Rational Numbers Class 8 Mathematics — Revision Notes

TL;DR: Rational numbers are numbers that can be expressed as the ratio of two integers. They form an important part of the number system and have properties…

Rational numbers are numbers that can be expressed as the ratio of two integers. They form an important part of the number system and have properties simil

Definition and Representation

• Rational number is a number that can be expressed as p/q where p and q are integers and q is not zero
• Examples include 2/3, -5/7, 4/1, and 0/5
• Every integer is a rational number since it can be written with denominator 1
• Rational numbers can be positive, negative, or zero
• Equivalent rational numbers represent the same value; 2/3 equals 4/6

Standard Form of Rational Numbers

• Standard form is when numerator and denominator have no common factor other than 1
• Denominator should be positive in standard form
• To reduce to standard form, divide both by their greatest common divisor
• Example: 8/12 in standard form is 2/3
• -2/3 is in standard form but -2/-3 should be written as 2/3

Operations on Rational Numbers

• Addition: a/b + c/d = (ad + bc)/bd
• Subtraction: a/b - c/d = (ad - bc)/bd
• Multiplication: a/b x c/d = ac/bd
• Division: a/b ÷ c/d = a/b x d/c = ad/bc
• Commutative and associative properties hold for addition and multiplication

Properties of Rational Numbers

• Closure property: sum, difference, product of rationals are rational
• Additive identity is 0; a + 0 = a for any rational a
• Multiplicative identity is 1; a x 1 = a for any rational a
• Additive inverse of a/b is -a/b; their sum is zero
• Multiplicative inverse of a/b is b/a; their product is one

Representation on Number Line

• Rational numbers can be plotted on a number line between integers
• Positive rationals are to the right of zero; negatives to the left
• Between any two rational numbers, infinitely many rational numbers exist
• Division method: divide the segment into equal parts to mark fractional positions
• 1/2 lies midway between 0 and 1

Key Terms

• Rational Number: A number that can be expressed as p/q where p and q are integers and q is not zero
• Standard Form: Rational number in form where numerator and denominator have no common factors
• Multiplicative Inverse: For a/b, the multiplicative inverse is b/a; their product equals 1
• Additive Inverse: For a number x, the additive inverse is -x; their sum equals 0

What makes a number rational

A rational number is any number that can be written in the form p over q, where p and q are integers and q is not zero. The word comes from ratio, not from reasonableness — a rational number is one expressible as a ratio of two integers.

The phrase can be written matters more than it looks. The number 4 is rational because it can be written as 4 over 1, even though it is usually not. The number 0.75 is rational because it equals 3 over 4. A number does not have to arrive in fraction form to be rational; it only has to be capable of being put in that form.

The condition that q is not zero is not a technicality to be skipped. Division by zero is undefined, so p over 0 is not a number at all, and every property in this chapter carries that restriction quietly along with it.

Why the chapter is about properties rather than calculation

You already know how to add and multiply fractions. What this chapter asks is a different kind of question: which rules of arithmetic survive when the number system is enlarged, and which break.

Whole numbers are not closed under subtraction, because 3 minus 5 is not a whole number. Integers fixed that but are not closed under division, because 3 divided by 5 is not an integer. Rational numbers are introduced precisely to close that last gap, and the tables in this chapter record exactly how much was gained.

So when a question asks you to verify a property with an example, it is asking you to check a claim about the whole system, not merely to do a sum. And to show a property fails, one counter-example is enough — that is a point worth internalising early, because it is how mathematics disproves things for the rest of your school career.

Closure — which operations keep you inside the system

A system is closed under an operation if performing that operation on any two members always produces a member of the same system.

• Rational numbers are closed under addition: the sum of any two rational numbers is rational.
• Rational numbers are closed under subtraction, which whole numbers are not.
• Rational numbers are closed under multiplication.
• Rational numbers are not closed under division, and the single reason is zero: dividing by zero is not defined. Excluding zero, the remaining rational numbers are closed under division.
• For comparison, whole numbers are closed only under addition and multiplication, and integers are closed under addition, subtraction and multiplication but not division.
• The pattern is worth seeing as a story: each enlargement of the number system was made to close an operation that had failed in the previous one.

