SOF IMO · Bite-size lessons

Number Systems

About 6 min · Mathematical Reasoning · English

Time worked out from the lesson's 370 words at 150 words a minute, plus a minute for each question.

Number Systems forms the foundation of algebra and higher mathematics. For SOF IMO, this topic tests your understanding of how real numbers are classified, how decimals behave, and how exponents follow specific laws. Questions may involve identifying number types, proving irrationality, converting fractions to decimals, simplifying expressions with exponents, and rationalizing denominators.

Key ideas

• Natural Numbers (N): Counting numbers {1, 2, 3, ...}. Whole Numbers (W) = N ∪ {0}. Integers (Z) include negative numbers too.
• Rational Numbers (Q): Any number expressible as p/q where p, q are integers and q ≠ 0. Includes terminating and repeating decimals.
• Irrational Numbers: Real numbers that cannot be written as p/q. Their decimal expansions are non-terminating and non-repeating (e.g., √2, π, e).
• Real Numbers (R): The union of all rational and irrational numbers. Every point on the number line represents a real number.
• Terminating Decimals: Rational numbers whose denominators (in simplest form) have only 2 and/or 5 as prime factors. Example: 7/8 = 0.875 terminates.

Watch out for

• Confusing terminating condition: Students think 7/12 terminates because it's a "nice" fraction. Wrong — 12 = 2² × 3 has prime 3, so 7/12 = 0.58333... repeats. Fix: Always factor denominator and check for primes other than 2 or 5.
• Adding exponents when multiplying different bases: Writing 2³ × 3² = 6⁵ is incorrect. Fix: You can only add exponents for the same base. Keep different bases separate: 2³ × 3² = 8 × 9 = 72.
• Forgetting negative sign in a^(−n): Writing 2⁻³ = −8 instead of 1/8. Fix: Negative exponent means reciprocal, not negative number. 2⁻³ = 1/(2³) = 1/8.

Quick reference

• Terminating decimal ↔ denominator = 2^m × 5^n only (in simplest form).
• a^m × a^n = a^(m+n) and (a^m)^n = a^(mn) — know the difference.
• a^(−n) = 1/a^n — negative exponent = reciprocal, not negative.
• Rationalize using conjugate: (a + √b) × (a − √b) = a² − b.
• Proof by contradiction: Assume rational, derive coprime factors both divisible by same prime → contradiction.

Check yourself: 3 questions

This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.

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