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?

RIRN 1
Source: Wikipedia

Numbers that can be represented in the form \frac{p}{q} , where p and q are integers and q \neq 0 . These numbers are represented as Q . We can also write them as:

Q = \{ \frac{p}{q} :  p, q \in Z \ and \ q \neq 0 \}

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: \frac{-3}{5}   and \frac{4}{5}

Step 1: First draw a number line.

Step 2: Mark 0 on the number line as O

Step 3: Mark 4 and -3 on the number line. Mark them P and Q respectively.

Step 4: Divide segment OP in 5 equal parts. Similarly, divide segment OQ in 5 equal parts.

Step 5: Therefore A represents \frac{4}{5} . Similarly, C will represent \frac{-3}{5}

Example 2: Find two rational number between x and y . Given x < y . 

Between two rational numbers x and y , such that x < y , there is a rational number \frac{x+y}{2} such that x < \frac{x+y}{2} < y . Hence \frac{x+y}{2} is first number.

Now find a rational number between \frac{x+y}{2}  and y , given \frac{x+y}{2} < y . Hence there is a number \frac{\frac{x+y}{2} + y}{2} = \frac{x+3y}{4} between \frac{x+y}{2}  and y

Hence the two number between x and y are: \frac{x+y}{2} and \frac{x+3y}{4} 

x < \frac{x+y}{2} < \frac{x+3y}{4} < y

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 1 .

In simple terms, just divide the numerator by the denominator to convert a rational number into a decimal.

For example, \frac{7}{8} = 0.875 or \frac{35}{16} = 2.1875

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 \frac{1}{3} = 0.3333333... or \frac{1}{6} = 0.166666666... or \frac{20}{11} = 1.81818181... Such representation are written as \frac{1}{3} = 0.\overline{3} or \frac{1}{6} = 0.1\overline{6} or \frac{20}{11} = 1.\overline{81}

 

How to convert a decimal into rational number in the form \frac{p}{q} ?

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 \frac{p}{q} :

Step 1: Determine the number of digits in the decimal part.

Step 2: Remove the decimal from the numerator. Write 1 in denominator along with 0's on the right side of 1 as the number of digits in the decimal part of the number.

Step 3: Reduce the numerator and denominator.

Example 3: Let’s convert 0.675 in the form of \frac{p}{q}

0.675 = \frac{675}{1000} = \frac{27}{40}

0.15 = \frac{15}{100} = \frac{3}{20}

 

(b) Conversion of a non-terminating decimal to the form \frac{p}{q} :

Step 1: Let say the number is x (e.g. x = 0.111111... or x = 1.232323... )

Step 2: Find the number of repeating digits. If the repeating digits are 1 , multiply by 10 . If the repeating digits are 2 , then multiply it by 100 . (e.g. 10x = 1.1111... or 100x = 123.232323... )

Step 3: Subtract the number in Step 1 from the number from Step 2.

Step 4: Solve for x

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) 0.\overline{1} and (ii) 1.\overline{23} in the form of \frac{p}{q} (iii) 0.7\overline{85}

(i) x = 0.1111111

10x = 1.11111

Subtracting 9x = 1 \Rightarrow x = \frac{1}{9}

(ii) x = 1.232323...

100x = 123.232323...

Subtracting 99x = 123 \Rightarrow x = \frac{123}{99}

(iii) x = 0.785858585...

10x = 7.85858585...

1000x = 785.858585...

Subtracting 990x = 778 \Rightarrow x = \frac{778}{990} 7s=2 = \frac{389}{495}

 

How do we know the nature of expansion of rational number? Is it terminating or non-terminating?

Theorem 1: If x is a rational number whose decimal expansion is terminating. Then we can express x in the form of \frac{p}{q} , where p and q are co-primes and the prime factorization of q is of the form 2^m \times 5^n , where m and n are non-negative integers.

Examples 5 (terminating decimals):

\frac{7}{8} = \frac{7}{2^3} = \frac{7 \times 5^3}{2^3 \times 5^3} = \frac{7 \times 125}{(2 \times 5)^3} = \frac{875}{10^3} = 0.875

\frac{2139}{1250} = \frac{2139}{2^1 \times 5^4} = \frac{2139 \times 2^3}{2^1 \times 5^4} = \frac{2139 \times 2^3}{(2 \times 5)^4} = \frac{17112}{10^4} = 1.7112

Theorem 2 (reverse of Theorem 1): If x = \frac{p}{q} is a rational  number such that q is of the form 2^m \times 5^n , where m and n are non-negative integers. Then x has a decimal expansion that terminates after k places of decimals, where k is larger of m and n .

Theorem 3 : If x = \frac{p}{q} is a rational  number such that q is NOT of the form 2^m \times 5^n , where m and n are non-negative integers. Then x has a decimal expansion that is NON-terminating repeating.

Example 6: (i) \frac{5}{3} = 1.66666...   (ii) \frac{17}{6} = 2.833333...    (iii) \frac{1}{7} = 0.142857142857...

In the above examples, the denominator of the rational numbers cannot be expresses in the form 2^m \times 5^n , where m and n are non-negative integers.

IRRATIONAL NUMBERS

What is an irrational number?

A number that cannot be written in the form of \frac{p}{q}, where p and q are both integers and q \new 0.

We can also say that a number is an irrational number if it has a non-terminating and non-repeating decimal representation.

Example 6: \sqrt{2} =1.4142135... or \sqrt{3} = 1.732050807...

 

How do you prove that \sqrt{n} is not a rational number, if n is not a perfect square?

If \sqrt{n} was a rational number, then we can write

\frac{p}{q} = \sqrt{n}

\Rightarrow \frac{p^2}{q^2} = n

\Rightarrow p^2 = nq^2

\Rightarrow n is a factor of p^2

\Rightarrow n is a factor of p

Assume p = nm for some natural number m

Therefore p = nm

\Rightarrow p^2 = n^2 m^2 

\Rightarrow nq^2 = n^2m^2

\Rightarrow q^2 = nm^2

\Rightarrow n is factor of q^2

\Rightarrow n is a factor of q

This means that n is a factor of both p and q . But for the number to be a rational number, p and q 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:

  1. Negative of a irrational number is also an irrational number.
  2. The sum of a rational number and an irrational number is an irrational number.
  3.  The product of a non-zero rational number and an irrational number is an irrational number.
  4. The sum, difference, product and quotient of two irrational numbers need not be an irrational number.

Notes:

  1. Every whole number is NOT a natural number.
  2. Every integer IS a rational number.
  3. Every rational number is NOT an integer.
  4. Every natural number IS a whole number.
  5. Every integer is NOT a whole number.
  6. Every rational number is NOT a whole number.

 

Notes:

  1. Every point on a number line corresponds to a real number which may be either a rational number or irrational number.
  2. The decimal representation of a rational number is either terminating  or repeating.
  3. Every real number is either terminating number or a non-terminating recurring number.
  4.  \pi is a irrational number.
  5. Irrational numbers cannot be represented by points on a number line.
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