In Class 8 we studied about Numbers. You could quickly revise Class 8: Lecture Notes on Numbers to refresh your memory.

RATIONAL NUMBERS

*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)

(i)

Subtracting

(ii)

Subtracting

(iii)

Subtracting

*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.

IRRATIONAL NUMBERS

*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

Therefore

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.

*Notes:*

*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.*

*Notes:*

*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.*

*Notes:*

*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.*