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 find rational numbers between two given numbers.
How do we represent rational numbers on a number line?
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.
Example 2: Find two rational number between and . Given .
Hence the two number between and are:
How do we convert a rational number into a decimal representation?
We know that the numerator and the denominator do not have a common factor other than .
In simple terms, just divide the numerator by the denominator to convert a rational number into a decimal.
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.
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.
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.
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.
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.
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.
- The negative of an 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 an irrational number.
- The decimal representation of a rational number is either terminating or repeating.
- Every real number is either a terminating number or a non-terminating recurring number.
- is a irrational number.
- Irrational numbers cannot be represented by points on a number line.