Q1:

Is zero a rational number? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$?

Step 1: Theoretical Foundation of Rational Numbers

In mathematics, the set of rational numbers, denoted by $\mathbb{Q}$, is defined as the set of all numbers that can be expressed as a quotient or fraction $\frac{p}{q}$ of two integers. [Per the fundamental axioms of Number Theory], the formal definition requires two strict conditions to be met:

• The numerator $p$ and the denominator $q$ must be integers ($p, q \in \mathbb{Z}$).
• The denominator $q$ must not be equal to zero ($q \neq 0$), because division by zero is mathematically undefined.

Step 2: Analyzing Zero ($0$) Against the Rational Criteria

To determine if zero is a rational number, we must test it against the two conditions established in Step 1.

• Condition 1 (Integer Numerator): We can set the numerator $p = 0$. Since zero is a whole number without a fractional or decimal component, it is an integer ($0 \in \mathbb{Z}$).
• Condition 2 (Non-Zero Integer Denominator): We can select any non-zero integer for the denominator $q$ (e.g., $1, 2, -5, 100$). For any chosen integer where $q \neq 0$, the second condition is perfectly satisfied.

Step 3: Constructing the $\frac{p}{q}$ Form

When zero is divided by any non-zero number, the quotient is always zero. [By the Zero Property of Division], we can express $0$ in infinitely many equivalent fractional forms:

$0 = \frac{0}{1} = \frac{0}{2} = \frac{0}{-7} = \frac{0}{999}$

In every single one of these representations, $p = 0$ and $q \in \{1, 2, -7, 999\}$. Because $p$ and $q$ are integers and $q \neq 0$, zero rigorously fulfills the definition of a rational number.

Step 4: Visual Representation on the Number Line

The number line below illustrates how zero sits among the integers, while simultaneously being representable as a rational fraction.

Final Solution: Yes, zero is a rational number. It can be written in the form $\frac{p}{q}$ (such as $\frac{0}{1}$, $\frac{0}{2}$, or $\frac{0}{-5}$), where $p = 0$ and $q$ is any non-zero integer ($q \neq 0$).