Class 9 — Mathematics (NCERT)

Number Systems: Irrational Numbers, Real Numbers, and Laws of Exponents

Class 9

✓To define irrational numbers and differentiate them from rational numbers.
✓To represent irrational numbers on the number line using geometric construction.
✓To understand the concept of real numbers and their representation on the number line.
✓To apply the laws of exponents for real numbers with rational exponents.
✓To simplify expressions involving real numbers and their exponents.

Key concepts

Irrational Numbers

A number 's' is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0. The decimal expansion of an irrational number is non-terminating and non-recurring. Examples include √2, √3, √5, π, and numbers like 0.101101110... .

Real Numbers

The collection of all rational numbers and irrational numbers together forms the set of Real Numbers. It is denoted by R. Every real number is represented by a unique point on the number line, and conversely, every point on the number line represents a unique real number. Hence, the number line is often called the Real Number Line.

Representing Irrational Numbers on the Number Line

Irrational numbers like √2, √3, √5 can be represented on the number line using the Pythagorean theorem. For example, to represent √2, we construct a right-angled triangle with perpendicular sides of length 1 unit each. The hypotenuse will be √(1^2 + 1^2) = √2 units. We then transfer this length to the number line using a compass.

Pythagorean Theorem: Hypotenuse^2 = Base^2 + Perpendicular^2

Laws of Exponents for Real Numbers

For positive real number 'a' and rational exponents 'p' and 'q', the following laws hold true:

a^p * a^q = a^(p+q)\n(a^p)^q = a^(pq)\na^p / a^q = a^(p-q)\na^p * b^p = (ab)^p\na^0 = 1 (where a ≠ 0)\na^(-p) = 1/a^p (where a ≠ 0)\na^(p/q) = (q√a)^p = q√(a^p)

Key facts to remember

1Rational numbers have decimal expansions that are either terminating or non-terminating recurring.
2Irrational numbers have decimal expansions that are non-terminating and non-recurring.
3The set of real numbers (R) is the union of all rational and irrational numbers.
4Every real number corresponds to a unique point on the number line, and vice-versa.
5For any positive real number 'a' and rational exponents 'p' and 'q', the laws of exponents hold true.
6√p is irrational if p is a positive integer that is not a perfect square.
7π is an irrational number, and its approximate value 22/7 or 3.14 is rational.

Worked examples

Example 1

Represent √5 on the number line.

IDraw a number line and mark a point O as 0 and A as 2 units from O.

IIAt A, draw a perpendicular AX. From AX, cut off AB = 1 unit.

IIIJoin OB. By Pythagoras theorem, OB = √(OA^2 + AB^2) = √(2^2 + 1^2) = √(4 + 1) = √5 units.

IVWith O as centre and OB as radius, draw an arc intersecting the number line at point P.

VPoint P represents √5 on the number line.

Answer

The point P on the number line represents √5.

This method can be extended to represent other irrational numbers like √6, √7, etc., by constructing a spiral.

Example 2

Simplify: (i) 2^(2/3) * 2^(1/3) (ii) (1/3^5)^4

I(i) 2^(2/3) * 2^(1/3)

IIUsing the law a^p * a^q = a^(p+q):

III= 2^((2/3) + (1/3))

IV= 2^(3/3)

V= 2^1

VI= 2

VII(ii) (1/3^5)^4

VIIIUsing the law (a^p)^q = a^(pq) and a^(-n) = 1/a^n:

9= (3^(-5))^4

10= 3^((-5) * 4)

11= 3^(-20)

12= 1/3^20

Answer

(i) 2 (ii) 1/3^20

Example 3

Simplify: (i) 7^(1/5) / 7^(1/3) (ii) 13^(1/5) * 17^(1/5)

I(i) 7^(1/5) / 7^(1/3)

IIUsing the law a^p / a^q = a^(p-q):

III= 7^((1/5) - (1/3))

IV= 7^((3 - 5)/15)

V= 7^(-2/15)

VI= 1 / 7^(2/15)

VII(ii) 13^(1/5) * 17^(1/5)

VIIIUsing the law a^p * b^p = (ab)^p:

9= (13 * 17)^(1/5)

10= (221)^(1/5)

Answer

(i) 1 / 7^(2/15) (ii) (221)^(1/5)

Common mistakes

✗Confusing rational numbers with non-terminating recurring decimals for irrational numbers.
✗Incorrectly applying the laws of exponents, especially when bases or exponents are different (e.g., adding exponents when bases are different).
✗Errors in geometric construction for representing irrational numbers on the number line, such as incorrect right-angled triangle dimensions.
✗Assuming that all numbers under a square root are irrational (e.g., √9 is rational, not irrational).
✗Mistakes in handling negative or fractional exponents, like writing a^(-n) as -a^n instead of 1/a^n.

Exam tips

★Practice the geometric construction for representing various irrational numbers (like √2, √3, √5, etc.) on the number line multiple times to ensure accuracy.
★Memorise all the laws of exponents thoroughly and practice solving a variety of problems to master their application.
★Always show complete step-by-step working for simplification problems involving exponents, clearly stating the law used at each step.
★Understand the definitions of rational and irrational numbers clearly to correctly classify given numbers.
★Pay close attention to the signs and fractions when dealing with exponents to avoid calculation errors.

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