Maharashtra Board Class 8 Maths part 1 Chapter 1 Rational and Irrational numbers PDF Download

Rational And Irrational Numbers

Let's Recall

We are familiar with Natural numbers, Whole numbers, Integers and Rational numbers.

Natural numbers: 1, 2, 3, 4, ...

Whole numbers: 0, 1, 2, 3, 4, ...

Integers: ..., -4, -3, -2, -1, 0, 1, 2, 3, ...

Rational numbers: \(\frac{-25}{3}\), \(\frac{10}{7}\), -4, 0, 3, 8, \(\frac{32}{3}\), \(\frac{67}{5}\), etc.

Rational numbers: The numbers of the form \(\frac{m}{n}\) are called rational numbers. Here, m and n are integers but n is not zero.

We have also seen that there are infinite rational numbers between any two rational numbers.

Teacher's Note

Rational numbers are numbers that can be written as fractions. Just like you divide a pizza into slices, rational numbers divide one whole number by another.

Exam Trick

Remember: Rational = Ratio = Fraction. Any number that can be written as a fraction is rational.

Points to Remember

Rational numbers have a numerator and denominator.
The denominator can never be zero.
There are infinite rational numbers between any two numbers.
All integers are also rational numbers.

To Show Rational Numbers On A Number Line

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

Let us draw a number line.

We can show the number 2 on a number line.

\(\frac{7}{3} = 7 \times \frac{1}{3}\), therefore each unit on the right side of zero is to be divided in three equal parts. The seventh point from zero shows \(\frac{7}{3}\); or \(\frac{7}{3} = 2 + \frac{1}{3}\), hence the point at \(\frac{1}{3}\) rd distance of unit after 2 shows \(\frac{7}{3}\).

To show \(\frac{2}{3}\) on the number line, first we show \(\frac{2}{3}\) on it. The number to the left of 0 at the same distance will show the number \(\frac{-2}{3}\).

Teacher's Note

To mark fractions on a number line, divide each unit into equal parts. For \(\frac{7}{3}\), divide each unit into 3 parts and count 7 parts from zero.

Exam Trick

Negative numbers are always on the left side of zero, and positive numbers on the right side. Mirror the positive number to find the negative number.

Points to Remember

Always divide the number line into equal parts for fractions.
Count carefully from zero to find the correct position.
Negative fractions are on the left of zero.
Positive fractions are on the right of zero.

Comparison Of Rational Numbers

We know that, for any pair of numbers on a number line the number to the left is smaller than the other. Also, if the numerator and the denominator of a rational number is multiplied by any non zero number then the value of rational number does not change. It remains the same. That is, \(\frac{a}{b} = \frac{ka}{kb}\), (k ≠ 0).

Example 1: Compare the numbers \(\frac{5}{4}\) and \(\frac{2}{3}\). Write using the proper symbol of <, =, >.

Solution: \(\frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}\) and \(\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)

\(\frac{15}{12} > \frac{8}{12}\) therefore \(\frac{5}{4} > \frac{2}{3}\)

Example 2: Compare the rational numbers \(\frac{-7}{9}\) and \(\frac{4}{5}\).

Solution: A negative number is always less than a positive number. Therefore, \(\frac{-7}{9} < \frac{4}{5}\).

To compare two negative numbers, let us verify that if a and b are positive numbers such that a < b, then -a > -b.

2 < 3 but -2 > -3

\(\frac{5}{4} < \frac{7}{4}\) but \(\frac{-5}{4} > \frac{-7}{4}\)

Example 3: Compare the numbers \(\frac{-7}{3}\) and \(\frac{-5}{2}\).

Solution: Let us first compare \(\frac{7}{3}\) and \(\frac{5}{2}\).

\(\frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6}\), \(\frac{5}{2} = \frac{5 \times 3}{2 \times 3} = \frac{15}{6}\)

and \(\frac{14}{6} < \frac{15}{6}\) therefore \(\frac{7}{3} < \frac{5}{2}\) so \(\frac{-7}{3} > \frac{-5}{2}\)

Example 4: \(\frac{3}{5}\) and \(\frac{6}{10}\) are rational numbers. Compare them.

Solution: \(\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}\) therefore \(\frac{3}{5} = \frac{6}{10}\)

The following rules are useful to compare two rational numbers.

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

(1) if a × d < b × c then \(\frac{a}{b} < \frac{c}{d}\)

(2) if a × d = b × c then \(\frac{a}{b} = \frac{c}{d}\)

(3) if a × d > b × c then \(\frac{a}{b} > \frac{c}{d}\)

Teacher's Note

To compare fractions easily, multiply the numerator of one by the denominator of the other. Like comparing prices of two shops to find which is cheaper.

Exam Trick

Use cross multiplication: \(\frac{a}{b}\) and \(\frac{c}{d}\). If a × d > b × c, then the first fraction is bigger. This is faster than making common denominators.

Points to Remember

Negative numbers are always smaller than positive numbers.
When comparing negative numbers, the bigger negative number is actually smaller.
Cross multiply to compare fractions quickly.
If the numerators are the same, the fraction with smaller denominator is bigger.

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