Question
Which of the following numbers is irrational? A. 0.3333... B. 0.2468 C. 3 D. \( \sqrt{2} \)
Ask by Schultz Schmidt. in the United States
Feb 19,2025
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Answer
D. \( \sqrt{2} \) is irrational.
Solution
To determine which of the given numbers is irrational, let's analyze each option:
A. \( 0.3333... \) - This is a repeating decimal, which can be expressed as the fraction \( \frac{1}{3} \). Since it can be represented as a fraction of two integers, it is a rational number.
B. \( 0.2468 \) - This is a terminating decimal, which can also be expressed as a fraction (for example, \( \frac{2468}{10000} \)). Therefore, it is a rational number.
C. \( 3 \) - This is an integer and can be expressed as \( \frac{3}{1} \). Thus, it is a rational number.
D. \( \sqrt{2} \) - This number cannot be expressed as a fraction of two integers. It is known to be an irrational number.
Based on this analysis, the irrational number among the options is:
**D. \( \sqrt{2} \)**.
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Simplify this solution Bonus Knowledge
The number \( \sqrt{2} \) is famous for being the first number proven to be irrational! This means it cannot be expressed as a fraction of two integers, unlike the other options listed, which can all be represented in such a way. The ancient Greeks, particularly the Pythagoreans, were shocked by this finding because they believed all numbers were rational. In real-world applications, understanding irrational numbers is crucial in fields like engineering and physics. For example, when calculating diagonal lengths in construction projects or working with circular areas, knowing that the square root of 2 won't fit nicely into a fraction helps in making precise measurements!
