This page is for Rational Numbers Class 8 — Chapter 1 of the NCERT Mathematics textbook for Class 8. The chapter runs about 13 printed pages, from the opening equations to Exercise 1.1 and the closing summary, and the official NCERT chapter PDF is right here.
Below the download link you will find a plain-language walkthrough of what the chapter teaches, so you can use the PDF faster — or skip it entirely if you only need a revision recap.
Download the Rational Numbers Class 8 PDF
Get the file directly from NCERT: download the official NCERT Rational Numbers Class 8 chapter PDF. It is the current NCERT Class 8 Mathematics book, chapter 1, published by the National Council of Educational Research and Training on ncert.nic.in.
The Rational Numbers Chapter at a Glance
The chapter tests one question over and over: do the usual operations stay inside the rational numbers, and do their rules match what you already know from whole numbers and integers? The table below shows the chapter’s shape — how many sections it holds, how many figures it carries, and how many exercise questions close it.
| Chapter element | Count |
|---|---|
| Chapter sections | 28 |
| Figures and diagrams | 5 |
| Worked examples | 0 (worked checks are built into the tables) |
| Exercise questions | 3 |
Why the Chapter Begins with Equations: Building the Rational Numbers
Here is the idea in one sentence: every time the numbers you already know cannot solve a simple equation, the number system expands to include a new kind of number. The chapter opens with this exact chain.
- \( x + 2 = 13 \) is solved by \( x = 11 \), a natural number (1, 2, 3, …).
- \( x + 5 = 5 \) needs \( x = 0 \), the whole number zero — natural numbers alone cannot solve it.
- \( x + 18 = 5 \) needs \( x = -13 \), which is an integer. The book notes that positive integers are the same collection as natural numbers.
- \( 2x = 3 \) and \( 5x + 7 = 0 \) need \( \frac{3}{2} \) and \( \frac{-7}{5} \) — the integers are not enough.
That last step is where rational numbers enter. A rational number is any number of the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). The book’s examples are \( \frac{-2}{3} \), \( \frac{6}{7} \) and \( \frac{9}{-5} \), and it adds that 0, −2 and 4 are rational too, because each can be written as \( \frac{p}{1} \).
That first figure is the chapter’s promise: you already know how to add, subtract, multiply and divide fractions. What the chapter adds is checking the properties — closure, commutativity, associativity, identities and distributivity — one operation at a time, for rational numbers and for the number systems you met earlier.
Closure Property: When the Answer Stays Inside the Number System
A set of numbers is closed under an operation when doing that operation on two members of the set always gives another member of the same set. If the answer escapes the set, the operation is not closed. Think of it as a door: the result must stay inside the room.
- Whole numbers are closed under addition and multiplication — but subtraction escapes, since \( 5 – 7 = -2 \) is not a whole number, and division escapes too, since \( 5 \div 8 = \frac{5}{8} \) is not a whole number.
- Integers become closed under subtraction as well — \( -6 – 8 = -14 \) is still an integer — but integers are still not closed under division, because \( 5 \div 8 \) is not an integer.
- Rational numbers are closed under addition, subtraction and multiplication. Rational numbers are not closed under division, and the reason is a single exception: \( a \div 0 \) is not defined (NCERT, section 1.2.1). Exclude zero, and the remaining rational numbers are closed under division.
The chapter’s habit is to ask “Is it a rational number?” after every operation instead of assuming. That habit is the whole method — check, then conclude.
Commutativity and Associativity: Order and Grouping Don’t Change the Result
Two more properties, in plain words: commutative means order does not matter, \( a + b = b + a \); associative means grouping does not matter, \( a + (b + c) = (a + b) + c \).
For rational numbers — and for every number system the chapter visits — addition and multiplication have both properties. Subtraction and division have neither. One counter-example is enough to prove it: \( \frac{2}{5} – \frac{1}{3} \neq \frac{1}{3} – \frac{2}{5} \), so subtraction is not commutative.
The pattern mnemonic: addition and multiplication behave the same in every number system the chapter visits, while subtraction and division do not. If you remember that one line, you can regenerate most of the chapter’s Yes/No tables from memory.
Why do these properties exist? Because they make computation faster. The book’s worked Examples 1 and 2 show a long sum and a long product being regrouped so that friendly pairs are combined before heavy LCM work — that is the payoff of commutativity and associativity.
