The World of Numbers — Class 9 MCQs with Answers

Key concepts: The World of Numbers (Class 9 Mathematics)

This chapter completes the number line. Rational numbers are dense but leave gaps; those gaps are filled by irrationals such as √2 and π, and together the two form the real numbers. It also covers converting decimals to fractions, rationalising denominators and the laws of exponents.

Rational number

Any number expressible as p/q with p and q integers and q ≠ 0. Every integer qualifies, since n can be written n/1.

Irrational number

A number that cannot be written as p/q, such as √2, √3 and π. Its decimal expansion never terminates and never settles into a repeating block.

Real numbers

The rationals and irrationals together. Every real number corresponds to exactly one point on the number line, and every point to one real number.

Density of rationals

Between any two distinct rational numbers there is another rational — their mean, for instance — so infinitely many rationals lie between any two.

Terminating decimal

A decimal that stops, such as 0.75. A fraction in lowest terms terminates exactly when its denominator's only prime factors are 2 and 5.

Recurring decimal

A decimal in which a block of digits repeats forever, written with a bar, as in 0.3̄. Every rational number is either terminating or recurring.

Decimal to p/q form

A terminating decimal is written over the matching power of 10 and reduced; a recurring decimal is converted by multiplying by a power of 10 and subtracting to cancel the repeating block.

Surd

An irrational root left in root form, such as √5. Surds are kept exact rather than rounded, so answers stay precise.

Rationalising the denominator

Removing a root from the denominator by multiplying numerator and denominator by a suitable factor — for √a by √a, and for a binomial by its conjugate.

Conjugate

The partner expression obtained by flipping the middle sign, so the conjugate of √5 − 2 is √5 + 2. Multiplying them gives a difference of squares with no root left.

Operations and closure

Rationals are closed under addition, subtraction, multiplication and division by a non-zero number. Irrationals are not: √2 × √2 = 2 is rational.

Laws of exponents

Rules for combining powers with the same base — add exponents when multiplying, subtract when dividing, multiply when raising a power to a power.

Representing roots on the number line

A length such as √2 is constructed geometrically as the hypotenuse of a right triangle with legs 1 and 1, then transferred to the line with compasses.