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.
Similar Questions
Identify the incorrect relationship(s) from the list below:
(i) \(\sqrt{6} + \sqrt{2} = \sqrt{5} + \sqrt{3}\)
(ii) \(\sqrt{6} + \sqrt{2} < \sqrt{5} + \sqrt{3}\)
(iii) \(\sqrt{6} + \sqrt{2} > \sqrt{5} + \sqrt{3}\)
- Which of the following numbers is both a Rational Number and a Terminating Decimal?
- Which of these is a number that is real but NOT rational?
- The decimal 0.875 is best classified as:
Important Questions from Rational or Irrational Numbers
Which of the following number is irrational?
Which of the following numbers will have an irrational square root?
What is the square root of 16 + 6√7?
A non-terminating but recurring decimal is:
Which of the following is false?
