Question
predict which type of number the product will be and explain why 4/5 . pi
Answers
GPT-4o mini
To determine the type of number that the product \( \frac{4}{5} \times \pi \) will be, we can analyze the components of the multiplication:
1. **Rational Number**: The number \( \frac{4}{5} \) is a rational number because it can be expressed as the quotient of two integers (4 and 5).
2. **Irrational Number**: The number \( \pi \) is an irrational number, meaning it cannot be expressed as a fraction of two integers; its decimal representation is non-repeating and non-terminating.
When multiplying a rational number by an irrational number, the product will always be an irrational number. This is because if the product were rational, it would imply that \( \pi \) could be expressed as a ratio of two integers, which is not true.
Thus, the product:
\[
\frac{4}{5} \times \pi
\]
is an irrational number.
In summary, the product \( \frac{4}{5} \cdot \pi \) will be irrational because a rational number multiplied by an irrational number results in an irrational number.
1. **Rational Number**: The number \( \frac{4}{5} \) is a rational number because it can be expressed as the quotient of two integers (4 and 5).
2. **Irrational Number**: The number \( \pi \) is an irrational number, meaning it cannot be expressed as a fraction of two integers; its decimal representation is non-repeating and non-terminating.
When multiplying a rational number by an irrational number, the product will always be an irrational number. This is because if the product were rational, it would imply that \( \pi \) could be expressed as a ratio of two integers, which is not true.
Thus, the product:
\[
\frac{4}{5} \times \pi
\]
is an irrational number.
In summary, the product \( \frac{4}{5} \cdot \pi \) will be irrational because a rational number multiplied by an irrational number results in an irrational number.