Sure, let's go through the given questions step-by-step to provide detailed explanations:
### (vii) If [tex]\(\frac{p}{q}\)[/tex] is a rational number, then [tex]\(q\)[/tex] cannot be ...
A rational number is any number that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers, and [tex]\(q \neq 0\)[/tex]. This condition is crucial because division by zero is undefined in mathematics. If the denominator [tex]\(q\)[/tex] were zero, the expression would be undefined, so [tex]\(q\)[/tex] must be a nonzero integer.
Thus, the blank should be filled with:
Answer: zero
### (viii) Two rational numbers with different numerators are equal, if their numerators are in the same ... as their denominators.
To determine when two fractions are equivalent, we can use the property of proportions. Two fractions [tex]\(\frac{p_1}{q_1}\)[/tex] and [tex]\(\frac{p_2}{q_2}\)[/tex] are equal if their cross-products are equal, i.e., [tex]\(p_1 \times q_2 = p_2 \times q_1\)[/tex]. This implies that the numerators and denominators are in the same proportion (or ratio). In other words, if [tex]\(\frac{p_1}{q_1} = \frac{p_2}{q_2}\)[/tex], then [tex]\(\frac{p_1}{p_2} = \frac{q_1}{q_2}\)[/tex].
Thus, the blank should be filled with:
Answer: ratio
