To determine which of the given options is irrational, let's analyze each one:
A - \(7 \frac{8}{37}\): This is a mixed number and can be expressed as a fraction. Converting it to an improper fraction gives us: \[ 7 \frac{8}{37} = \frac{7 \cdot 37 + 8}{37} = \frac{259 + 8}{37} = \frac{267}{37} \] Since this is a fraction of integers, it is rational.
B - \(3.6363636363636363...\): This is a repeating decimal (where "63" repeats). Repeating decimals are always rational numbers because they can be expressed as a fraction. Therefore, this option is rational.
C - \(52.781654292\): This is a non-repeating, finite decimal, which means it can also be expressed as a fraction. Thus, it is rational.
D - \(\frac{\sqrt{3}}{4}\): The square root of 3 (\(\sqrt{3}\)) is an irrational number. Since the division of an irrational number by a non-zero rational number (4 in this case) remains irrational, \(\frac{\sqrt{3}}{4}\) is also irrational.
From this analysis, the only irrational option is D: \[ \text{D: } \frac{\sqrt{3}}{4} \]
4 answers