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.