In Class 8 we studied about Numbers. You could quickly revise Class 8: Lecture Notes on Numbers to refresh your memory.
What is a rational number?
Numbers that can be represented in the form , where and are integers and . These numbers are represented as . We can also write them as:
Rational numbers also include Natural Numbers, Whole Numbers and Integers. Note: Whole Numbers are also Integers.
We need to learn two simple concepts: 1) how to represent rational numbers on a number line and 2) how to fine rational numbers between two given numbers.
How do we represent rational numbers on a number line?
Example 1: Represent on a number line: and
Step 1: First draw a number line.
Step 2: Mark on the number line as
Step 3: Mark and on the number line. Mark them and respectively.
Step 4: Divide segment in equal parts. Similarly, divide segment in equal parts.
Step 5: Therefore represents . Similarly, will represent
Example 2: Find two rational number between and . Given .
Between two rational numbers and , such that , there is a rational number such that . Hence is first number.
Now find a rational number between and , given . Hence there is a number between and
Hence the two number between and are: and
How do we convert a rational number into a decimal representation?
We know that the numerator and the denominator do not have common factor other than .
In simple terms, just divide the numerator by the denominator to convert a rational number into a decimal.
For example, or
In the above two examples, we see that the division eventually terminates. Such rational numbers are called finite or terminating decimals.
But then there are rational numbers non-terminating but the repeating decimals. For example or or Such representation are written as or or
How to convert a decimal into rational number in the form ?
There can be two scenarios… 1) when the decimal number is of terminating nature and 2) when the decimal number is of non-terminating nature.
(a) Conversion of a terminating decimal to the form :
Step 1: Determine the number of digits in the decimal part.
Step 2: Remove the decimal from the numerator. Write in denominator along with on the right side of as the number of digits in the decimal part of the number.
Step 3: Reduce the numerator and denominator.
Example 3: Let’s convert in the form of
(b) Conversion of a non-terminating decimal to the form :
Step 1: Let say the number is (e.g. or )
Step 2: Find the number of repeating digits. If the repeating digits are , multiply by . If the repeating digits are , then multiply it by . (e.g. or )
Step 3: Subtract the number in Step 1 from the number from Step 2.
Step 4: Solve for
Note: In case all digits are not recurring, bring the non recurring digits to the left of the decimal point by multiplying with appropriate multiple of 10. (iii) is such a case.
Example 4: Express (i) and (ii) in the form of (iii)
How do we know the nature of expansion of rational number? Is it terminating or non-terminating?
Theorem 1: If is a rational number whose decimal expansion is terminating. Then we can express in the form of , where and are co-primes and the prime factorization of is of the form , where and are non-negative integers.
Examples 5 (terminating decimals):
Theorem 2 (reverse of Theorem 1): If is a rational number such that is of the form , where and are non-negative integers. Then has a decimal expansion that terminates after k places of decimals, where is larger of and .
Theorem 3 : If is a rational number such that is NOT of the form , where and are non-negative integers. Then has a decimal expansion that is NON-terminating repeating.
Example 6: (i) (ii) (iii)
In the above examples, the denominator of the rational numbers cannot be expresses in the form , where and are non-negative integers.
What is an irrational number?
A number that cannot be written in the form of , where and are both integers and .
We can also say that a number is an irrational number if it has a non-terminating and non-repeating decimal representation.
Example 6: or
How do you prove that is not a rational number, if is not a perfect square?
If was a rational number, then we can write
is a factor of
is a factor of
Assume for some natural number
is factor of
is a factor of
This means that is a factor of both and . But for the number to be a rational number, and should be co-primes, i.e. they should have no common factor. Hence are starting assumption is wrong. Hence the number is not a rational number. It is a Irrational Number.
- Negative of a irrational number is also an irrational number.
- The sum of a rational number and an irrational number is an irrational number.
- The product of a non-zero rational number and an irrational number is an irrational number.
- The sum, difference, product and quotient of two irrational numbers need not be an irrational number.
- Every whole number is NOT a natural number.
- Every integer IS a rational number.
- Every rational number is NOT an integer.
- Every natural number IS a whole number.
- Every integer is NOT a whole number.
- Every rational number is NOT a whole number.
- Every point on a number line corresponds to a real number which may be either a rational number or irrational number.
- The decimal representation of a rational number is either terminating or repeating.
- Every real number is either terminating number or a non-terminating recurring number.
- is a irrational number.
- Irrational numbers cannot be represented by points on a number line.