.
for example 3 ÷1 = 3.
Prime numbers/ composite numbers/ even numbers/ odd numbers
We use numbers in Every Walk of life.A life without numbers is difficult. We use numbers for almost everything like time, Money, date, years, mobile numbers and so on. There are two different parts of numbers there
Prime and composite numbers
odd and even numbers
Prime and composite numbers:
Definition of prime numbers:
Prime numbers are natural numbers which have only two factors, ie 1 and the number itself. Every prime number is an odd number except number 2.
For example: 2, 5 ,7 ,11, 13, 17 ,19 ,23…….
Definition of composite numbers:
composite numbers are natural numbers other than prime numbers. composite numbers have more than two factors, these numbers are divisible by other numbers as well.
For example: 4,6,8,9,12,15…….
Facts of prime and composite numbers:
1 is neither Prime nor composite.
2 is the only even prime number
2 is the smallest prime number.
Except 2 all the other prime numbers are odd.
Chart of prime numbers from 1 -100:
Even and Odd numbers :
Even and Odd numbers are set of integers which do not have fractional parts; they can be positive, negative or zero.
Definition of even numbers:
Even numbers are the numbers which are divisible by 2. Even Numbers have the unit place as 0,2,4,6,8. In other words we can tell Even Numbers as the numbers which can be grouped into pairs.
For example: 24,788, 356,294…..
Definition of odd numbers:
In contrast to the even numbers, odd numbers are the set of integers which are not divisible by 2. They have 1,3,5,7,9 in the unit's place. Odd numbers always leave a remainder of 1 by when divided by 2. In other words odd numbers are the numbers which cannot be grouped into pairs, they always have 1 number left out without a pair.
For example: 13,67, 989,247……
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