Maharashtra Board Class 9 Maths Part I Chapter 2 Real Numbers PDF Download

Real Numbers

In previous classes we have learnt about Natural numbers, Integers and Real numbers.

N = Set of Natural numbers = {1, 2, 3, 4, ...}

W = Set of Whole numbers = {0, 1, 2, 3, 4,...}

I = Set of Integers = {..., -3, -2, -1, 0, 1, 2, 3,...}

Q = Set of Rational numbers = \(\left\{ \frac{p}{q} \mid p, q \in I, q \neq 0 \right\}\)

R = Set of Real numbers.

N ⊆ W ⊆ I ⊆ Q ⊆ R

Order Relation On Rational Numbers

\(\frac{p}{q}\) and \(\frac{r}{s}\) are rational numbers where q > 0, s > 0

(i) If p × s = q × r then \(\frac{p}{q} = \frac{r}{s}\)

(ii) If p × s > q × r then \(\frac{p}{q} > \frac{r}{s}\)

(iii) If p × s < q × r then \(\frac{p}{q} < \frac{r}{s}\)

Properties Of Rational Numbers

If a, b, c are rational numbers then:

Teacher's Note

Rational numbers are numbers that can be written as fractions. In India, when you divide money like rupees into parts, you are using rational numbers every day.

Exam Trick

Remember: The four properties (Commutative, Associative, Identity, Inverse) work for both addition and multiplication. Just learn them once and use them for both operations.

Points To Remember

Rational numbers can be written as fractions \(\frac{p}{q}\) where p and q are integers and q is not zero.
Natural numbers, whole numbers, and integers are all rational numbers.
When you compare two fractions, cross multiply to check which is bigger.
Every rational number has four properties for addition and multiplication.
Zero is the identity for addition, and one is the identity for multiplication.

Decimal Form Of Rational Numbers

The decimal form of any rational number is either terminating or non-terminating recurring type.

Terminating type examples:

\(\frac{2}{5} = 0.4\)

\(-\frac{7}{64} = -0.109375\)

\(\frac{101}{8} = 12.625\)

Non-terminating recurring type examples:

\(\frac{17}{36} = 0.472222... = 0.\overline{472}\)

\(\frac{33}{26} = 1.2692307692307... = 1.\overline{2692307}\)

\(\frac{56}{37} = 1.513513513... = 1.\overline{513}\)

To Express The Recurring Decimal In \(\frac{p}{q}\) Form

Example 1: Express the recurring decimal 0.777.... in \(\frac{p}{q}\) form.

Solution: Let x = 0.777... = \(0.\overline{7}\)

∴ 10x = 7.777... = \(7.\overline{7}\)

∴ 10x - x = \(7.\overline{7} - 0.\overline{7}\)

∴ 9x = 7

∴ x = \(\frac{7}{9}\)

∴ 0.777... = \(\frac{7}{9}\)

Example 2: Express the recurring decimal 7.529529529... in \(\frac{p}{q}\) form.

Solution: Let x = 7.529529... = \(7.\overline{529}\)

∴ 1000x = 7529.529529... = \(7529.\overline{529}\)

∴ 1000x - x = \(7529.\overline{529} - 7.\overline{529}\)

∴ 999x = 7522.0

∴ x = \(\frac{7522}{999}\)

∴ 7.529 = \(\frac{7522}{999}\)

Teacher's Note

To convert recurring decimals to fractions, count the repeating digits and multiply by 10, 100, or 1000. In India, shop prices are often written as decimals like Rs. 5.50, which is easy to convert to fractions.

Exam Trick

Remember: If one digit repeats, multiply by 10. If two digits repeat, multiply by 100. If three digits repeat, multiply by 1000. Then subtract to remove the repeating part.

Points To Remember

Count the number of repeating digits carefully before multiplying.
When you subtract, the repeating part cancels out completely.
Always simplify the final fraction if possible.
Check if the denominator has only factors 2 and 5 for terminating decimals.
If the denominator has other factors, the decimal will be non-terminating and repeating.

Irrational Numbers

The numbers \(\sqrt{2}\) and \(\sqrt{3}\) shown on a number line are not rational numbers. This means they are irrational numbers.

On a number line OA = 1 unit. Point B which is left to the point O is at a distance of 1 unit. The coordinate of point B is -1. The coordinate of point P is \(\sqrt{2}\) and its opposite number \(-\sqrt{2}\) is shown by point C. The coordinate of point C is \(-\sqrt{2}\). Similarly, opposite of \(\sqrt{3}\) is \(-\sqrt{3}\) which is the coordinate of point D.

D C B O A P

-√3 -√2 -1 0 1 √2 √3

Teacher's Note

\(\sqrt{2}\) and \(\sqrt{3}\) cannot be written as simple fractions. In real life, the diagonal of a square with side 1 unit is \(\sqrt{2}\), which is irrational.

Exam Trick

Remember: If a square root is not a perfect square (like \(\sqrt{2}, \sqrt{3}, \sqrt{5}\)), it is always irrational. But \(\sqrt{4} = 2\) and \(\sqrt{9} = 3\) are rational because they are perfect squares.

Points To Remember

Irrational numbers cannot be written as fractions \(\frac{p}{q}\).
Square roots of non-perfect squares are always irrational.
Every point on a number line represents a real number.
Real numbers include both rational and irrational numbers.
Numbers like π, \(\sqrt{2}\), \(\sqrt{3}\) are examples of irrational numbers.

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