Rational and irrational numbers are two major types of real numbers, and the difference comes from whether a number can be written as a ratio of integers. This distinction matters because it explains why some decimals end or repeat while others continue forever without a pattern. Fractions, decimals, square roots, and constants like pi all fit into this classification.
Understanding the split helps students compare values, simplify expressions, and recognize exact versus approximate answers.
A rational number can always be written in the form a/b, where a and b are integers and b is not zero. Its decimal form either terminates, like 0.75, or repeats, like 0.333.... An irrational number cannot be written as a fraction of integers, and its decimal expansion is nonterminating and nonrepeating.
Numbers such as pi and sqrt(2) are irrational because their decimal digits continue forever with no repeating block.
Key Facts
- A rational number has the form a/b, where a and b are integers and b != 0.
- Terminating decimals are rational, such as 0.8 = 8/10 = 4/5.
- Repeating decimals are rational, such as 0.333... = 1/3.
- Irrational decimals are nonterminating and nonrepeating, such as pi = 3.14159...
- sqrt(n) is irrational when n is not a perfect square, such as sqrt(2) and sqrt(5).
- All rational and irrational numbers are real numbers, but no number can be both rational and irrational.
Vocabulary
- Rational number
- A number that can be written as a fraction a/b, where a and b are integers and b is not zero.
- Irrational number
- A real number that cannot be written as a fraction of two integers.
- Terminating decimal
- A decimal that ends after a finite number of digits.
- Repeating decimal
- A decimal with a digit or block of digits that repeats forever.
- Perfect square
- A number that is the square of an integer, such as 1, 4, 9, 16, or 25.
Common Mistakes to Avoid
- Calling every long decimal irrational is wrong because some long decimals repeat and can be written as fractions.
- Thinking pi equals 3.14 is wrong because 3.14 is only an approximation of pi, not its exact value.
- Assuming every square root is irrational is wrong because square roots of perfect squares are integers, such as sqrt(49) = 7.
- Forgetting that integers are rational is wrong because any integer n can be written as n/1.
Practice Questions
- 1 Convert 7/8 to a decimal and classify it as rational or irrational.
- 2 Write 0.666... as a fraction and classify the number.
- 3 Explain why sqrt(36) is rational but sqrt(37) is irrational.