Overview
This lesson explains the types of decimal numbers, focusing on distinguishing between terminating and repeating (non-terminating) decimal quotients, as well as rational and irrational numbers.
Rational and Irrational Numbers
- A rational number can be written as a fraction or decimal and includes all integers and zero.
- Rational numbers as decimals either terminate (end) or repeat in a pattern.
- Examples of rational numbers: -3, 0, 1, 2, 3, 1/1,000, 333/1,000.
- An irrational number cannot be written as a fraction and its decimal form neither terminates nor repeats.
- Square roots of non-perfect squares (e.g., √2, √3) are irrational numbers.
Types of Decimals
- A terminating decimal ends after a finite number of digits (e.g., 0.15, 0.225).
- A repeating decimal continues infinitely, but a group of digits repeats in a pattern (e.g., 0.181818...).
- A non-terminating decimal does not end, and if it does not repeat, it is also irrational.
Examples and Practice
- 3 ÷ 20 = 0.15 (terminating decimal)
- 9 ÷ 40 = 0.225 (terminating decimal)
- 2 ÷ 11 = 0.181818... (repeating decimal)
- 6 ÷ 13 = 0.4615384615... (repeating decimal)
- 5 ÷ 16 = 0.3125 (terminating decimal)
- 5/9 = 0.555... (repeating decimal)
- 3/20 = 0.15 (terminating decimal)
- 2 3/4 = 2.75 (terminating decimal)
- 0.25 as a fraction = 1/4
- 0.333 as a fraction = 333/1,000
Key Terms & Definitions
- Rational Number — a number that can be written as a fraction or terminating/repeating decimal.
- Irrational Number — a number with a decimal that never terminates or repeats and cannot be written as a fraction.
- Terminating Decimal — a decimal that ends after a finite number of digits.
- Repeating Decimal — a decimal with an infinite group of repeated digits.
- Vinculum — a bar placed over digits to indicate they repeat in a decimal.
Action Items / Next Steps
- Classify numbers as rational or irrational in your notebook.
- Solve division problems and identify decimals as terminating or repeating.
- Convert fractions to decimals and vice versa for practice.
- Complete Learning Task 4: Identify the type of decimal in given examples.