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 $\frac{p}{q}$, 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.