Why a Denominator Is Never Allowed to Be Zero

Because nothing can be divided into zero parts. Asking what $\frac{5}{0}$ equals is asking which number multiplied by 0 gives 5 — and no number does, since anything times zero is zero.

So the condition $q\ne 0$ is not a technicality; it is the reason the definition works at all. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: what a rational number is, standard form, equivalent forms, the number line, and comparing rational numbers.

A rational number is any number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.

So all of these are rational:

- $\frac{3}{4}$ and $\frac{-5}{7}$ — ordinary fractions, positive and negative
- $5$, because it can be written as $\frac{5}{1}$
- $-8$, because it is $\frac{-8}{1}$
- $0$, because it is $\frac{0}{1}$
- $2\frac{1}{3}$, because it is $\frac{7}{3}$

That means every integer is a rational number, and so is every fraction and every mixed number. The rational numbers contain all of them.

A rational number is positive when $p$ and $q$ have the same sign, and negative when they have opposite signs. So $\frac{-3}{-4}$ is positive and equals $\frac{3}{4}$, while $\frac{3}{-4}$ is negative.

Market prices show them daily — half a kilo is $\frac{1}{2}$, and a loss of ₹5 is $-5$, which is $\frac{-5}{1}$.

What is not rational is anything with a zero denominator, such as $\frac{7}{0}$ or $\frac{0}{0}$. Note the difference carefully: $\frac{0}{7}=0$ is a perfectly good rational number, while $\frac{7}{0}$ is not a number at all. Zero on top is fine; zero underneath is not.

A rational number is in standard (lowest) form when its denominator is positive and the numerator and denominator have no common factor except 1.

Two steps get you there: make the denominator positive, then divide both parts by their HCF.

Worked example 1. Reduce $\frac{-12}{18}$. The HCF of 12 and 18 is 6:

$$\frac{-12}{18}=\frac{-12÷6}{18÷6}=\frac{-2}{3}$$

Worked example 2. Reduce $\frac{9}{-15}$. The denominator is negative, so multiply both parts by $-1$ first:

$$\frac{9}{-15}=\frac{-9}{15}=\frac{-9÷3}{15÷3}=\frac{-3}{5}$$

Equivalent rational numbers are got by multiplying or dividing both parts by the same non-zero integer.

To write $\frac{-2}{3}$ with denominator 21: since $3\times 7=21$, multiply both by 7:

$$\frac{-2}{3}=\frac{-2\times 7}{3\times 7}=\frac{-14}{21}$$

To write $\frac{-2}{3}$ with **numerator $-8$**: since $-2\times 4=-8$, multiply both by 4, giving $\frac{-8}{12}$.

The rule that must never be broken is both parts, same number. Multiplying only the numerator changes the value entirely, and adding the same number to both parts — a frequent mistake — does too: $\frac{2}{3}$ and $\frac{2+1}{3+1}=\frac{3}{4}$ are not equal.

Divide the unit to the left of zero into as many equal parts as the denominator, then count that many parts leftward.

To plot $\frac{-3}{4}$: mark 0 and $-1$, divide the gap between them into 4 equal parts, and count 3 parts to the left of 0. The third mark is $\frac{-3}{4}$.

To plot $\frac{5}{4}$, which is an improper fraction: it lies between 1 and 2, since $\frac{5}{4}=1\frac{1}{4}$. So divide the gap from 1 to 2 into 4 parts and take the first mark.

For $\frac{-7}{3}=-2\frac{1}{3}$, it lies between $-2$ and $-3$, so divide that gap into 3 parts and take the first one beyond $-2$.

A ruler is the everyday number line, with each centimetre divided into ten equal millimetre parts.

The habit that keeps the plotting right is converting an improper fraction to a mixed number first. That tells you immediately which two whole numbers it sits between, so you divide the correct unit rather than counting from zero and overshooting.

Two methods work, and both give the same answer.

Method 1 — equal denominators. Write both with the LCM of the denominators, then compare numerators. Compare $\frac{-3}{4}$ and $\frac{-2}{3}$, with LCM 12:

$$\frac{-3}{4}=\frac{-9}{12},\frac{-2}{3}=\frac{-8}{12}$$

Since $-9<-8$, we get $\frac{-3}{4}<\frac{-2}{3}$.

Method 2 — cross-multiplication. With positive denominators, cross-multiply and compare the products:

$$(-3)\times 3=-9\text{and}(-2)\times 4=-8$$

Again $-9<-8$, so $\frac{-3}{4}<\frac{-2}{3}$.

Ordering a set. Arrange $\frac{-1}{2},\frac{3}{4},\frac{-5}{6},\frac{1}{3}$ in ascending order. The LCM of 2, 4, 6 and 3 is 12:

$$\frac{-6}{12},\frac{9}{12},\frac{-10}{12},\frac{4}{12}$$

Ordering the numerators gives $-10<-6<4<9$, so the ascending order is

$$\frac{-5}{6},\frac{-1}{2},\frac{1}{3},\frac{3}{4}$$

The trap is the same one negatives always set. With negative numbers the larger-looking value is the smaller number, so $\frac{-5}{6}$ is less than $\frac{-1}{2}$. And every negative rational number is less than every positive one, so sorting the signs first saves work.