Commutativity and associativity

Commutativity is about the order of two numbers; associativity is about the grouping of three. They are different claims and are tested separately.

• Addition is commutative for rational numbers: a plus b equals b plus a.
• Multiplication is commutative: a times b equals b times a.
• Subtraction is not commutative. Two thirds minus one half is one sixth, while one half minus two thirds is minus one sixth, and those are different.
• Division is not commutative either — a divided by b is generally not b divided by a.
• Addition is associative: the grouping of three numbers does not affect the sum.
• Multiplication is associative in the same way.
• Subtraction and division are not associative, so brackets genuinely change the answer and cannot be moved.

Identities and inverses — the roles of 0 and 1

Two numbers behave in a special way under the two main operations, and the vocabulary for describing them is examined directly.

• Zero is the additive identity: adding zero to any rational number leaves it unchanged. It plays the same role for whole numbers and integers.
• One is the multiplicative identity: multiplying any rational number by one leaves it unchanged.
• The additive inverse of a number, also called its negative, is the number that added to it gives zero. The additive inverse of three fifths is minus three fifths.
• The multiplicative inverse, or reciprocal, is the number that multiplied by it gives one. The reciprocal of a over b is b over a, provided a is not zero.
• Zero has no reciprocal. There is no number that multiplied by zero gives one, which is the same fact as division by zero being undefined.
• Note that 1 is its own reciprocal, and minus 1 is its own reciprocal too — a small point that appears in objective questions.

Distributivity, and rational numbers between rational numbers

Distributivity is the one property that links the two operations, and it is the property that makes clever simplification possible.

• Multiplication distributes over addition: a times the quantity b plus c equals ab plus ac.
• Multiplication also distributes over subtraction: a times the quantity b minus c equals ab minus ac.
• Read backwards, distributivity lets you take out a common factor, which is usually how it saves work in an examination.
• Between any two rational numbers there are infinitely many rational numbers. This is not true of integers, where nothing lies between 3 and 4.
• One reliable method is the mean: the average of two rational numbers always lies between them, and the process can be repeated as often as you like.
• A second method is to write both numbers with a common denominator and then enlarge that denominator until enough numerators fit between them.

Solved Numericals — Step by Step

Worked Examples

• 4 is a rational number — It can be written as 4 over 1, and both 4 and 1 are integers with a non-zero denominator. A number need not look like a fraction to be rational.
• 0 is rational but has no reciprocal — Zero can be written as 0 over 1, so it is rational. But no number multiplied by 0 gives 1, so it has no multiplicative inverse.
• Between 3 and 4 there is no integer, but infinitely many rationals — 3.5, 3.1, 3.01 and endlessly more all lie between them. This density is the property that distinguishes rationals from integers.
• −1 is its own reciprocal — Since (−1) × (−1) = 1, the reciprocal of −1 is −1. The only other number that is its own reciprocal is 1.

Common Mistakes Students Make

• Forgetting the condition that the denominator must not be zero. p over q is rational only when q is not zero, and every property in the chapter depends on that.
• Saying rational numbers are closed under division without qualification. Division by zero is undefined, so the system is closed under division only after zero is excluded.
• Claiming zero has a reciprocal, or writing the reciprocal of 0 as 0. Zero has no multiplicative inverse at all.
• Confusing additive inverse with multiplicative inverse. The additive inverse of 3/4 is −3/4; its reciprocal is 4/3. One gives 0, the other gives 1.
• Dropping the sign when taking a reciprocal. The reciprocal of −5/8 is −8/5, not 8/5 — the sign is retained.
• Assuming subtraction is associative and moving brackets freely. Both subtraction and division fail associativity, so the brackets must be respected.
• Answering that there are only a few rational numbers between two given ones. There are infinitely many, and a question asking for three is asking for any three.

Where This Shows Up in Real Life

Why distributivity is worth spotting in an exam

An expression such as one containing two products that share a factor can usually be collapsed by taking the common factor out, turning three or four fraction multiplications into one. In the worked example above, recognising the shared factor of minus three sevenths reduces the whole calculation to multiplying by 1. Marks in these questions are given for naming the property used, so the technique earns credit twice: once for the shorter route and once for identifying it.

