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Number Systems Practice

Solve chapter-level practice questions for Number Systems with reveal-only solutions and quick revision support.

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Practice Set 1 — Rational, Irrational, Decimals

Classifying numbers, converting recurring decimals, number line.

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Q1. Is zero a rational number? Justify.
Q2. Find three rational numbers between 3/5 and 4/5.
Q3. Express $0.\overline{47}$ as a fraction in lowest terms.
Q4. Without long division, state whether 17/8 has a terminating decimal expansion.
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Q5. Classify as rational or irrational: $\sqrt{225}$, $\sqrt{5}$, $7.478478\ldots$, $\pi$, $1.101001000\ldots$
Q6. Simplify $(\sqrt{5}+\sqrt{2})^2$.

Practice Set 2 — Surds and Exponents

Rationalising denominators and using the laws of exponents.

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Q1. Rationalise the denominator of $\dfrac{1}{7+3\sqrt{2}}$.
Q2. Evaluate $64^{1/2}$, $32^{1/5}$, and $125^{1/3}$.
Q3. Simplify $2^{2/3} \cdot 2^{1/3}$.
Q4. Simplify $\dfrac{7^{1/2}}{7^{1/4}}$.
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Q5. Find the value of $(216)^{-2/3}$.
Q6. If $\sqrt{2} = 1.414$, find the value of $\dfrac{1}{\sqrt{2}}$ correct to three decimal places.

Quick Q&A Before You Revise

Is every real number rational?

No. Every rational number is real, but numbers like √2 and π are real and irrational. Real numbers are the union of rational and irrational numbers.

Are the square roots of all positive integers irrational?

No. √4 = 2, √9 = 3, √25 = 5 are rational. Only the square root of a positive integer that is not a perfect square is irrational.

What does it mean to rationalise a denominator?

It means to rewrite the fraction so that its denominator has no surd (irrational root). We multiply the numerator and denominator by a suitable factor (often the conjugate).

Why is 0.101001000100001… irrational?

Its decimal expansion never ends and never settles into a repeating block (the number of zeros keeps increasing). A decimal that is non-terminating and non-recurring represents an irrational number.

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