Hello Students!
Before learning any topic in mathematics we should start with it’s basic facts to make our understanding more clear. All the students should be very clear about this chapter as the concept of number system is applied in almost all mathematical chapters. In this chapter we will learn the following facts:
Learning Points:
- Definition of Number
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Types of Numbers with Definition and examples.
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Properties of Numbers.
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Operations of Numbers.
Definition:
A number is a mathematical value that is used for counting or measuring any object by using various mathematical operations like addition, subtraction, multiplication and division. In number system, numerical digits and some mathematical symbols are used for representing number.
Types of Numbers:
The various types of numbers in mathematics are as below:
- Whole Numbers
- Natural Numbers
- Integers
- Rational Numbers
- Irrational Numbers
- Decimal Numbers
- Real Numbers
Whole Number:
All positive numbers from zero to infinity without decimal or fraction are called Whole Numbers. A set of whole number is represented by W, i.e W={0,1,2,3,4,5,6,…………}
The numbers 2.5, 3.45, 2/3 are not whole numbers.
Natural Number:
All positive numbers starting from 1 to infinity (∞) without any decimal or fraction are called Natural Numbers. A set of Natural Number is represented by the symbol N, i.e N = {1,2,3,4,5,6……..∞}. Here we can say that all natural number are called whole numbers but all whole numbers are not natural numbers, but all the whole numbers excluding “0” are natural numbers.
Integers:
A set of all the positive and negative whole numbers including zero is called Integer and is denoted by the symbol Z, i.e Z= {….., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ………}. Here we can say that all Whole Numbers and Natural Numbers are Integers but all whole numbers and natural numbers are not integers.
The above numbers are further sub-divided into the following numbers:-
(a) Prime Number, (b) Composite Number, (c) Even Number and (d) Odd Number
Prime Number: All the positive integers having only two factors (1 and the number itself) are called Prime Numbers. For example 1, 3, 5, 7 etc., i.e when we divide such numbers by 1 and the number itself we get the reminder 0.
Composite Number:All the positive integers having more than two factors are called composite numbers. For example 2, 4, 6, 8, 9, 10, 12, 14, 15 etc.,
Even Numbers: All the integers which are divisible by 2 are called even numbers. For example +2, -2, +4, -4, +6, -6, +8, -8 etc.
Odd Number: All the integers which are not divisible by 2 are called odd numbers. For example +1, -1, +3, -3, +5, -5, +7, -7, +9, -9, +11, -11 etc.
Rational Number:
Any number expressed in the form of fraction or ratio of two integer is called Rational Number. This is also called a fractional number and the set of rational number is represented by the symbol Q.
Here Rational Number Q = p/q , where p is called numerator and q is called denominator and the denominator q is a non zero integer, i.e. q≠0
For Example: 3/4, -3/2, 5, -7 and 0 are rational numbers.
Here we see that all whole numbers, natural numbers and integers are also Rational Numbers as their denominators is 1.
For Example 5/1, 7/2, -3/1, 0/1 etc.
We can express rational numbers in two forms like in ratio forms as 3/4, -2/5 and in decimal forms as 3.5, -2.7 etc.
Types of Rational Numbers: Following are the types of Rational Number.
1. Proper Rational Number : A rational Number where numerator is less than denominator is called proper rational number and the value lies between -1 to 1. For example 2/5, 3/8, -1/5 etc.
2. Improper Rational Number : A rational number where numerator is greater than denominator is called improper rational number and the value is grater than or equal to 1 or less than or equal to -1. For example 5/3, -3/2, 11/4, 5/1 etc.
3. Rational number in the form of Decimal Number : Rational numbers can be expressed in the form of following decimal numbers:-
- Terminating Decimals: When we divide the numerator with denominator of a rational number, if the quotient ends with a finite number of digits after decimal point, then this is called terminating decimal number. For example 1/2 = 0.5, 3/4=0.75 etc.
- Non-terminating Repeating Decimals: If the digits of a quotient repeat after decimal point, then this called non-terminating repeating decimal number. For example, 1/3 = 0.33333333…, 2/3= 0.66666666…., 2/11= 0.18181818…. etc.
- Non-terminating Non-repeating Decimals: In the above case if the digits of quotient after decimal does not end, then this is called non-terminating non-repeating decimal number. For example, 4/17 = 0.2352941176…, 7/19 = 0.3684210526… etc.
Properties of Numbers:
The following are the key properties of number systems which are essential for performing mathematical operations and solving equations.
- Closure Property
- Commutative Property
- Associative Property
- Distributive Property
- Identity Property
- Inverse Property
Now let us define each of the above numbers with their properties using mathematical operations like addition, subtraction, multiplication and division.
Properties of Whole number:
Whole Numbers are used in arithmetic operations like addition, subtraction, multiplication, and division. These numbers do not include negative numbers, decimals, or fractions.
Properties Under Arithmetical Operations
Closure Properties:
Under Addition
The sum of any two whole numbers is also a whole number. Let ‘a’ and ‘b’ be any two whole number, then a+b is also a whole number. This is called closure property of Whole number under addition and we can say that whole number is closed under addition.
For example: 0 and 4 are two whole numbers, hence 0+4=4 is also a whole number. Similarly 3 and 8 are two whole numbers, hence 3+8=11 is also a whole number.
Under Subtraction
For any two whole numbers ‘a’ and ‘b’ the value of a-b is a whole number only if a>b, unless the value of a-b is not a whole number. Hence whole number is closed under subtraction when a>b and not closed under subtraction when a<b.
For example: Let a=4 and b=3 are two whole numbers(where a>b), hence 4-3=1 is a whole number, but if a=3 and b=4, then 3-4=-1 is not a whole number since a<b.
Under Multiplication
Multiplication of any two whole numbers is also a whole number. Let a and b be any two whole numbers then the value of a×b is also a whole number. Hence whole number is closed under multiplication.
For example: 0 and 4 are two whole numbers, hence 0×4=0 is also a whole number. 3 and 8 are two whole numbers, hence 3×8=24 is also a whole number.
Under Division
For any two whole numbers ‘a’ and ‘b’ the value of a÷b in a/b form or in decimal number form is not a whole numbers. Hence whole number is not closed under division.
For example: 2 and 5 are two whole numbers, hence 2÷5=2/5 is not a whole number. Similarly 3 and 8 are two whole numbers, but 3÷8= 3/8 is not a whole number.
I hope the above explanation about closure property of the whole number may help the students to make their understanding clear.