Which number produces a rational number when added to 0.5?
A. –1.73205081…
B. 1/4
C. 0.54732871…
D. √3
The Correct Answer and Explanation is:
The correct answer is B. 1/4.
Explanation:
To determine which number produces a rational number when added to 0.5, we need to review the definitions of rational and irrational numbers.
- Rational numbers are numbers that can be expressed as the quotient of two integers (i.e., as fractions) in the form ( \frac{a}{b} ), where ( a ) and ( b ) are integers and ( b \neq 0 ). Examples of rational numbers include ( 1/4, -2, 0.5, ) and ( 7 ). Rational numbers have either terminating or repeating decimal expansions.
- Irrational numbers are numbers that cannot be written as the quotient of two integers. They cannot be expressed as a fraction, and their decimal expansions neither terminate nor repeat. Examples include ( \pi ), ( \sqrt{2} ), and ( \sqrt{3} ).
Now, let’s evaluate each option and see which one, when added to 0.5, results in a rational number.
- Option A: -1.73205081…
This number is the decimal approximation of ( -\sqrt{3} ), which is an irrational number. Adding an irrational number to a rational number (0.5) results in an irrational number. Therefore, the sum is irrational. - Option B: 1/4
This is a rational number because it can be written as the fraction ( \frac{1}{4} ). When we add a rational number (1/4) to another rational number (0.5), the result is still a rational number. Specifically, ( 0.5 + 1/4 = 0.5 + 0.25 = 0.75 ), which is rational. - Option C: 0.54732871…
This is a non-repeating, non-terminating decimal, which indicates that the number is irrational. When you add an irrational number to a rational number, the result is irrational. Therefore, the sum is irrational. - Option D: ( \sqrt{3} )
The square root of 3 is an irrational number, as its decimal expansion does not terminate or repeat. Adding an irrational number to a rational number results in an irrational sum, so the result is irrational.
Thus, the only option that produces a rational sum when added to 0.5 is B. 1/4, since the sum is 0.75, which is a rational number.