Question

Prove that the product of 2π and 3/4 is an irrational number. Find the product and explain why the value is irrational. Explain your reasoning in 1–2 sentences.

Answer 1

To find the product of \(2\pi\) and \(\frac{3}{4}\), we calculate:

\[
2\pi \times \frac{3}{4} = \frac{3 \times 2\pi}{4} = \frac{3\pi}{2}.
\]

Now, to explain why \(\frac{3\pi}{2}\) is irrational, we note that \(\pi\) is known to be an irrational number, which means it cannot be expressed as a fraction of two integers. The product of a nonzero rational number (in this case, \(\frac{3}{2}\)) and an irrational number (\(\pi\)) is always irrational. Therefore, \(\frac{3\pi}{2}\) is irrational.