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The Real Number System

A free Pre-Algebra lesson from the “Decimals, Roots, and the Real Number System” unit, with a worked example and practice problems including step-by-step solutions.

Rational numbers can be written as a fraction p/q where p and q are integers and q is not zero — their decimals either terminate or repeat. Irrational numbers (like pi and sqrt(2)) cannot be written that way; their decimals go on forever without repeating. Together, rationals and irrationals form the real numbers. In Decimals, Roots, and the Real Number System, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.

What you'll learn

Why it matters: Engineering and construction use rational approximations of irrational values constantly — like 3.14 for pi or 1.414 for sqrt(2). Knowing a number is irrational tells you that the decimal is an approximation, not the exact value.

Worked example

Problem. Is sqrt(2) rational or irrational?

  1. sqrt(2) is approximately 1.41421356... and its decimal never terminates or repeats.
  2. Because it cannot be written as a fraction p/q of integers, sqrt(2) is irrational.
  3. Connect the calculation back to The Real Number System so the method, not just the arithmetic, is clear.
  4. Check the result against the original question before writing the final answer.

Answer: irrational

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. Which number is irrational?

Choices: 0.5 · pi · 3/7 · 0

Show solution
  1. pi is approximately 3.14159... with a non-repeating, non-terminating decimal.
  2. The others can all be written as fractions of integers.

Answer: pi

2. Is 7/3 rational or irrational?

Choices: Rational · Irrational

Show solution
  1. 7/3 is a ratio of two integers with a nonzero denominator.
  2. That is the definition of rational.

Answer: Rational

3. Is sqrt(16) rational or irrational?

Choices: Rational · Irrational

Show solution
  1. sqrt(16) = 4, which is an integer.
  2. Every integer is rational.

Answer: Rational

4. Is the repeating decimal 0.3333... rational or irrational?

Choices: Rational · Irrational

Show solution
  1. Repeating decimals can always be written as fractions.
  2. 0.3333... = 1/3.

Answer: Rational

5. Is sqrt(11) rational or irrational?

Choices: Rational · Irrational

Show solution
  1. 11 is not a perfect square.
  2. Square roots of non-perfect squares are irrational.

Answer: Irrational

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