Division Is the One Operation Rational Numbers Cannot Always Do
Learn to add and subtract rationals using the LCM, multiply and divide them with reciprocals, test the closure, commutative, associative and distributive properties, and solve word problems using the additive inverse.
Which operation can rational numbers not always perform?
Division — and for exactly one reason.
Add, subtract or multiply any two rational numbers and the answer is always another rational number. Divide one by another and the answer is rational too — unless the divisor is zero, in which case there is no answer at all.
So rationals are closed under addition, subtraction and multiplication, and not closed under division. One awkward number spoils the whole property, and no amount of care with the rest of the arithmetic gets round it. This page covers the second part of the ICSE Class 8 Mathematics chapter on rational numbers: the four operations, the properties they obey, and where those properties fail.
Add, subtract or multiply any two rational numbers and the answer is always another rational number. Divide one by another and the answer is rational too — unless the divisor is zero, in which case there is no answer at all.
So rationals are closed under addition, subtraction and multiplication, and not closed under division. One awkward number spoils the whole property, and no amount of care with the rest of the arithmetic gets round it. This page covers the second part of the ICSE Class 8 Mathematics chapter on rational numbers: the four operations, the properties they obey, and where those properties fail.
How do you add and subtract rational numbers with unlike denominators?
Convert both to the LCM of the denominators, then add or subtract the numerators only.
Worked example 1. Find .
The LCM of and is :
Notice the answer is positive, even though the larger-looking fraction was negative. Comparing with shows why.
Worked example 2. Find .
Subtracting a negative is adding the positive:
The LCM of and is :
As a mixed number that is .
Worked example 3 — three terms. Find .
The LCM of and is :
Always simplify the final answer: divided through by gives .
Why the LCM and not just any common denominator. Multiplying the denominators together always works — would have served in the first example — but it gives larger numbers and more simplifying at the end. The LCM is the smallest common denominator, so it is the least work.
The one mistake to watch for. Once the denominators are equal, only the numerators are combined; the denominator stays as it is. Writing by adding the denominators too is a genuine and common slip.
Worked example 1. Find .
The LCM of and is :
Notice the answer is positive, even though the larger-looking fraction was negative. Comparing with shows why.
Worked example 2. Find .
Subtracting a negative is adding the positive:
The LCM of and is :
As a mixed number that is .
Worked example 3 — three terms. Find .
The LCM of and is :
Always simplify the final answer: divided through by gives .
Why the LCM and not just any common denominator. Multiplying the denominators together always works — would have served in the first example — but it gives larger numbers and more simplifying at the end. The LCM is the smallest common denominator, so it is the least work.
The one mistake to watch for. Once the denominators are equal, only the numerators are combined; the denominator stays as it is. Writing by adding the denominators too is a genuine and common slip.
Formula
How do you multiply and divide rational numbers?
Multiplication needs no common denominator at all:
Division is multiplication by the reciprocal of the divisor:
Worked example 1 — multiplying. Find .
Dividing numerator and denominator by the HCF gives .
Worked example 2 — dividing. Find .
Invert the divisor and multiply:
Dividing through by the HCF gives .
The reciprocal, also called the multiplicative inverse, is what you get by interchanging the numerator and the denominator. Its defining property is that a number times its reciprocal equals ****.
- The reciprocal of is .
- The reciprocal of is , and the sign stays the same. Checking: .
- The reciprocal of is , since .
- The reciprocal of is , and of is . These are the only two numbers that are their own reciprocals.
- Zero has no reciprocal, because is undefined.
That last point is the whole reason for this page's title. Every rational number except zero has a reciprocal, so every division except division by zero can be carried out — which is exactly the gap in the closure property.
The sign rules, which are the same as for integers. Two like signs give a positive result; two unlike signs give a negative one. So is negative, and would have been positive.
A shortcut worth using. Cancel common factors before multiplying. In example 1, cancelling the into the and the into the gives directly, with no large numbers to simplify afterwards.
Division is multiplication by the reciprocal of the divisor:
Worked example 1 — multiplying. Find .
Dividing numerator and denominator by the HCF gives .
Worked example 2 — dividing. Find .
Invert the divisor and multiply:
Dividing through by the HCF gives .
The reciprocal, also called the multiplicative inverse, is what you get by interchanging the numerator and the denominator. Its defining property is that a number times its reciprocal equals ****.
- The reciprocal of is .
- The reciprocal of is , and the sign stays the same. Checking: .
- The reciprocal of is , since .
- The reciprocal of is , and of is . These are the only two numbers that are their own reciprocals.
- Zero has no reciprocal, because is undefined.
That last point is the whole reason for this page's title. Every rational number except zero has a reciprocal, so every division except division by zero can be carried out — which is exactly the gap in the closure property.