Worked addition (original numbers): find \( \frac{7}{12} + \frac{-5}{8} + \frac{5}{12} \).
\[ \frac{7}{12} + \left(\frac{-5}{8}\right) + \frac{5}{12} = \left[\frac{7}{12} + \frac{5}{12}\right] + \left(\frac{-5}{8}\right) = 1 + \frac{-5}{8} = \frac{3}{8} \]
Direct check with one LCM: \( \frac{14}{24} – \frac{15}{24} + \frac{10}{24} = \frac{9}{24} = \frac{3}{8} \). Regrouping is faster because \( \frac{7}{12} + \frac{5}{12} \) makes a whole 1.
Worked multiplication (original numbers): find \( \frac{-8}{15} \times \frac{9}{4} \times \frac{5}{8} \times \frac{2}{9} \).
\[ \left(\frac{-8}{15} \times \frac{5}{8}\right) \times \left(\frac{9}{4} \times \frac{2}{9}\right) = \frac{-1}{3} \times \frac{1}{2} = \frac{-1}{6} \]
Read the associativity figure as one spoken sentence: \( \frac{-7}{3} \times \left(\frac{5}{4} \times \frac{2}{9}\right) = \left(\frac{-7}{3} \times \frac{5}{4}\right) \times \frac{2}{9} = \frac{-35}{54} \). The grouping changed; the result did not. That equality is what “associative” means.
The Role of 0 and 1: Additive and Multiplicative Identities
An identity is a number that leaves other numbers unchanged under an operation. This chapter has exactly two identities, and they pair with the two operations that behave.
- 0 is the additive identity: \( a + 0 = 0 + a = a \). Adding zero changes nothing, for whole numbers, integers and rational numbers alike (NCERT, section 1.2.4).
- 1 is the multiplicative identity: \( a \times 1 = 1 \times a = a \). Multiplying by one changes nothing (NCERT, section 1.2.5).
The classic slip is swapping the pair: writing \( a \times 0 = a \) or \( a + 1 = a \). Test with a real number — \( 5 \times 1 = 5 \) and \( 5 + 0 = 5 \) — and the pairing sticks. The book also asks whether 1 is the multiplicative identity for integers and whole numbers; the answer is yes, because the same property \( a \times 1 = a \) holds there.
Distributivity: How Multiplication Spreads Over Addition and Subtraction
The earlier properties connected addition to addition and multiplication to multiplication. Distributivity is the only property that links the two operations: multiplication can be spread across a sum or a difference.
\[ a(b + c) = ab + ac \quad \text{and} \quad a(b – c) = ab – ac \]
The book verifies this using \( \frac{-3}{4} \), \( \frac{2}{3} \) and \( \frac{-5}{6} \): it computes \( \frac{-3}{4} \times \left(\frac{2}{3} + \frac{-5}{6}\right) \) first inside the bracket, then spread out, and both routes give \( \frac{1}{8} \). The useful direction for examinations is the reverse — factoring. When the same fraction multiplies both terms, pull it out in front.
Worked example (original numbers):
\[ \frac{5}{6} \times \frac{-7}{9} + \frac{5}{6} \times \frac{11}{9} = \frac{5}{6} \times \left[\frac{-7}{9} + \frac{11}{9}\right] = \frac{5}{6} \times \frac{4}{9} = \frac{20}{54} = \frac{10}{27} \]
Direct check: \( \frac{-35}{54} + \frac{55}{54} = \frac{20}{54} \). When you are unsure whether you used distributivity correctly, compute both routes fully and compare — the chapter does exactly that.
The distributivity figure shows the book’s verification style: first \( \frac{-3}{4} \times \frac{2}{3} = \frac{-1}{2} \), then the other product, then the two partial results are added. That piece-by-piece route is how you check your own work.
Reading the Property Tables: How the Chapter Builds Each Result
Half the chapter runs on one table skeleton: an Operation column, a Numbers column of worked checks, and a Remarks column carrying the conclusion. Once you can read that skeleton, every table in the chapter opens itself.
The trick is to look for the exception row first. In the whole-numbers closure table, the subtraction row deliberately shows \( 5 – 7 = -2 \) — an answer that escapes the set — and the division row shows \( 5 \div 8 = \frac{5}{8} \). That escape is the whole argument: one counter-example proves the property fails.
The second figure isolates the chapter’s single exception: \( a \div 0 \) is not defined, so “not closed under division” has one cause — zero in the denominator. Remove zero, and the rest of the rational numbers are closed under division.
The TRY THESE grids in the book are the same table with the answers blanked out. By the end of the chapter you should be able to regenerate the full Yes/No picture for closure, commutativity and associativity from memory — the consolidated tables below are that finished picture.
