Which number produces an irrational number when multiplied by -1.25
A. 3/4
B. 5.21
C.-0.454545
D. sqaure root 17
The Correct Answer and Explanation is:
To determine which number produces an irrational number when multiplied by (-1.25), we first need to understand the nature of rational and irrational numbers:
- Rational Numbers: These are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. For example, numbers like (\frac{3}{4}), (5.21) (which can be expressed as (\frac{521}{100})), and (-0.454545) (which can be expressed as (\frac{-45}{99})) are all rational numbers.
- Irrational Numbers: These cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal representations. An example of an irrational number is (\sqrt{17}).
Now, we can analyze each option:
- A. (\frac{3}{4}):
[
-1.25 \times \frac{3}{4} = -\frac{15}{16} = -0.9375 \quad (\text{Rational})
] - B. (5.21):
[
-1.25 \times 5.21 = -6.5125 \quad (\text{Rational})
] - C. (-0.454545):
[
-1.25 \times -0.454545 = 0.56818125 \quad (\text{Rational})
] - D. (\sqrt{17}):
[
-1.25 \times \sqrt{17} \quad (\text{Irrational})
]
Since (-1.25) is a rational number and (\sqrt{17}) is an irrational number, the product of a rational number and an irrational number is always irrational. Therefore, when we multiply (-1.25) by (\sqrt{17}), the result remains irrational.
Thus, the correct answer is D. (\sqrt{17}).
In summary, the product of a rational number (like (-1.25)) and an irrational number (like (\sqrt{17})) will always yield an irrational number. The other options consist of rational numbers that, when multiplied by (-1.25), result in rational outputs, confirming that (\sqrt{17}) is the only number in the list that meets the criteria of producing an irrational number when multiplied by (-1.25).