Which number is rational?
1. -2.1010010001…
2. -0.8974512…
3. 1.2547569…
4. 5.3333333…
The correct answer and explanation is:
The number that is rational among the options is:
4. 5.3333333…
Explanation:
A rational number is any number that can be expressed as a fraction pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0. Rational numbers have decimal expansions that either terminate (end after a finite number of digits) or repeat a pattern indefinitely (repeating decimals).
Let’s analyze each number:
- -2.1010010001… This decimal appears to have a non-repeating, irregular pattern: the digits after the decimal are 1,0,1,0,0,1,0,0,0,1… with no obvious repeating cycle. Such a number is likely irrational, meaning it cannot be expressed as a simple fraction. Irrational numbers have non-terminating, non-repeating decimals.
- -0.8974512… This number shows a decimal expansion with digits that do not repeat or terminate visibly. Without a repeating pattern, it is likely an irrational number.
- 1.2547569… Similar to option 2, this number has a decimal part that does not terminate or show a repeating pattern. Hence, it is likely irrational.
- 5.3333333… This decimal expansion is a clear repeating decimal: the digit 3 repeats indefinitely. This repeating decimal can be written exactly as the fraction 163\frac{16}{3} (since 5.3333…=5+13=153+13=1635.3333… = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3}).
Because this number can be expressed as a fraction of two integers, it is rational.
Summary
- Rational numbers = decimals that terminate or have a repeating pattern.
- Irrational numbers = decimals that neither terminate nor repeat.
- Option 4 has a repeating decimal and is therefore rational.
- Options 1, 2, and 3 appear non-repeating and non-terminating, so are irrational.
Final answer: 4. 5.3333333… is rational.