Which operations are closed, commutative and associative?
| Number system | Closed under +, −, ×, ÷ | Commutative for +, −, ×, ÷ | Associative for +, −, ×, ÷ |
|---|---|---|---|
| Natural numbers | +, × only | +, × only | +, × only |
| Whole numbers | +, × only | +, × only | +, × only |
| Integers | +, −, × (not ÷) | +, × only | +, × only |
| Rational numbers | +, −, × (not ÷) | +, × only | +, × only |
These three grids are the chapter’s core in compressed form. Notice the same mnemonic at work: addition and multiplication behave the same everywhere; subtraction and division do not.
Key Definitions from the Rational Numbers Chapter
This is the chapter’s glossary in one place. If a term trips you later in the chapter, it lives here.
| Term | Meaning | Form |
|---|---|---|
| Rational number | A number expressible as one integer divided by another, with a non-zero denominator | \( \frac{p}{q} \), where \( p, q \) are integers and \( q \neq 0 \) |
| Closed | The result of the operation stays in the same number system | \( a + b \) is in the set whenever \( a, b \) are |
| Commutative | Order does not matter | \( a + b = b + a \), \( a \times b = b \times a \) |
| Associative | Grouping does not matter | \( a + (b + c) = (a + b) + c \) |
| Additive identity | The number that leaves every number unchanged under addition | 0, because \( a + 0 = a \) |
| Multiplicative identity | The number that leaves every number unchanged under multiplication | 1, because \( a \times 1 = a \) |
| Distributivity | Multiplication spreads over a sum or difference | \( a(b + c) = ab + ac \), \( a(b – c) = ab – ac \) |
Common Mistakes Students Make with Rational Numbers
Each row below is a wrong statement students write under pressure, followed by the correction grounded in the chapter’s own reasoning.
| Mistake | Correct rule | How to check your answer |
|---|---|---|
| “Rational numbers are closed under division.” | Not closed, because \( a \div 0 \) is not defined. Excluding zero, the rest of the rational numbers are closed under division. | Test \( a \div 0 \) — there is no rational number it equals. |
| Swapping the identities: “\( a \times 0 = a \)”. | 0 is the additive identity; 1 is the multiplicative identity. | Test a number: \( 5 + 0 = 5 \), \( 5 \times 1 = 5 \) — never the reverse. |
| Treating subtraction and division like addition and multiplication. | Subtraction and division are neither commutative nor associative. | One counter-example settles it: \( \frac{2}{5} – \frac{1}{3} \neq \frac{1}{3} – \frac{2}{5} \). |
| Rejecting \( \frac{9}{-5} \) as “not a rational number”. | It is rational: the denominator is a non-zero integer (NCERT, section 1.2.1). | Check the definition — only \( q \neq 0 \) matters, not the sign of q. |
| Forgetting that every integer is rational. | Any integer \( p \) is \( \frac{p}{1} \), so 0, −2 and 4 qualify. | Write the integer over 1 and compare with the definition. |
| Dropping the negative sign when adding negative fractions. | Combine the signs in the numerator: \( \frac{3}{8} + \frac{-5}{7} = \frac{21 – 40}{56} = \frac{-19}{56} \). | If both numerators fight, the answer takes the sign of the larger absolute value sum. |
Exercise 1.1: Which Property Each Question Tests
Exercise 1.1 has just three questions, and each is a property-spotting test — the skill is naming the property, not computing. A full-marks answer names the property and states the operation it belongs to.
| Question | What it shows | Property tested |
|---|---|---|
| 1(i) — \( \frac{-4}{5} \times 1 = 1 \times \frac{-4}{5} = \frac{-4}{5} \) | Multiplying by 1 leaves the number unchanged | Multiplicative identity (1) |
| 1(ii) — \( \frac{-13}{17} \times \frac{-2}{7} = \frac{-2}{7} \times \frac{-13}{17} \) | Reversing the order of factors | Commutativity of multiplication |
| 1(iii) — \( \frac{-19}{29} \times \frac{29}{-19} = 1 \) | Multiplying a number by its reciprocal gives 1 | Reciprocal product (this edition’s chapter text does not use the term “multiplicative inverse”) |
| 2 — computing \( \frac{1}{3} \times (6 \times \frac{4}{3}) \) as \( (\frac{1}{3} \times 6) \times \frac{4}{3} \) | Moving the brackets, order unchanged | Associativity of multiplication |
| 3 — the product of two rational numbers is always a ____ | The product stays rational | Closure under multiplication |
Before attempting the exercise, study the chapter’s worked Examples 1 to 3: Example 1 regroups a sum, Example 2 regroups a product, and Example 3 factors out a common fraction. The exercise asks you to name those moves, not invent new ones.
