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

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

\[
2\pi \cdot \frac{3}{4} = \frac{3 \cdot 2\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2}.
\]

The number \( \pi \) is known to be irrational, and since the product of a non-zero rational number (in this case, \( \frac{3}{2} \)) and an irrational number (\( \pi \)) is also irrational, \( \frac{3\pi}{2} \) is irrational. Therefore, the product \( 2\pi \cdot \frac{3}{4} \) is an irrational number.