Measurement and everyday division

Any measurement made with a real instrument is a rational number, because it is read to a finite number of decimal places and any terminating decimal can be written as a fraction over a power of ten. Sharing a bill, scaling a recipe, converting between units and computing an average all stay inside the rational numbers, which is a practical statement of the closure properties in this chapter.

The gap that rational numbers do not fill

Rational numbers are dense — between any two of them lie infinitely many more — which makes it surprising that they still leave gaps on the number line. The length of the diagonal of a unit square cannot be written as a ratio of two integers, and neither can the ratio of a circle's circumference to its diameter. Those gaps are what irrational numbers are introduced for in Class 9, so this chapter is the last one in which every number on the line is a ratio.

Board-Style Questions with Model Answers

Define a rational number and state why the denominator cannot be zero. (2 marks)

A rational number is a number that can be expressed in the form p over q, where p and q are integers and q is not equal to zero. The denominator cannot be zero because division by zero is not defined in mathematics: there is no number which, multiplied by zero, gives a non-zero result, and any number multiplied by zero gives zero, so the expression p over zero cannot be assigned a value.

Are rational numbers closed under division? Justify your answer. (2 marks)

Not entirely. For any two rational numbers, the quotient is again rational provided the divisor is not zero, but division by zero is not defined, so the result of dividing by zero is not a rational number and indeed is not a number at all. Therefore the set of all rational numbers is not closed under division, while the set of rational numbers excluding zero is closed under division.

State the additive identity and the multiplicative identity for rational numbers, and give the additive inverse and the reciprocal of −7/9. (3 marks)

The additive identity is 0, because adding zero to any rational number leaves it unchanged. The multiplicative identity is 1, because multiplying any rational number by one leaves it unchanged. The additive inverse of −7/9 is 7/9, since their sum is zero. The reciprocal, or multiplicative inverse, of −7/9 is −9/7, since their product is one; note that the negative sign is retained when the numerator and denominator are interchanged.

Verify by taking suitable rational numbers that subtraction is neither commutative nor associative. (5 marks)

For commutativity take a = 2/3 and b = 1/2. Then a minus b equals 4/6 minus 3/6, which is 1/6, while b minus a equals 3/6 minus 4/6, which is minus 1/6. Since these differ, subtraction is not commutative. For associativity take a = 1/2, b = 1/3 and c = 1/6. Then the quantity a minus b, all minus c, equals 1/2 minus 1/3 minus 1/6, which is 3/6 minus 2/6 minus 1/6, that is 0. But a minus the quantity b minus c equals 1/2 minus the quantity 1/3 minus 1/6, which is 1/2 minus 1/6, that is 1/3. Since 0 is not equal to 1/3, subtraction is not associative. In each case a single counter-example is sufficient to disprove the property.

How many rational numbers lie between 2/5 and 3/5? Find any two of them. (3 marks)

There are infinitely many rational numbers between any two distinct rational numbers, so infinitely many lie between 2/5 and 3/5. To find some, write both with a larger common denominator: 2/5 equals 20/50 and 3/5 equals 30/50. Any numerator strictly between 20 and 30 gives a rational number in the required range, for example 21/50 and 25/50, the latter simplifying to 1/2. The process can be repeated with a still larger denominator to obtain as many such numbers as desired, which is why the count is infinite.

Use a suitable property to simplify 3/7 × (−5/9) + 3/7 × (−4/9), naming the property used. (3 marks)

Both terms share the common factor 3/7, so distributivity of multiplication over addition can be used in reverse to take it out. The expression becomes 3/7 multiplied by the quantity minus 5/9 plus minus 4/9. Inside the bracket, minus 5/9 plus minus 4/9 equals minus 9/9, which is minus 1. Therefore the expression equals 3/7 multiplied by minus 1, which is minus 3/7. The property used is the distributive property of multiplication over addition.

Assertion–Reason Questions (CBSE Format)

For each question choose: (a) both A and R are true and R is the correct explanation of A; (b) both are true but R is not the correct explanation; (c) A is true but R is false; (d) A is false but R is true.

