• A number that can be written in the form of \(\dfrac{a}{b}\), is called rational number, where \(a\) and \(b\) are integers and  \(b\neq\) zero.

Rational numbers can be written in different forms.

(1) Rational numbers as fractions:

A rational number is also a fraction, as it can be written in \(\dfrac{a}{b}\) form, where \(b\ne0\).

For example: \(\dfrac{3}{4}\)

(2) Rational numbers as mixed numbers:

Mixed numbers can be converted into improper fractions.

For example: \(3\dfrac{1}{2}\)

\(=\dfrac{(3×2)+1}{2}=\dfrac{6+1}{2}=\dfrac{7}{2}\)

\(\therefore\;\dfrac{7}{2}\) is a rational number.

(3) Rational numbers as ratio:

A ratio can be written as a fraction.

\(\therefore\) It is also a rational number.

For example: \(3:2\) \(=\dfrac{3}{2}\)

\(\dfrac{3}{2}\) is a rational number.

(4) Rational numbers as percent:

A percent can be changed into a fraction by putting it over \(100\).

For example: \(25\,\%\)

\(=\dfrac{25}{100}\)

\(=\dfrac{25\div25}{100\div25}=\dfrac{1}{4}\)

Thus, \(\dfrac{1}{4}\) is a rational number.

• A rational number is in the form of \(\dfrac{a}{b}\) where \(b\neq0\) and, \(a\) and \(b\) are integers.

Example: \(\dfrac{3}{2}\)

• An irrational number can not be written in the form of \(\dfrac{a}{b}\).

Example: \(\sqrt {15}\)

• If we need to represent a decimal as a rational or an irrational number, we should know about the types of decimals.

The types are described as follows:

(1) Terminating decimals: The decimals that end with a finite (limited) number of digits after decimal are known as terminating decimals.

For example: \(0.8,\;0.25,\;0.46374\) etc.

(2) Repeating decimals:  The decimals which do not end but one or more digits keep on repeating after decimal are known as repeating decimals or non-terminating but repeating decimals.

• We can also write a repeating decimal by using a 'bar' \((-)\).

Here, 'bar' represents the repetition.

For example: \(0.6262...=0.\overline{62}\)

• We can only put the bar over the digits which repeat.

\(0.888=0.\overline8\)

\(2.1575757...=2.1\overline{57}\)

(3) Non-terminating and non-repeating decimals:

The decimals which neither end with a finite number of digits nor the digits keep on repeating are called non-terminating and non-repeating decimals.

For example: \(2.42152434...\)

Note : Those squares roots which are not perfect square numbers, always give non-terminating and non-repeating decimals.

For example: \(\sqrt 3=1.732.....\)

• A decimal is a rational number if it is-

1. Terminating or
2. Repeating

• A decimal is an irrational number if it is non-terminating and non-repeating.

• Non-terminating but repeating decimals are the decimals which do not end but one or more digits keep on repeating.
• We can convert a rational number into a decimal by dividing the numerator by the denominator.
• If the remainder always repeats or does not come out to be zero, the quotient will be a non-terminating but repeating decimal.
• Consider the two cases:

Case I: When only \(1\) digit after the decimal repeats.

For example: \(\dfrac{16}{45}\)

Denominator \(45\) cannot be changed into \(10\) or \(100\) or \(1000\,.....\), thus, we should use long division method.

Here, \(25\) is the repeating remainder and \(0.3555\,.....\) is a repeating decimal or repetend.

Thus, \(\dfrac{16}{45}=0.3555\,.....\)

\(=0.3\overline5\)

Case II: When more than \(1\) digit repeat.

For example:

\(\dfrac{1}{11}\)

Here, repeating remainder is \(1\) and \(.090909\,.....\) is a repeating decimal or repetend.

Thus, \(\dfrac{1}{11}=0.090909\,.....\)

\(=0.\overline{09}\)

Here, repeating remainder is \(1\) and \(.090909\,.....\) is a repeating decimal or repetend.

