Question

According to the Closure Property, which of these sums results in a rational number? (1 point)
0.6 +
9
10
π + √4
0.643892553... +
3
4
36 +221

Answer 1

The closure property of rational numbers states that the sum of two rational numbers is always a rational number. To evaluate which of the given sums results in a rational number, let's identify the rational and irrational numbers in each case:

1. \( 0.6 + 9 \)
- Both \( 0.6 \) (which can be written as \( \frac{6}{10} \) or \( \frac{3}{5} \)) and \( 9 \) (a whole number) are rational.
- **Result:** Rational

2. \( \frac{10}{1} + \pi + \sqrt{4} \)
- \( \pi \) is irrational, but \( \frac{10}{1} \) (10) and \( \sqrt{4} \) (which equals 2) are both rational.
- Since the sum includes the irrational number \( \pi \), the overall sum is irrational.
- **Result:** Irrational

3. \( 0.643892553... + \frac{3}{4} \)
- \( 0.643892553... \) is a decimal that represents a rational number (it is not a repeating or terminating decimal).
- \( \frac{3}{4} \) is rational.
- The sum of two rational numbers is rational.
- **Result:** Rational

4. \( 36 + 221 \)
- Both \( 36 \) and \( 221 \) are whole numbers (rational).
- **Result:** Rational

**Summary of Results:**
- \( 0.6 + 9 \): Rational
- \( \pi + \sqrt{4} \): Irrational
- \( 0.643892553... + \frac{3}{4} \): Rational
- \( 36 + 221 \): Rational

Thus, the sums that result in a rational number are the first, third, and fourth expressions.