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?

Source: Wikipedia

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

\displaystyle 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 find rational numbers between two given numbers.

How do we represent rational numbers on a number line?

\displaystyle \text{Example 1: Represent on a number line:  } \frac{-3}{5}  \text{ and } \frac{4}{5} 

Step 1: First draw a number line.

Step 2: Mark \displaystyle 0 on the number line as \displaystyle O

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

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

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

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

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

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

Hence the two number between \displaystyle x and \displaystyle y are:

\displaystyle \frac{x+y}{2}  \text{ and } \frac{x+3y}{4}   

\displaystyle 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 a common factor other than \displaystyle 1 .

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

\displaystyle \text{For example, }  \frac{7}{8} = 0.875 \text{ 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.

\displaystyle \text{For example } \frac{1}{3} = 0.3333333... \text{ or } \frac{1}{6} = 0.166666666... \text{ or } \frac{20}{11} = 1.81818181...

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

 

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

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.

\displaystyle \text{(a) Conversion of a terminating decimal to the form  } \frac{p}{q}  \text{ : }

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

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

Step 3: Reduce the numerator and denominator.

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

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

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

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

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

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

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

Step 4: Solve for \displaystyle 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.

\displaystyle \text{Example 4: Express (i) }  0.\overline{1}  \text{ and } (ii)  1.\overline{23}  \text{in the form of }  \frac{p}{q}  \text{ (iii) } 0.7\overline{85}

(i) \displaystyle x = 0.1111111

\displaystyle 10x = 1.11111

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

(ii) \displaystyle x = 1.232323...

\displaystyle 100x = 123.232323...

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

(iii) \displaystyle x = 0.785858585...

\displaystyle 10x = 7.85858585...

\displaystyle 1000x = 785.858585...

\displaystyle \text{Subtracting } 990x = 778 \Rightarrow x = \frac{778}{990} = \frac{389}{495}   

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

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

Examples 5 (terminating decimals):

\displaystyle \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   

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

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

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

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

IRRATIONAL NUMBERS

What is an irrational number?

A number that cannot be written in the form of \displaystyle \frac{p}{q}  , where \displaystyle p and \displaystyle q are both integers and \displaystyle q \neq 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: \displaystyle \sqrt{2} =1.4142135... or \displaystyle \sqrt{3} = 1.732050807...

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

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

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

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

\displaystyle \Rightarrow p^2 = nq^2

\displaystyle \Rightarrow n is a factor of \displaystyle p^2

\displaystyle \Rightarrow n is a factor of \displaystyle p

Assume \displaystyle p = nm for some natural number \displaystyle m

Therefore \displaystyle p = nm

\displaystyle \Rightarrow p^2 = n^2 m^2

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

\displaystyle \Rightarrow q^2 = nm^2

\displaystyle \Rightarrow n is factor of \displaystyle q^2

\displaystyle \Rightarrow n is a factor of \displaystyle q

This means that \displaystyle n is a factor of both \displaystyle p and \displaystyle q . But for the number to be a rational number, \displaystyle p and \displaystyle 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. The negative of an 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 an irrational number.
  2. The decimal representation of a rational number is either terminating or repeating.
  3. Every real number is either a terminating number or a non-terminating recurring number.
  4.  \displaystyle \pi is a irrational number.
  5. Irrational numbers cannot be represented by points on a number line.