Thus, \(\dfrac{1}{11}=0.90909\,.....\)

\(=0.\overline{09}\)

• We can plot a terminating decimal at the exact place on the number line but in case of non-terminating, it is quite different.
• Consider the following two cases:

Case 1: Terminating decimals on the number line:

Example: \(0.3\)

\(0.3\) means \(3\) tenths.

\(\therefore\) Draw a number line from \(0\) to \(1\) and divide the interval into \(10\) equal segments.

Thus, count \(3\) segments and plot \(0.3.\)

Case 2: Non-terminating decimals on the number line:

Example: \(0.\overline3\)

\(0.\overline3\) means \(0.333\,...\)

\(\therefore\;0.3<0.\overline3<0.4\), which means \(0.\overline3\) lies in between \(0.3\) and \(0.4\) on the number line.

Thus, plotting \(0.\overline3\) between \(0.3\) and \(0.4.\)

• Terminating decimals are the decimals that end with a finite (limited) number of digits after decimal(point).
• We can convert a rational number into a decimal by dividing the numerator by the denominator.
• After dividing, if the remainder comes out to be zero, then the quotient will be a terminating decimal.
• There are two methods of converting a rational number into a decimal.

I. By Using Long Division Method

For example: \(\dfrac{14}{8}\)

Divide \(14\) by \(8.\)

Thus, \(\dfrac{14}{8}\) gives the quotient as \(1.75\).

Note:

This method can be used in both terminating and non-terminating decimals.

II. By Using Equivalent Fraction Method

• This method is used:

(i) When we have \(10\) or \(100\) or \(1000\,.....\) (multiples of \(10\)) in the denominator.

(ii) Fractions having the denominators that can be written in the form of powers of  \(2\,[like\,\,4,8,16,...]\) or \(5\,[like\,\,25,625,...]\) result in terminating decimals.

•  It means that the denominator should have only 2 or 5 or both as its factors. 

For example: \(\dfrac{13}{25}\)

For the multiples of \(10\) in the denominator, we multiply both numerator and the denominator by \(4.\)

\(=\dfrac{13×4}{25×4}\)

\(=\dfrac{52}{100}\)

\(=0.52\)

Thus, \(\dfrac{13}{25}\) can be written as \(0.52.\)

• This method always gives results as terminating decimals.

Note:

If we have a mixed number to be converted into a decimal, consider the following two methods:

Method 1:

First convert it into an improper fraction and then convert it into a decimal.

For example: \(1\dfrac{1}{2}\)

\(=\dfrac{(2×1)+1}{2}\)

\(=\dfrac{2+1}{2}\)

\(=\dfrac{3}{2}\)

\(=1.5\)

Method 2:

Keep aside the whole number and convert only the fraction part into a decimal.

For example: \(-2\dfrac{2}{3}=-\left(2+\dfrac{2}{3}\right)\)

\(=-2-\dfrac{2}{3}\)

\(\implies-\dfrac{2}{3}=-0.66\)

Write the whole number and the decimal part together.

\(=-2.66\)

• As we can convert a rational number into a decimal number, we can also convert a decimal number into a rational number.
• Here, we are converting a terminating decimal into a rational number.
• Consider the following two cases:

Case 1: Terminating decimal as a rational number:

For example: \(0.36\)

According to the place value chart, \(0.36\) is thirty-six hundredths.

So, it can be written as-

\(=\dfrac{36}{100}\)

Now, simplify the fraction obtained.

\(=\dfrac{36\div4}{100\div4}\)

\(=\dfrac{9}{25}\)

\(9\) and \(25\) do not have any common factor other than \(1\).

\(\therefore\;\dfrac{9}{25}\) is in its simplest form.

Thus, \(0.36\) can be written as \(\dfrac{9}{25}\) in the rational form.

Case 2: Terminating decimal as mixed numbers:

For example:

\(6.05\)

According to the place value chart, \(6.05\) is six and five hundredths.

