To determine which of the given numbers is irrational, we first need to understand what irrational numbers are. An irrational number is any real number that cannot be expressed as a fraction or ratio of two integers. This means it has a non-repeating, non-terminating decimal expansion.
Let's analyze each of the given options:
-
52.781654292: This is a decimal representation. It is a finite decimal and can be expressed as a fraction (for example, it can be written as \(\frac{52781654292}{1000000000}\)). Therefore, this number is rational.
-
52.781654292 - (-7): This simplifies to \(52.781654292 + 7\), which is also a finite decimal. The result will be a finite decimal and can also be expressed as a fraction, making it rational.
-
8/37: This is a fraction where both the numerator and the denominator are integers, hence it is rational.
-
-7: This is an integer and can be expressed as a fraction (\(-7 = \frac{-7}{1}\)), which means it is also rational.
-
3.6363636363636363...: This decimal has a repeating part (the "63"), indicating that it can be expressed as a fraction. It can be represented as \(\frac{3636}{1000} + \frac{63}{9900}\) or other equivalent forms, confirming that it is rational.
-
\( \frac{\sqrt{3}}{4} \): Here, \( \sqrt{3} \) is an irrational number, and when divided by a non-zero integer (4), it remains irrational. Therefore, \( \frac{\sqrt{3}}{4} \) cannot be expressed as a fraction of two integers.
Based on this analysis, \( \frac{\sqrt{3}}{4} \) is the only irrational number among the provided options.