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Q1:
Is zero a rational number? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$?
Solution :
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 ($q \neq 0$).
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.1
- Q2: Find six rational numbers between 3 and 4.
- Q3: Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.
- Q4(i): State whether the following statements are true or false. Give reasons for your answers. (i) Every natural number is a whole number.
- Q4(ii): State whether the following statements are true or false. Give reasons for your answers. (ii) Every integer is a whole number.
- Q4(iii): State whether the following statements are true or false. Give reasons for your answers. (iii) Every rational number is a whole number.
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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I am a strong believer in concept-based gain of knowledge. I start from scratch while teaching the very basic concepts and take my students to a level where they have a deep understanding of the subject. Once they understand the concept, the next step is that practice a lot of questions/numerical/problem in the class itself as practice is the key to retain knowledge. I always refer to prescribed books by CBSE/ICSE board but always encourage students to refer additional 1 or 2 books based on their speed to practice. In last I provide them enough homework to practice at home and send me the doubts before the next class. Also for 10th students, we prepare right from beginning by covering the previous year questions and this approach till date have been fruitful as most of my students excel in the board exams
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