Correct Answer
Rational and Irrational Numbers
A rational number is any number that can be written as a fraction, where both the top (numerator) and bottom (denominator) are whole numbers, like one half \( \frac{1}{2} \) or 3 (which can be written as fraction with numerator of 3 and denominator of 1). \( \frac{3}{1}). \) These numbers either end or repeat when written as a decimal. An irrational number, on the other hand, can't be written as a simple fraction. Its decimal form goes on forever without repeating, like pi \( \pi \) (3.14159 . . . ) or the square root of 2 \( \sqrt{2} \) (1.41421 . . . ).
Explanation
For r plus s, \( r + s, \) since r and s are both rational numbers, their sum is also rational. This is because rational numbers are closed under addition, meaning adding two rational numbers always produces another rational number.
For r plus w \( r + w, \) adding a rational number to an irrational number always results in an irrational number. This is because if r plus w \( r + w \) were rational, subtracting r (which is rational) from r plus w \( r + w \) would make w rational, contradicting the fact that w is irrational.
For r times s \( r\,s, \) since both r and s are rational numbers, their product is also rational. This follows from the fact that rational numbers are closed under multiplication.
For r times w, \( r\,w, \) r is a rational number (nonzero) and w is an irrational number. The product of a nonzero rational number and an irrational number is always irrational. If r times w \( r\,w \) were rational, then dividing by the nonzero rational r would imply that w is rational, which contradicts the fact that w is irrational.
