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
- Distinguish rational numbers from irrational numbers
- Classify numbers as natural, whole, integer, rational, or irrational
- Approximate irrational numbers between consecutive integers
Worked example
Problem. Is sqrt(2) rational or irrational?
- sqrt(2) is approximately 1.41421356... and its decimal never terminates or repeats.
- Because it cannot be written as a fraction p/q of integers, sqrt(2) is irrational.
- Connect the calculation back to The Real Number System so the method, not just the arithmetic, is clear.
- Check the result against the original question before writing the final answer.
Answer: irrational
Practice problems
1. Which number is irrational?
Choices: 0.5 · pi · 3/7 · 0
Show solution
- Warm-up: First identify exactly what the question is asking: Which number is irrational?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- pi is approximately 3.14159... with a non-repeating, non-terminating decimal.
- The others can all be written as fractions of integers.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: pi
2. Is 7/3 rational or irrational?
Choices: Rational · Irrational
Show solution
- Warm-up: First identify exactly what the question is asking: Is 7/3 rational or irrational?
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- 7/3 is a ratio of two integers with a nonzero denominator.
- That is the definition of rational.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Rational
3. Is sqrt(16) rational or irrational?
Choices: Rational · Irrational
Show solution
- Warm-up: First identify exactly what the question is asking: Is sqrt(16) rational or irrational?
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- sqrt(16) = 4, which is an integer.
- Every integer is rational.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Rational
4. Is the repeating decimal 0.3333... rational or irrational?
Choices: Rational · Irrational
Show solution
- Core Practice: First identify exactly what the question is asking: Is the repeating decimal 0.3333... rational or irrational?
- For decimals, keep place value aligned and use estimation to make sure the decimal point is reasonable.
- Repeating decimals can always be written as fractions.
- 0.3333... = 1/3.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Rational
5. Is sqrt(11) rational or irrational?
Choices: Rational · Irrational
Show solution
- Core Practice: First identify exactly what the question is asking: Is sqrt(11) rational or irrational?
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- 11 is not a perfect square.
- Square roots of non-perfect squares are irrational.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Irrational
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