Number System (Classes VI-VIII) • Topic 5 of 6

Rational Numbers

By Class 7 and 8, children formalise rational numbers: any number that can be written as p/q where p and q are integers and q is not zero. This sweeps up integers, fractions, terminating and recurring decimals under one roof, and the CTET items test whether candidates grasp the structure rather than just compute. Common misconceptions: thinking a rational number must be a fraction less than 1 (so 5 or -3 are wrongly excluded, even though 5 = 5/1), and the surprising-to-children density property, that between any two rational numbers there are infinitely many more (between 1/2 and 1/3 there is no 'next' rational number the way there is a next whole number). Children also stumble on negative rationals, applying integer sign rules unevenly. A teacher emphasises the p/q definition, the q not equal to 0 condition, representation on the number line, and finding a rational number between two given ones by averaging. Expect a definition-check or a 'find a number between these two' item.

✅ Solved examples

1. Find a rational number between 1/4 and 1/2.
Take their average: (1/4 + 1/2) / 2 = (1/4 + 2/4) / 2 = (3/4) / 2 = 3/8. So 3/8 lies between them.
2. A child says 7 is not a rational number because it is not a fraction. Is the child right?
No. A rational number is any number of the form p/q with integers p, q and q not zero. 7 = 7/1, so it is rational. The misconception is treating rational as only proper fractions.
3. Is 0 a rational number?
Yes. 0 = 0/1 (or 0/q for any non-zero q), which fits the p/q form, so 0 is rational. Only q must be non-zero, not p.
4. How many rational numbers lie between 1/3 and 1/2?
Infinitely many. Between any two distinct rational numbers there are infinitely many rationals; this is the density property.

✏️ Practice — try these, take hints as needed

1. Find a rational number between 2 and 3.
Average the two numbers.
(2 + 3) / 2.
5/2 (i.e. 2.5)
2. Write -4 as a rational number in p/q form.
Any integer is rational.
Use a denominator of 1.
-4/1
3. A child claims there is a rational number immediately next to 1/2 with nothing between. Why is this wrong?
Think about the density property.
Try averaging 1/2 with any nearby rational.
Between any two rationals lie infinitely many rationals, so there is no immediate next one.
4. Is every integer a rational number?
Can you write n as n/1?
Check the p/q condition.
Yes, since any integer n = n/1 fits the p/q form.

📝 Topic test — 8 questions

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