Result Type: Rational x Irrational Multiplication

To determine the type of number resulting from multiplying a Rational Number by an Irrational Number, we examine their definitions and properties.

Number Definitions Explained

• Rational Numbers: Can be written as $p/q$, where $p, q$ are integers, $q \neq 0$. Examples: $5$, $1/3$, $-0.25$.
• Irrational Numbers: Cannot be written as $p/q$. Decimal form is non-terminating and non-repeating. Examples: $\sqrt{2}$, $\pi$.

Multiplication Cases Analysis

Let $r$ be a rational number and $i$ be an irrational number.

• Case 1: $r \neq 0$
  The product $r \times i$ is generally irrational. Example: $3 \times \sqrt{5} = 3\sqrt{5}$ (irrational). Reasoning: If $r \times i = q$ (rational), then $i = q/r$. Since $q, r$ are rational ($r \neq 0$), $q/r$ must be rational, contradicting $i$'s irrationality.
• Case 2: $r = 0$
  The product $0 \times i$ is always 0. Example: $0 \times \pi = 0$. Reasoning: 0 is a rational number ($0/1$).

Result Type Conclusion

Since the product can be irrational (Case 1) or rational (Case 2), the result is Sometimes Rational, Sometimes Irrational.