To prove that the product of \(3\pi\) and \(\frac{2}{5}\) is an irrational number, we start by calculating the product:
\[ 3\pi \times \frac{2}{5} = \frac{6\pi}{5} \]
Next, we need to establish why \(\frac{6\pi}{5}\) is irrational.
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Understanding the components:
- \(\pi\) (pi) is known to be an irrational number. By definition, an irrational number cannot be expressed as a fraction of two integers, meaning it cannot be represented in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).
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Properties of irrational numbers:
- The product of a non-zero rational number and an irrational number is always irrational. Since \(\frac{6}{5}\) is a rational number (where \(6\) and \(5\) are integers, and \(5 \neq 0\)), we can use this property.
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Applying the property:
- We know that \( \pi \) is irrational and \( \frac{6}{5} \) is rational. Thus, we compute the product: \[ \frac{6\pi}{5} = \frac{6}{5} \cdot \pi \]
- Since \(\frac{6}{5}\) is rational and \(\pi\) is irrational, \(\frac{6\pi}{5}\) must also be irrational.
Therefore, \(\frac{6\pi}{5}\) is an irrational number because it is the product of a rational number \( \frac{6}{5} \) and an irrational number \( \pi \).
Conclusion
The product \(3\pi \times \frac{2}{5} = \frac{6\pi}{5}\) is irrational because it results from multiplying a rational number by an irrational number.
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