The sign rules, which are the same as for integers. Two like signs give a positive result; two unlike signs give a negative one. So is negative, and would have been positive.
A shortcut worth using. Cancel common factors before multiplying. In example 1, cancelling the into the and the into the gives directly, with no large numbers to simplify afterwards.
Which properties do rational numbers obey, and which do they break?
Four properties, and each holds for some operations and fails for others.
Closure — is the answer always a rational number?
- Addition: yes. Subtraction: yes. Multiplication: yes.
- Division: no, because division by zero is undefined.
Commutative — does changing the order leave the answer unchanged? Written .
- Addition: yes. Multiplication: yes.
- Subtraction: no. Division: no.
Checking the failure for subtraction:
The two results are and , which are not equal. One counterexample is enough to disprove a property, and this is the one to quote.
Associative — does the grouping matter? Written .
- Addition: yes. Multiplication: yes.
- Subtraction: no. Division: no.
Distributive — multiplication distributes over addition and subtraction:
Verifying it. Take , , .
Left-hand side — bracket first:
Right-hand side — multiply separately, then add:
Both sides give , so the property is verified.
The identities and inverses:
- ** is the additive identity**: for every rational .
- ** is the multiplicative identity**: .
- The additive inverse of is , and their sum is : .
- The multiplicative inverse of is its reciprocal, and their product is — except for zero, which has none.
The pattern behind all of it. Addition and multiplication are the well-behaved operations, obeying every property. Subtraction and division are the awkward ones, failing commutativity and associativity — because each is really the inverse of a well-behaved operation rather than an operation in its own right.
Closure — is the answer always a rational number?
- Addition: yes. Subtraction: yes. Multiplication: yes.
- Division: no, because division by zero is undefined.
Commutative — does changing the order leave the answer unchanged? Written .
- Addition: yes. Multiplication: yes.
- Subtraction: no. Division: no.
Checking the failure for subtraction:
The two results are and , which are not equal. One counterexample is enough to disprove a property, and this is the one to quote.
Associative — does the grouping matter? Written .
- Addition: yes. Multiplication: yes.
- Subtraction: no. Division: no.
Distributive — multiplication distributes over addition and subtraction:
Verifying it. Take , , .
Left-hand side — bracket first:
Right-hand side — multiply separately, then add:
Both sides give , so the property is verified.
The identities and inverses:
- ** is the additive identity**: for every rational .
- ** is the multiplicative identity**: .
- The additive inverse of is , and their sum is : .
- The multiplicative inverse of is its reciprocal, and their product is — except for zero, which has none.
The pattern behind all of it. Addition and multiplication are the well-behaved operations, obeying every property. Subtraction and division are the awkward ones, failing commutativity and associativity — because each is really the inverse of a well-behaved operation rather than an operation in its own right.
How do you solve word problems with rational numbers?
Translate the words into an operation, carry it out with the LCM, and then check the answer against the question.
Worked example 1 — a subtraction. A tank is full. If of the tank is drained out, what fraction remains?
So of the tank remains. A sensible check: is less than , as it must be after draining.
Worked example 2 — finding a missing addend. The sum of two rational numbers is . If one of them is , find the other.
The other number is the sum minus the known one:
Checking by adding it back:
which is the required sum. Correct.
Worked example 3 — using the additive inverse. What should be added to to get ?
The required number is the target minus what you have:
And the special case of the same question: what must be added to to get zero? The additive inverse, .
Worked example 4 — a division problem. The product of two rational numbers is . If one of them is , find the other.
The other is the product divided by the known one:
Checking: . Correct.
The habit that makes all of these reliable. Every one of the four was checked by reversing the operation — adding the answer back, or multiplying it back. Those checks take one line each and catch almost every arithmetic and sign error, which is why they are worth writing out rather than doing mentally.
Worked example 1 — a subtraction. A tank is full. If of the tank is drained out, what fraction remains?
So of the tank remains. A sensible check: is less than , as it must be after draining.
Worked example 2 — finding a missing addend. The sum of two rational numbers is . If one of them is , find the other.
The other number is the sum minus the known one:
Checking by adding it back:
which is the required sum. Correct.
Worked example 3 — using the additive inverse. What should be added to to get ?
The required number is the target minus what you have:
And the special case of the same question: what must be added to to get zero? The additive inverse, .
Worked example 4 — a division problem. The product of two rational numbers is . If one of them is , find the other.
The other is the product divided by the known one:
Checking: . Correct.
The habit that makes all of these reliable. Every one of the four was checked by reversing the operation — adding the answer back, or multiplying it back. Those checks take one line each and catch almost every arithmetic and sign error, which is why they are worth writing out rather than doing mentally.
