prove that the product of 2pi and 3/4 is an irrational number

To prove that the product of \( 2\pi \) and \( \frac{3}{4} \) is an irrational number, we will use the fact that the product of a rational number and an irrational number is always irrational.

1. **Identify the two numbers**:
- \( 2\pi \) is known to be an irrational number (since \( \pi \) is irrational).
- \( \frac{3}{4} \) is a rational number (since it can be expressed as the quotient of two integers).

2. **Use the property of rational and irrational numbers**:
- Let \( r = \frac{3}{4} \) (a rational number).
- Let \( x = 2\pi \) (an irrational number).

3. **Consider the product**:
- We want to examine \( r \cdot x \):
\[
r \cdot x = \frac{3}{4} \cdot 2\pi = \frac{3 \cdot 2\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2}
\]

4. **Determine the nature of \( \frac{3\pi}{2} \)**:
- Since \( \pi \) is irrational, any non-zero multiple of \( \pi \) (including \( \frac{3}{2} \)) is also irrational.

Thus, we conclude that \( \frac{3\pi}{2} \) is an irrational number.

**Conclusion**: The product of \( 2\pi \) and \( \frac{3}{4} \) is \( \frac{3\pi}{2} \), which is irrational. Therefore, we can state that the product \( 2\pi \cdot \frac{3}{4} \) is indeed an irrational number.