One honest caution: textbook contents and the examinable syllabus are not always identical — check the current official syllabus for your class.
What the Chapter Concludes: Six Takeaways for Rational Numbers
The chapter ends with its own summary, “What Have We Discussed?”, and these points are what it concludes. Restated in fresh words:
- Rational numbers are closed under addition, subtraction and multiplication.
- Addition and multiplication are commutative for rational numbers.
- Addition and multiplication are associative for rational numbers.
- The rational number 0 is the additive identity.
- The rational number 1 is the multiplicative identity.
- Distributivity holds: \( a(b + c) = ab + ac \) and \( a(b – c) = ab – ac \).
The summary leaves you with a seventh idea: between any two given rational numbers there are countless rational numbers, and the mean gives you one of them. Worked example: between \( \frac{3}{4} \) and \( \frac{5}{6} \), the mean is \( \left(\frac{3}{4} + \frac{5}{6}\right) \div 2 = \left(\frac{9}{12} + \frac{10}{12}\right) \div 2 = \frac{19}{24} \), and \( \frac{3}{4} \lt \frac{19}{24} \lt \frac{5}{6} \).
Take the mean again and you get another rational number — repeat forever, and you see why they are countless.
Related Resources: Class 8 Maths Book and the Next Chapter
This listing is maintained for the 2026-27 academic session using the NCERT textbook information available to us. NCERT remains the authority for confirming the latest edition.
Where to go next, in order:
- The full Class 8 Mathematics book page — Chapter 1 is this book’s opening chapter, so the book page is also the “previous chapter” slot.
- Chapter 2: Linear Equations in One Variable — the next chapter, which builds directly on the rational number arithmetic from this one.
- Class 8 Mathematics revision notes — for quick revision once you have finished the chapter.
- Class 8 hub — all Class 8 subjects and notes in one place.
- Class 8 Mathematics Power Play — practice material that checks whether you can apply the properties, not just name them.
A note for parents: this page describes the current NCERT Class 8 Mathematics textbook, Chapter 1 (Rational Numbers); the official PDF is linked at the top of the page.
Sources and Data Verification
- The contents and figures described on this page come from the NCERT Class 8 Mathematics textbook, Chapter 1 (Rational Numbers).
- This page covers that one chapter, not the whole CBSE syllabus.
- The page is maintained for the current academic session using the NCERT information available to us.
- NCERT settles textbook editions, contents and official PDFs; CBSE settles the curriculum, syllabus and examinations.
Textbook files are published on the official NCERT website.
| What the chapter holds | Count | Where it is used |
|---|---|---|
| Printed pages | 14 | |
| Sections in the chapter | 8 | |
| Figures with NCERT captions | 5 | |
| Tables | 10 | |
| Exercise questions | 3 | answered in our NCERT Solutions |
| Official NCERT PDF | Download the chapter PDF | the chapter exactly as NCERT publishes it |
Frequently Asked Questions About Rational Numbers Class 8
Is zero a rational number?
Yes. Zero can be written as \( \frac{0}{1} \), so it fits the definition \( \frac{p}{q} \) with \( q \neq 0 \). The book lists 0 among numbers writable as \( \frac{p}{q} \).
Why are rational numbers not closed under division?
Because \( a \div 0 \) is not defined for any number a. Once zero is excluded from the denominator, the remaining rational numbers are closed under division.
What are the additive and multiplicative identities of rational numbers?
0 is the additive identity (\( a + 0 = a \)) and 1 is the multiplicative identity (\( a \times 1 = a \)). Keep the pairing with the operation: 0 for addition, 1 for multiplication.
How do you find a rational number between two rational numbers?
Take the mean of the two numbers. The chapter’s summary says the mean idea helps, and repeating the mean gives another rational number every time — which is why there are countless rational numbers between any two.
Are all integers rational numbers?
Yes. Every integer \( p \) can be written as \( \frac{p}{1} \), which is the form \( \frac{p}{q} \) with \( q \neq 0 \). So 0, −2 and 4 are all rational numbers.
Is subtraction commutative for rational numbers?
No. Order matters in subtraction — for example, \( \frac{2}{3} – \frac{5}{4} \neq \frac{5}{4} – \frac{2}{3} \). Only addition and multiplication are commutative.
Reference: NCERT Class 8 Mathematics textbook, chapter 1, official edition on ncert.nic.in.
Explore Class 8 Mathematics Books
- Next: Power Play
Related chapters:
- A Square and a Cube
- Linear Equations in One Variable
- A Story of Numbers