Exam tip
Exam tip: one counterexample disproves a property
To show a property fails, you need one counterexample with both sides worked out. *Subtraction is not commutative because but * is complete. To show a property holds, verify it with an example, but say the property is general.
For closure, the answer for division is not closed, and the reason is division by zero. That reason is the mark.
When adding or subtracting, use the LCM and combine only the numerators. The denominator does not change.
Subtracting a negative is adding: .
For division, write the reciprocal step explicitly: .
Remember a reciprocal keeps the sign — the reciprocal of is — and that zero has no reciprocal.
When verifying the distributive property, compute the two sides separately and fully, then state that they are equal.
Simplify every final answer and cancel before multiplying where you can.
And check each answer by reversing the operation. It costs one line and it is the difference between a right answer and a hoped-for one.
For closure, the answer for division is not closed, and the reason is division by zero. That reason is the mark.
When adding or subtracting, use the LCM and combine only the numerators. The denominator does not change.
Subtracting a negative is adding: .
For division, write the reciprocal step explicitly: .
Remember a reciprocal keeps the sign — the reciprocal of is — and that zero has no reciprocal.
When verifying the distributive property, compute the two sides separately and fully, then state that they are equal.
Simplify every final answer and cancel before multiplying where you can.
And check each answer by reversing the operation. It costs one line and it is the difference between a right answer and a hoped-for one.
Did you know
Why does zero alone spoil a property that otherwise always holds?
Division of rationals works beautifully. Any rational divided by any other gives a rational, and the method — multiply by the reciprocal — never fails.
Except once. Zero has no reciprocal, so cannot be carried out.
It is worth seeing why there is no answer rather than simply accepting it. Suppose equalled some number . Then, reversing the division, would have to equal . But anything multiplied by zero is zero, so no such exists. The question has no answer, not a difficult one.
And one exception is enough. A property of a set must hold for every member of it, so a single case where it fails means the set is not closed under that operation — even though the operation succeeds for every other pair of numbers you could name.
That is why the tables in this chapter have a single no sitting among the yeses. Mathematics is unforgiving about exceptions in a way that most rules of thumb are not.
Except once. Zero has no reciprocal, so cannot be carried out.
It is worth seeing why there is no answer rather than simply accepting it. Suppose equalled some number . Then, reversing the division, would have to equal . But anything multiplied by zero is zero, so no such exists. The question has no answer, not a difficult one.
And one exception is enough. A property of a set must hold for every member of it, so a single case where it fails means the set is not closed under that operation — even though the operation succeeds for every other pair of numbers you could name.
That is why the tables in this chapter have a single no sitting among the yeses. Mathematics is unforgiving about exceptions in a way that most rules of thumb are not.
Key takeaways
Operations and properties of rational numbers: quick revision
- Add or subtract using the LCM of the denominators, combining only the numerators: , and .
- Subtracting a negative is adding. Always simplify the result: .
- Multiply straight across: . Cancel before multiplying where possible.
- Divide by multiplying by the reciprocal: .
- The reciprocal interchanges numerator and denominator, keeps the sign, and gives a product of ****. and are their own reciprocals, and zero has none.
- Closure — rationals are closed under addition, subtraction and multiplication, but not division, because of division by zero.
- Commutative and associative hold for addition and multiplication and fail for subtraction and division. The counterexample: but .
- Distributive: . Verified with , , — both sides give .
- ** is the additive identity, the multiplicative identity. The additive inverse** of is with sum ; the multiplicative inverse is the reciprocal with product .
- Word problems: remains; a sum of with one part leaves ; adding takes to ; a product of with one factor leaves .
- Check every answer by reversing the operation.
Work through a set of property verifications, giving a full counterexample wherever a property fails — producing the counterexample is the skill being tested, not reciting the table.
- Subtracting a negative is adding. Always simplify the result: .
- Multiply straight across: . Cancel before multiplying where possible.
- Divide by multiplying by the reciprocal: .
- The reciprocal interchanges numerator and denominator, keeps the sign, and gives a product of ****. and are their own reciprocals, and zero has none.
- Closure — rationals are closed under addition, subtraction and multiplication, but not division, because of division by zero.
- Commutative and associative hold for addition and multiplication and fail for subtraction and division. The counterexample: but .
- Distributive: . Verified with , , — both sides give .
- ** is the additive identity, the multiplicative identity. The additive inverse** of is with sum ; the multiplicative inverse is the reciprocal with product .
- Word problems: remains; a sum of with one part leaves ; adding takes to ; a product of with one factor leaves .
- Check every answer by reversing the operation.
Work through a set of property verifications, giving a full counterexample wherever a property fails — producing the counterexample is the skill being tested, not reciting the table.