To determine which of the given numbers is irrational, we need to understand the definitions:
- A rational number is a number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \).
- An irrational number cannot be expressed as a simple fraction, and it has non-repeating, non-terminating decimal expansions.
Now let's analyze each option:
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52.781654292 - This is a terminating decimal. Therefore, it is a rational number.
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−7837 - This is an integer. All integers are rational numbers, so this is rational.
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3.6363636363636363... - This is a repeating decimal (specifically, it can be represented as \( 3.636363... = \frac{109}{30} \)). Therefore, it is a rational number.
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3√4 - This is the cube root of 4. The value of \( \sqrt{4} = 2 \), and the cube root of 2, \( \sqrt[3]{2} \), is known to be an irrational number.
Based on the above analysis, the only irrational number in the list is 3√4.
1 answer