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.
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.
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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