Rational and Irrational Numbers Question 1 of 10 2 Points Which number produces a rational number when added to
? A.
B.
C.
D. -1.41421356…
The Correct Answer and Explanation is:
To answer this question accurately, I need to know which number is being added to the choices (A, B, C, D) — specifically, the number referenced in the blank after “when added to ___?”. Since the question doesn’t display the number clearly, I’ll make a reasonable assumption that the number being added to the choices is an irrational number, most likely √2 ≈ 1.41421356…, because the choice (D) is -1.41421356…, which looks like -√2.
Assumed Question:
Which number produces a rational number when added to √2?
Choices:
- A. (Unknown)
- B. (Unknown)
- C. (Unknown)
- D. -1.41421356…
Let’s focus on choice D:
- D = -1.41421356… ≈ -√2
Now evaluate:
√2 + (–√2) = 0
This result, 0, is a rational number, because it can be written as a fraction:
0 = 0/1
✅ Correct Answer: D. -1.41421356…
📘 Explanation
Rational and irrational numbers are two major classifications of real numbers.
A rational number is any number that can be expressed as a ratio of two integers, like 1/2, -3, or 7. In decimal form, rational numbers either terminate (e.g., 0.75) or repeat (e.g., 0.333…).
An irrational number, on the other hand, cannot be written as a simple fraction. Its decimal expansion is non-terminating and non-repeating. Examples include √2, π, and e.
In this quiz question, you are asked to find a number that, when added to an irrational number (likely √2), gives a rational number. To achieve this, you must cancel out the irrational part.
The only way to cancel an irrational number like √2 is by adding its additive inverse, which is –√2. When you do:
√2 + (–√2) = 0
The irrational parts cancel out completely, resulting in 0, which is rational.
Among the choices, –1.41421356… approximates –√2. Although it’s not exact, this decimal is meant to represent the irrational value of –√2. Therefore, adding it to √2 results in a rational number.
This concept is fundamental in algebra and number theory and highlights the relationship between different types of real numbers.
In conclusion, choice D is correct because adding –√2 to √2 gives 0, a rational number.
