Real Number System Practice: Level 3
Difficulty: 🔴 Challenge | Focus: Proofs, Complex Expressions, & Critical Thinking
Section 1: Mastery Challenges
These problems require deep understanding of number sets and properties.
1. True or False: The sum of two irrational numbers is always irrational?
Click to see Answer & Proof
Answer: False.
Proof: Consider √2 and -√2. Both are irrational, but their sum is 0 (√2 + (-√2) = 0), and 0 is a Rational number.
2. Simplify and Classify: 3(x + 4) – 3x
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Answer: 12.
Classification: Natural, Whole, Integer, Rational, Real.
Step: Using Distributive Property: 3x + 12 – 3x. The 3x and -3x cancel out, leaving 12.
3. Which property justifies the statement: If a = b, then b = a?
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Answer: Symmetric Property of Equality.
Section 2: The Ultimate Challenge
4. Is the number 0.121121112… (where the number of 1s increases each time) Rational or Irrational?
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Answer: Irrational.
Reason: Although there is a pattern, the decimal is non-terminating and non-repeating. To be rational, the same block of digits must repeat infinitely.