Case Study for Class 9 Maths Chapter 3 The World of Numbers

The Case Study Questions for Class 9 Maths Chapter 3 The World of Numbers include short, real life problem situations that have clear answers and step by step solutions to help students gain confidence for exams. It covers important topics including identifying types of numbers (natural, whole, integers, rational, irrational), locating numbers on a number line, understanding properties of real numbers, writing fractions as decimals, simplifying radical expressions, applying exponent rules, and identifying terminating and non-terminating decimals. These practice questions help the students in better understanding of the concepts, handling the numbers smoothly and to be faster and accurate for their board exams. A free PDF is included for offline timed practice.

Introduction to Case Study on The World of Numbers

Natural Numbers and Whole Numbers

Natural numbers are the counting numbers starting from 1: 1, 2, 3, 4… Whole numbers include zero along with all natural numbers: 0, 1, 2, 3… Every natural number is a whole number, but zero is a whole number that is not a natural number.

Integers and Rational Numbers

Integers include all whole numbers and their negative counterparts: …−3, −2, −1, 0, 1, 2, 3… A rational number is any number expressible as p/q where p and q are integers and q ≠ 0. Every integer is rational (write n as n/1), but not every rational number is an integer.

Irrational Numbers

Irrational numbers cannot be written as p/q for any integers. Their decimal expansions are non-terminating and non-repeating. The most commonly tested irrational numbers in Class 9 are √2, √3, √5, √7, π, and e.

Real Numbers and Their Properties

Real numbers = all rational numbers ∪ all irrational numbers. Every point on the number line is a real number. The real number system is closed under addition, subtraction, multiplication, and division (except by zero).

Case Study 1: Understanding Rational and Irrational Numbers

Priya is helping her mother choose square tiles for their kitchen floor. She finds two types of tiles at the shop. Tile A has an area of 25 cm² and Tile B has an area of 7 cm². She wants to know whether the side length of each tile is a rational or an irrational number.

Questions:

(i) What is the side length of Tile A? Is it rational or irrational?

(ii) What is the side length of Tile B? Is it rational or irrational?

(iii) Tile A's side length can be written as p/q. What are the values of p and q?

(iv) What type of decimal expansion does the side of Tile B have?

(v) Are both side lengths real numbers? Justify your answer.

Solution:

(i) Side of Tile A = √25 = 5 cm. Since 5 = 5/1, and both 5 and 1 are integers with denominator ≠ 0, it is a rational number.

(ii) Side of Tile B = √7 = 2.6457513… This decimal never ends and never repeats. It cannot be written as p/q. It is an irrational number.

(iii) Side of Tile A = 5 = 5/1. So p = 5 and q = 1.

(iv) The side of Tile B (√7) has a non-terminating, non-repeating decimal expansion the defining characteristic of all irrational numbers.

(v) Yes. Both are real numbers because the real number system includes all rational numbers and all irrational numbers. Every measurable length is a real number.

Case Study 2: Number Representation on the Number Line

Rahul's maths teacher draws a number line on the board and marks several points. She asks the class to identify the type of each marked number and verify whether each point is correctly placed.

Questions:

(i) Which of the five points represent irrational numbers?

(ii) Point Q is at 0. To which number sets does 0 belong?

(iii) Point R is at ¾. What kind of decimal does ¾ produce?

(iv) Between which two consecutive integers does √2 lie?

(v) Can all five points be called real numbers? Why?

Solution:

(i) Points S (√2) and T (π) are irrational numbers. Their decimal expansions never terminate and never repeat.

(iI) Zero belongs to whole numbers, integers, rational numbers (0 = 0/1), and real numbers. It does not belong to natural numbers.

(iii) ¾ = 0.75. This is a terminating decimal it ends after two decimal places. All fractions with denominators that have only 2 and 5 as prime factors produce terminating decimals.

(iv) 1² = 1 and 2² = 4. Since 1 < 2 < 4, we get 1 < √2 < 2. So √2 lies between 1 and 2. More precisely, √2 ≈ 1.414, which is closer to 1.

(v) Yes. All five integers, whole numbers, rational, and irrational numbers are subsets of real numbers. Every number that can be placed on the number line is a real number.

Case Study 3: Decimal Expansions of Rational Numbers

A science teacher gives students five numbers and asks them to classify each based on the behaviour of their decimal expansion. The numbers are: 1/4, 1/3, 7/11, √5, and 22/7.

Questions:

(i) What is the decimal expansion of 1/4? What type is it?

(ii) What is the decimal expansion of 1/3? What type is it?

(iII) Is 22/7 exactly equal to π? Explain.

(iv) Which of the five numbers is irrational?

(v) How can you identify a rational number from its decimal expansion alone?

Solution:

(i) 1/4 = 0.25. This is a terminating decimal it ends after a finite number of digits. It is rational.

(ii) 1/3 = 0.333… = 0.3̄. This is a non-terminating repeating decimal the digit 3 repeats forever in a pattern. It is rational.

(iii) No. 22/7 = 3.142857142857… (a repeating decimal), while π = 3.14159265358… (non-terminating, non-repeating). They are close in value but are entirely different numbers. π is irrational; 22/7 is rational and is only an approximation of π.

(iv) √5 is the only irrational number among the five. Its decimal is 2.2360679…non-terminating and non-repeating.

(v) If the decimal either terminates (ends) or has a repeating block of digits, the number is rational. If it goes on forever with no repeating block whatsoever, it is irrational.

Case Study 4: Real Life Applications of Irrational Numbers

An architect is designing a circular fountain and a square park. The fountain has radius 7 metres. The square park has area 50 square metres. She needs to find the circumference of the fountain and the side of the park.

Questions:

(i) Write the formula for the circumference of a circle. Is π rational or irrational?

(ii) What is the side length of the square park?

(iii) Is the side of the square park a rational or irrational number?

(iv) The architect uses π ≈ 22/7. Does this make the circumference exactly correct?

(v) Name two other real-life situations where irrational numbers appear.

Solution:

(i) Circumference = 2πr. π is irrational its decimal expansion is 3.14159265… with no repeating pattern.

(iI) Area = side² = 50, so side = √50 = √(25 × 2) = 5√2 metres.

(iii) 5√2 is irrational. Since √2 is irrational and 5 is a non-zero rational number, their product 5√2 is also irrational. (A non-zero rational multiplied by an irrational is always irrational.)

(iv) No. Using 22/7 gives an approximation, not the exact answer, because 22/7 ≠ π. The result is accurate enough for practical construction but not mathematically exact.

(v) Irrational numbers appear in: the diagonal of a square with side 1 (= √2), the ratio of circumference to diameter of any circle (= π), the golden ratio in art and nature (φ = (1 + √5)/2), and the side of an equilateral triangle (involving √3).

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