Question

according to the closure property which of these sums results in a rational number

a. /0.6 + 9/10
b. π + √4
c. 0.643892553... + 3/4
d. √36 + √221

Answer 1

The closure property states that a set is closed under an operation if performing that operation on elements of the set always produces an element of the same set. The set of rational numbers is closed under addition, which means that the sum of any two rational numbers is also a rational number.

Let's analyze each option:

a. \(0.6 + \frac{9}{10}\)

- \(0.6\) is a rational number (it can be expressed as \(\frac{3}{5}\)), and \(\frac{9}{10}\) is also a rational number.
- The sum \(0.6 + \frac{9}{10} = \frac{3}{5} + \frac{9}{10} = \frac{6}{10} + \frac{9}{10} = \frac{15}{10} = 1.5\), which is rational.

b. \(\pi + \sqrt{4}\)

- \(\pi\) is irrational, and \(\sqrt{4} = 2\) is rational.
- The sum \(\pi + 2\) is irrational.

c. \(0.643892553... + \frac{3}{4}\)

- \(0.643892553...\) is a non-terminating, non-repeating decimal, which is likely irrational unless it repeats later, but typically we can't assume it’s rational.
- \(\frac{3}{4}\) is rational.
- The sum of an irrational number and a rational number is irrational.

d. \(\sqrt{36} + \sqrt{221}\)

- \(\sqrt{36} = 6\) is rational, but \(\sqrt{221}\) is irrational.
- The sum \(6 + \sqrt{221}\) is irrational.

The only sum that results in a rational number is:

**a. \(0.6 + \frac{9}{10}\)**.