Notes-Class 8-Math-Chapter 1-Rational and Irrational Numbers-Maharashtra Board

Topics to be learn : To show Rational Numbers on a Number line Comparison of Rational Numbers Decimal Representation of Rational Numbers Irrational Numbers.

Number System :

Natural numbers : The counting numbers 1, 2, 3, 4,  are called natural numbers.

Whole numbers : It is a set of natural numbers and zero.

• Ex. :  0, 1, 2, 3, 4, are called whole numbers.

Integers :  It is a set of natural numbers, zero and opposite of all natural numbers.

• Ex. : −4, −3, −2, −1,0,1, 2, 3,  are called integers.

Rational Numbers : If m is any integer and n is any non-zero integer, then the number \(\frac{m}{n}\) is called a rational number.

• Ex. : \(\frac{-17}{3}\), \(\frac{10}{-3}\) −7, 0, 5, \(\frac{27}{4}\) etc. are rational numbers.
• Here we can write 5 as a \(\frac{5}{1}\), therefore we can say 5 is a rational  number.
• There are infinite rational numbers between any two rational numbers.

Example : find the rational numbers between \(\frac{3}{7}\) and \(\frac{4}{7}\)

Now \(\frac{3}{7}=\frac{3×2}{7×2}=\frac{6}{14}\)  and \(\frac{4}{7}=\frac{4×2}{7×2}=\frac{8}{14}\) are equivalent to \(\frac{3}{7}\) and \(\frac{4}{7}\)

So \(\frac{7}{14}\) is the number between \(\frac{6}{14}\) and \(\frac{8}{14}\) i.e between \(\frac{3}{7}\) and \(\frac{4}{7}\)

Simillarly \(\frac{3}{7}=\frac{3×3}{7×3}=\frac{9}{21}\) and \(\frac{4}{7}=\frac{4×3}{7×3}=\frac{12}{21}\)  are equivalent to \(\frac{3}{7}\) and \(\frac{4}{7}\) . Therefore \(\frac{10}{21}\) and \(\frac{11}{21}\) are the number between \(\frac{3}{7}\) and \(\frac{4}{7}\)

Simillarly, \(\frac{13}{28}\), \(\frac{14}{28}\), \(\frac{15}{28}\) are the number between \(\frac{3}{7}\) and \(\frac{4}{7}\)

To show Rational Numbers on a Number line :

1) Let us see how to show \(\frac{9}{4},\frac{3}{4},\frac{-3}{4}\) on a number line.

Here, we have shown \(\frac{9}{4},\frac{3}{4},\frac{-3}{4}\) on the number line.

To show these rational numbers, we divide each unit distance in 4 equal parts, which is the denominator of the rational numbers.

The third point on the right of zero shows \(\frac{3}{4}\).

\(\frac{9}{4}=2\frac{1}{4}=2+\frac{1}{4}\)

So the point next to 2 on the right hand side is \(\frac{9}{4}\)

To show \(\frac{-3}{4}\). on the number line, first we show \(\frac{3}{4}\). on it. The number to the left of zero at the same distance of \(\frac{3}{4}\). will show the number \(\frac{-3}{4}\)..

Similarly, to show any rational number on the number line, divide the unit distance into as many equal parts as the denominators of the rational numbers.

2) Let us see how to show \(\frac{7}{3},2,\frac{-2}{3}\) on a number line.

Here, we have shown \(\frac{7}{3},2 \,\, and \,\frac{-2}{3}\) on the number line.

To show these rational numbers, we divide each unit distance in 3 equal parts, which is the denominator of the rational numbers.

The seventh point on the right of zero shows \(\frac{7}{3}\).

\(\frac{7}{3}=2\frac{1}{3}=2+\frac{1}{3}\)

So the point next to 2 on the right hand side is \(\frac{7}{3}\).

To show \(\frac{-2}{3}\) on the number line, first we show \(\frac{2}{3}\) on it. The number to the left of zero at the same distance of \(\frac{2}{3}\)  will show the number \(\frac{-2}{3}\) .

The sixth point on the right of zero shows \(\frac{6}{3}=2\).

