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

Answers

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Answered by GPT-4o mini
To find the product of \( 2\pi \) and \( \frac{3}{4} \):

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

The number \( \pi \) is known to be irrational, meaning it cannot be expressed as a fraction of two integers. Thus, any non-zero rational number (like \( \frac{3}{2} \)) multiplied by an irrational number (like \( \pi \)) results in an irrational number. Therefore, \( \frac{3\pi}{2} \) is irrational, confirming that the product of \( 2\pi \) and \( \frac{3}{4} \) is irrational.

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