Question

which of the following is an irrational number? -1 1/2, π, 51.80 (with a bar over 80), 11

Explanation:

Step1: Recall irrational number definition

An irrational number is a real number that cannot be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q
eq0\), and it has a non - repeating, non - terminating decimal expansion.

Step2: Analyze \(-1\frac{1}{2}\)

\(-1\frac{1}{2}=-\frac{3}{2}\), which is a fraction of two integers, so it is a rational number.

Step3: Analyze \(\pi\)

The decimal expansion of \(\pi\) is \(3.1415926535\cdots\), which is non - repeating and non - terminating. It cannot be expressed as a fraction of two integers, so \(\pi\) is an irrational number.

Step4: Analyze \(51.\overline{80}\)

A repeating decimal can be expressed as a fraction. Let \(x = 51.808080\cdots\), then \(100x=5180.808080\cdots\), subtracting \(x\) from \(100x\) gives \(99x = 5129\), so \(x=\frac{5129}{99}\), which is a rational number.

Step5: Analyze \(11\)

\(11=\frac{11}{1}\), which is a fraction of two integers, so it is a rational number.

Answer:

\(\pi\) (the second option)