Comparison of Rational Numbers :

• If the numerator and the denominator of a rational number is multiplied by any non-zero number, then the value of the rational number does not change.

That is \(\frac{m}{n}=\frac{m×k}{n×k}\), (k ≠ 0)

• For any pair of numbers on a number line, the number to the left is smaller than the other.
• A negative rational number is always less than a positive rational number.

Ex. \(\frac{-6}{7}<\frac{3}{11}\)

• To compare two negative numbers : If a and b are positive numbers such that a < b, then −a >−b. i.e. 2 < 3 but −2 > −3

Examples :

1) Compare the numbers \(\frac{-7}{3}\) and \(\frac{-5}{4}\)

Let us compare \(\frac{7}{3}\) and \(\frac{5}{4}\)

Let us make equal denominator of both number

\(\frac{7}{3}=\frac{7×4}{3×4}=\frac{28}{12}\) and \(\frac{5×3}{4×3}=\frac{15}{12}\)

\(\frac{28}{12}>\frac{15}{12}\),

∴ \(\frac{7}{3}>\frac{5}{4}\)    , ∴ \(\frac{-7}{3}<\frac{-5}{4}\) 

Rules to compare two rational numbers.

If \(\frac{a}{b}\) and \(\frac{c}{d}\) are rational numbers such that b and d are positive and if

(i) a x d < b x c, then \(\frac{a}{b}<\frac{c}{d}\)

(ii) a x d = b x c, then \(\frac{a}{b}=\frac{c}{d}\)

(iii) a x d > b x c, then \(\frac{a}{b}>\frac{c}{d}\)

Decimal Representation of Rational Numbers :

Terminating decimal form :

The decimal representation of a rational number is obtained by dividing its numerator by the denominator

• The decimal form of \(\frac{3}{5}\)   is 0.6.
• The decimal form of \(\frac{9}{2}\) is 4.5.
• The decimal form of \(\frac{11}{8}\) is 1.375.

In all the above cases, the remainder is zero. Here, the process of division ends.

Such a decimal form of a rational number is called a terminating decimal form.

Non-terminating recurring decimal form :

• \(\frac{5}{6}=0.8333...=0.8\dot 3\)

• \(\frac{-5}{3}=-1.666...=-1.\dot 6\)
• \(\frac{2}{11}=0.181818...=0.\overline{18}\)
• \(\frac{1234}{999}=1.235235235...=1.\overline{235}\)

In the above cases, the process is unending. Here, either a digit or a group of digits is repeated. Such a decimal form of a rational number is called a non-terminating recurring decimal form.

Remember : Similarly, a terminating decimal form can be written as a non-terminating recurring decimal form.

• For example,  \(\frac{7}{4}=1.75=1.7500... = 1.75\dot 0\)

Irrational Numbers :

In addition to rational numbers, there are many more numbers on a number line.

They are not rational numbers, that is, they are irrational numbers. \(\sqrt{2}\) is such an irrational number

Let us represent the number \(\sqrt{2}\)  on a number line.

• Draw a number line. Take a point O corresponding to the number O. Take a point A on the number line. It shows the number 1.
• Draw a line l perpendicular to the number line through the point A.
• Take a point P on the line l such that OA = AP = 1 unit.
• Draw seg OP.
• Δ OAP is a right angled triangle.

By Pythagoras’ theorem,

OP² = OA² + AP² = (1)² + (1)² = 1 + 1 = 2

OP² = 2

∴ OP = \(\sqrt{2}\)  (Taking square roots on both the sides)

• Now, draw an arc with centre O and radius OP.
• Let it intersect the number line at the point Q.
• Distance OQ = \(\sqrt{2}\),
• ∴ the point Q on the number line represents the number \(\sqrt{2}\)
• Point R on the number line to the left of O at the same distance as OQ, will represent the number \(\sqrt{2}\).
• The numbers which can be shown by points on a number line are called real numbers.
• We know that π is not a rational number. It means it is an irrational number.
• All rational numbers and irrational numbers are real numbers.
• We can show various irrational numbers like  on a number line.

Remember :

• The decimal form of an irrational number is non-terminating, non-recurring.
• π is an irrational number. For calculation we take the value of π as \(\frac{22}{7}\) or 3.14.