Types of Numbers
Natural Numbers
- Set of counting numbers is called the Natural Numbers.
Whole numbers
- Set of Natural numbers plus Zero is called the Whole Numbers
Note:
All natural Number are whole number but all whole numbers are not natural numbers
Examples:
-2 is not a Natural number
0 is a Whole number
Integers
In Number System, Integers is the set of all the whole number plus the negative of Natural Numbers
Note
- Integers contains all the whole number plus negative of all the natural numbers.
- The natural numbers without zero are commonly referred to as positive integers.
- The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer.
- Natural numbers with zero are referred to as non-negative integers.
- The natural numbers form a subset of the integers.
What are Rational Numbers ?
In Number system, a number is called rational number if it can be expressed in the form where p and q are integers ( q> 0).
Example : etc.
- In Number system; every integers, natural and whole number is a rational number as they can be expressed in terms of p/q.
- Rational Numbers are denoted by the alphabet 'Q'.
- ‘Rational’ comes from the word ‘ratio’, and Q comes from the word ‘quotient’.
- There are infinite rational number between two rational numbers.
- They either have termination decimal expression or repeating non terminating decimal expression.So if a number whose decimal expansion is terminating or non-terminating recurring then it is rational.
- The sum, difference and the product of two rational numbers is always a rational number.
- The quotient of a division of one rational number by a non-zero rational number is a rational number.
- Rational numbers satisfy the closure property under addition, subtraction, multiplication and division.
Equivalent Rational Numbers
- By multiplying or dividing the numerator and denominator of a rational number by the same integer, we can obtain another rational number equivalent to the given rational number.
- Numbers are said to be equivalent if they are proportionate to each other.
- We can find infinite number of Equivalent Rational Numbers for a given Rational Number.
Example
Therefore 1/2, 2/4, 4/8 are equivalent to each other as they are equal to each other.
Positive and Negative Rational Numbers
1. Positive Rational Numbers are the numbers whose both the numerator and denominator are positive.
Example: 3/4, 12/24 etc.
2. Negative Rational Numbers are the numbers whose one of the numerator or denominator is negative.
Example: (-2)/6, 36/(-3) etc.
Remark: The number 0 is neither a positive nor a negative rational number.
Rational Numbers on the Number Line
Representation of whole numbers, natural numbers and integers on a number line is done as follows
Rational Numbers can also be represented on a number line like integers i.e. positive rational numbers are on the right to 0 and negative rational numbers are on the left of 0.
Representation of rational numbers can be done on a number line as follows
Rational Numbers in Standard Form
- A rational number is in the standard form if its denominator is a positive integer and there is no common factor between the numerator and denominator other than 1.
- If any given rational number is not in the standard form then we can reduce it to its standard form or the lowest form by dividing its numerator and denominator by their HCF ignoring its negative sign.
Comparison of Rational Numbers
- To compare the two positive rational numbers we need to make their denominator same, then we can easily compare them.
- To make their denominator same, we need to take the LCM of the denominator of both the numbers.
case 2
- To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order.
- To compare, we need to compare them as normal numbers.
- If we have to compare one negative and one positive rational number then it is clear that the positive rational number will always be greater as the positive rational number is on the right to 0 and the negative rational numbers are on the left of 0.
Rational Numbers between Rational Numbers
To find the rational numbers between two rational numbers, we have to make their denominator same then we can find the rational numbers.
Example: Find Rational Numbers between 1 and 2.
There are “infinite” numbers of rational numbers between any two rational numbers.