Quick Self-Check

Key Terms Glossary

Rational number

A number that can be expressed as p over q, where p and q are integers and q is not zero.

Closure

A system is closed under an operation if applying that operation to any two of its members always gives a member of the same system.

Commutative property

A property by which the order of two numbers does not change the result; it holds for addition and multiplication but not for subtraction or division.

Associative property

A property by which the grouping of three numbers does not change the result; it holds for addition and multiplication only.

Distributive property

The rule that a times the quantity b plus c equals ab plus ac, linking multiplication with addition and subtraction.

Additive identity

The number 0, which added to any rational number leaves it unchanged.

Multiplicative identity

The number 1, which multiplied by any rational number leaves it unchanged.

Additive inverse

The negative of a number; the number that added to it gives 0. The additive inverse of a over b is minus a over b.

Multiplicative inverse

The reciprocal of a number; the number that multiplied by it gives 1. Zero has none.

Reciprocal

Another name for the multiplicative inverse, obtained by interchanging the numerator and the denominator.

Counter-example

A single case that shows a general claim to be false; one is sufficient to disprove a property.

Standard form of a rational number

The form in which the denominator is positive and the numerator and denominator have no common factor other than 1.

One-Minute Revision

• Rational number = p/q with p, q integers and q ≠ 0. The 'can be written as' is what matters — 4 and 0.75 are both rational.
• Closed under addition, subtraction and multiplication. Division fails only because of zero.
• Commutative and associative: addition and multiplication yes; subtraction and division no.
• 0 is the additive identity; 1 is the multiplicative identity.
• Additive inverse gives 0; reciprocal gives 1. Do not swap the two.
• Zero has no reciprocal — the same fact as division by zero being undefined.
• Keep the sign when taking a reciprocal: the reciprocal of −5/8 is −8/5.
• Distributivity: a(b + c) = ab + ac. Read backwards it takes out a common factor and saves most of the work.
• Between any two rational numbers lie infinitely many rational numbers; the average method proves it.
• One counter-example is enough to disprove a property — you never need more.

Syllabus reference: NCERT Class 8 Mathematics — Rational Numbers. Extends the integer properties met in Class 7 and is assumed by Class 9 Number Systems, where irrational numbers are added.

Frequently Asked Questions

Is every integer a rational number?

Yes, every integer can be expressed as a rational number with denominator 1. For example, 5 equals 5/1.

How many rational numbers exist between 0 and 1?

Infinitely many. Between any two rational numbers, infinitely more rational numbers can be found.

Is every integer a rational number?

Yes. Any integer n can be written as n over 1, where both the numerator and the denominator are integers and the denominator is not zero, which satisfies the definition exactly. So the integers form a subset of the rational numbers. The converse is not true: 3 over 4 is rational but is not an integer, which is why the rational numbers form a genuinely larger system.

What is the difference between the additive inverse and the multiplicative inverse?

They answer two different questions. The additive inverse is what you add to the number to get the additive identity 0, so the additive inverse of 3 over 4 is minus 3 over 4. The multiplicative inverse, or reciprocal, is what you multiply the number by to get the multiplicative identity 1, so the reciprocal of 3 over 4 is 4 over 3. Every rational number has an additive inverse, but zero has no multiplicative inverse.

How can I be sure there are infinitely many rational numbers between two given ones?

Take their average, which always lies strictly between them and is itself rational. Now you have a new pair — the original smaller number and this average — and you can take their average too. Nothing ever stops this process, so it produces an endless supply of distinct rational numbers in the original interval. This is why the answer to how many is always infinitely many, no matter how close together the two numbers are.

Why is division by zero not simply defined as zero or as infinity?

Because any definition would break the arithmetic that already works. If a divided by 0 had some value k, then by the meaning of division k multiplied by 0 would have to equal a. But k multiplied by 0 is 0 for every k, so this can only work when a is 0, and then every value of k would qualify equally. There is either no possible answer or no unique one, so the expression is left undefined rather than assigned a value that would make the other rules inconsistent.

More Class 8 Mathematics Revision Notes

• Squares and Square Roots
• Mensuration
• Exponents and Powers
• Linear Equations in One Variable
• Squares and Square Roots
• Cubes and Cube Roots

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