Class 9 Maths Ganita Manjari Chapter 3 MCQ – The World of Numbers

Class 9 Maths Ganita Manjari Part 1 Chapter 3 builds the number system step by step:

1. Natural numbers (ℕ): The counting numbers {1, 2, 3, …}.
2. Integers (ℤ): {…, −2, −1, 0, 1, 2, …}, including zero and the negative numbers.
3. Rational numbers (ℚ): Numbers that can be written as p/q, where p and q are integers and q ≠ 0.
4. Irrational numbers: Numbers that cannot be written as p/q, such as √2, √5 and π.
5. Real numbers (ℝ): All rational and irrational numbers together, filling the whole number line.
6. Brahmagupta’s rules: a + 0 = a, a − 0 = a, a × 0 = 0 and the product of two debts is a fortune: (−) × (−) = (+).
7. Density: Between any two rational numbers a and b, there is always another one, such as (a + b)/2.
8. Absolute value: |x| is the distance of x from 0 and the distance between a and b is |a − b|.
9. Terminating test: p/q in lowest terms terminates only if q has no prime factors other than 2 and 5.
10. Decimal signature: Rational numbers terminate or repeat; irrational numbers never terminate and never repeat.

Several ideas in this chapter go against what students expect. Here’s what is actually true:

• Myth: 0.999… is slightly less than 1.
  Fact: It is exactly 1. If x = 0.999…, then 10x − x = 9, so x = 1.
• Myth: π = 22/7.
  Fact: 22/7 is only an approximation. π is irrational, which Lambert proved in 1761.
• Myth: A decimal that goes on forever must be irrational.
  Fact: 5/11 = 0.4545… goes on forever but repeats, so it is rational.
• Myth: Every square root is irrational.
  Fact: √81 = 9 is a natural number. Only square roots of non-perfect squares, such as √2 and √12, are irrational.
• Myth: Rational numbers are so dense that they cover the whole number line.
  Fact: There are gaps, such as √2, that only irrational numbers fill.
• Myth: Irrational numbers cannot be shown on the number line.
  Fact: √2 can be constructed with a ruler and compass using a right triangle with sides 1 and 1.
• Myth: Natural numbers are closed under subtraction.
  Fact: 3 − 5 = −2 is not a natural number.
• Myth: You can tell if a decimal terminates only by dividing.
  Fact: Check the prime factors of the denominator. For example, 3/20 has 20 = 2² × 5, so it terminates as 0.15.
• Myth: A rational number has only one form.
  Fact: 1/3, 2/6 and 10/30 are all the same rational number.

Most maths chapters give you rules to follow. This one shows you how those rules were discovered. The notches on the Lebombo bone, the prime-number groupings on the Ishango bone, the Vedic names for powers of 10, the bindu for zero in the Bakhśhālī Manuscript, Brahmagupta’s fortunes and debts and Baudhāyana running into a length that no fraction could measure all mark steps in that story. Each new kind of number arrived because the old ones could not answer a real question. MCQs from this chapter check whether you understand why each kind of number exists and how they fit inside one another, from ℕ to ℤ to ℚ to ℝ. That understanding will support nearly every chapter you study in Classes 10, 11 and 12.

Before an exam, check that you can do each of these without looking at your notes. Place a fraction such as 9/4 or −7/4 on a number line. Find three rational numbers between any two given numbers. Decide from the denominator alone whether a fraction terminates. Convert decimals such as 0.4̄5̄ or 2.35̄7 into p/q form using the 10ⁿ method. Write out the proof that √2 is irrational from memory. Wherever you hesitate, reread that section of the textbook and then attempt the related MCQs again. If a problem from Exercise Sets 3.1 to 3.5 or the End-of-Chapter Exercises still doesn’t make sense, including the square root spiral in Question 16, our NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 3 walk through every answer in full.