So, it can be written as-

\(6.05=6\dfrac{5}{100}\)

\(=6+\dfrac{5}{100}\)

Now, we simplify only the fraction part i.e. \(\dfrac{5}{100}.\)

\(=\dfrac{5\div5}{100\div5}\)

\(=\dfrac{1}{20}\)

Thus,

\(6+\dfrac{5}{100}=6+\dfrac{1}{20}\)

or \(6\dfrac{5}{100}=6\dfrac{1}{20}\)

or \(6.05=6\dfrac{1}{20}\)

• Negative numbers are always smaller than the positive numbers.
• Let's consider an example to understand the comparison of two rational numbers.

Example: Compare \(1\dfrac{3}{4}\) and \(\dfrac{5}{12}\)

(1) First, convert them into the same form i.e. rational number.

\(1\dfrac{3}{4}=1+\dfrac{3}{4}\)

\(=\dfrac{(1×4)+3}{4}\)

\(=\dfrac{4+3}{4}\)

\(=\dfrac{7}{4}\)

\(\dfrac{5}{12}\) is already a rational number.

(2) For comparing \(\dfrac{5}{12}\) and \(\dfrac{7}{4}\), convert them into their equivalent fractions having the same denominator.

\(\dfrac{7}{4}=\dfrac{7×3}{4×3}=\dfrac{21}{12}\)

\(\dfrac{5}{12}\) already has the denominator

(3) \(21>5\)

\(\therefore\;\dfrac{21}{12}>\dfrac{5}{12}\) 

Note:

On the number line, the number which is to the left of another number is always smaller than the other, even if it looks bigger.

For example:

\(-11\) is at the left of \(-9\).

\(\therefore\;-11<-9\)

• Ordering refers to a method of arranging the numbers either from least to greatest or greatest to least.
• While ordering, first we convert all the numbers into the same form and then compare.
• Here, we are converting all the forms into the rational numbers.
• Let us order the following numbers from least to greatest.

\(3:4,\;50\,\%,\;-12.5,\;2^3\) and \(\dfrac{-8}{6}\)

• Since all the given numbers are in different forms, so first we convert them into the same form, i.e. rational number.

\(3:4\) can be written as  \(\dfrac{3}{4}\)

\(50\,\%\) can be written as \(\dfrac{50}{100}=\dfrac{1}{2}\)

\(-12.5\) can be written as \(\dfrac{-125}{10}=\dfrac{-25}{2}\)

\(2^3\) can be written as \(2^3=\dfrac{8}{1}\)

\(\dfrac{-8}{6}\) can be written as \(\dfrac{-8}{6}=\dfrac{-4}{3}\)

• Here, the denominators of all the rational numbers are not same.
• To compare all the rational numbers, we convert them into their equivalent rational numbers such that all have the same denominator.
• To do so, we calculate L.C.M. of \(4,\;2\) and \(3.\)
• Thus, \(12\) is the common denominator (L.C.M.)

• Now converting all the rational numbers into equivalent rational numbers with denominator as \(12\).

\(\dfrac{3}{4}=\dfrac{3×3}{4×3}=\dfrac{9}{12}\)

\(\dfrac{1}{2}=\dfrac{1×6}{2×6}=\dfrac{6}{12}\)

\(\dfrac{-25}{2}=\dfrac{-25×6}{2×6}=\dfrac{-150}{12}\)

\(\dfrac{8}{1}=\dfrac{8×12}{1×12}=\dfrac{96}{12}\)

\(\dfrac{-4}{3}=\dfrac{-4×4}{3×4}=\dfrac{-16}{12}\)

• Thus, we have numbers as:

\(\dfrac{9}{12},\;\dfrac{6}{12},\;\dfrac{-150}{12},\;\dfrac{96}{12},\;\dfrac{-16}{12}\)

• Now, we arrange them from least to greatest by comparing their numerators.
• So, the order is:

\(\dfrac{-150}{12}<\dfrac{-16}{12}<\dfrac{6}{12}<\dfrac{9}{12}<\dfrac{96}{12}\)

• Thus, the order of original numbers is:

\(-12.5<\dfrac{-8}{6}<50\,\%<3:4<2^3\)