The Bar Over a Sum Must Be Cleared Before Anything Else
Learn the sign rules for multiplying and dividing integers, verify the six properties with real numbers, simplify nested brackets and the vinculum in the right order, and solve integer word problems.
Which bracket do you open first when there are four of them?
The bar — the vinculum — before any of the others. The order is vinculum, then ( ), then { }, then [ ], and working outside-in instead is the surest way to get a wrong answer.
That order is non-negotiable and is worth learning before anything else in this chapter. This page covers everything in the ICSE Class 7 Mathematics chapter on integers: multiplication and division sign rules, the properties of integers, nested brackets, and word problems.
That order is non-negotiable and is worth learning before anything else in this chapter. This page covers everything in the ICSE Class 7 Mathematics chapter on integers: multiplication and division sign rules, the properties of integers, nested brackets, and word problems.
How do the sign rules work for multiplying and dividing integers?
Like signs give a positive result; unlike signs give a negative one. The rule is the same for multiplication and for division.
For a product of several factors, count the negatives. An even number of negative factors gives a positive product; an odd number gives a negative one:
Three negatives, so the answer is negative, and gives the value.
A bank statement is the everyday version: four withdrawals of ₹150 each change the balance by .
Two facts about zero are examined. Zero divided by any non-zero integer is zero, so . But division by zero is undefined — no integer multiplied by 0 gives 5, so names nothing at all.
For a product of several factors, count the negatives. An even number of negative factors gives a positive product; an odd number gives a negative one:
Three negatives, so the answer is negative, and gives the value.
A bank statement is the everyday version: four withdrawals of ₹150 each change the balance by .
Two facts about zero are examined. Zero divided by any non-zero integer is zero, so . But division by zero is undefined — no integer multiplied by 0 gives 5, so names nothing at all.
What are the properties of addition and multiplication on integers?
Six properties are verified in this chapter, and each needs a numerical example.
Closure — adding, subtracting or multiplying two integers always gives an integer. and are both integers. Division is not closed: is not an integer.
Commutative — order does not matter for addition and multiplication:
But not for subtraction or division: while .
Associative — grouping does not matter:
Identity — 0 is the additive identity, since ; 1 is the multiplicative identity, since .
Inverse — the additive inverse of an integer is the number that cancels it: .
Distributive — multiplication spreads over a sum:
Checking directly: . The same answer, both ways.
The property most often claimed too widely is commutativity. It holds for addition and multiplication only, and writing is simply false.
Closure — adding, subtracting or multiplying two integers always gives an integer. and are both integers. Division is not closed: is not an integer.
Commutative — order does not matter for addition and multiplication:
But not for subtraction or division: while .
Associative — grouping does not matter:
Identity — 0 is the additive identity, since ; 1 is the multiplicative identity, since .
Inverse — the additive inverse of an integer is the number that cancels it: .
Distributive — multiplication spreads over a sum:
Checking directly: . The same answer, both ways.
The property most often claimed too widely is commutativity. It holds for addition and multiplication only, and writing is simply false.
How do you simplify an expression with nested brackets and a vinculum?
Remove the brackets in the fixed order — vinculum (bar), then ( ), then { }, then [ ] — and inside each one follow BODMAS: division and multiplication before addition and subtraction.
Worked example.
Clear the vinculum first: .
Then the curly bracket: .
A second example with mixed operations:
Here the multiplication inside the curly bracket came before the addition, as BODMAS requires.
The habit that prevents most errors is rewriting the whole expression after each step, as above, rather than crossing parts out. And watch a minus sign in front of a bracket: in the answer is 6, not 30 — the minus applies to everything the bracket contained.
Worked example.
Clear the vinculum first: .
Then the curly bracket: .
A second example with mixed operations:
Here the multiplication inside the curly bracket came before the addition, as BODMAS requires.
The habit that prevents most errors is rewriting the whole expression after each step, as above, rather than crossing parts out. And watch a minus sign in front of a bracket: in the answer is 6, not 30 — the minus applies to everything the bracket contained.
How do you solve integer word problems?
Decide what each direction means as a sign, then write one expression and evaluate it.
Temperature. The temperature in Shimla is and falls by every hour for 6 hours:
Sea level. A submarine at below sea level rises . Taking below sea level as negative:
so it is below sea level.
Gain and loss. A shopkeeper gains ₹40 on each of 25 items and loses ₹15 on each of 12 others:
an overall gain of ₹820.
Floors. A lift starts at the 3rd floor, goes down 5 levels, then up 2:
the ground floor.
The step that decides the answer is the sign convention, so state it in your working: taking below sea level as negative. Once the directions are fixed, the arithmetic is ordinary — and an answer of must then be reported in words as 170 m below sea level, since a bare negative number does not answer the question asked.
Temperature. The temperature in Shimla is and falls by every hour for 6 hours:
Sea level. A submarine at below sea level rises . Taking below sea level as negative:
so it is below sea level.
Gain and loss. A shopkeeper gains ₹40 on each of 25 items and loses ₹15 on each of 12 others:
an overall gain of ₹820.
Floors. A lift starts at the 3rd floor, goes down 5 levels, then up 2:
the ground floor.
The step that decides the answer is the sign convention, so state it in your working: taking below sea level as negative. Once the directions are fixed, the arithmetic is ordinary — and an answer of must then be reported in words as 170 m below sea level, since a bare negative number does not answer the question asked.
Exam tip
Exam tip: clearing brackets in the right order
Simplification questions in this chapter are pure method marks, awarded line by line.
Write out the full expression again after removing each bracket. Four clean lines earn more than one crowded line with the right answer.
Remove brackets in order — bar, round, curly, square — and never start from the outside. If a bar appears, it goes first even when it is buried deep inside.
When a minus sign sits before a bracket, evaluate the bracket completely, then subtract the whole result.
For property questions, always verify with numbers. *Commutative: .* Stating the property without an example earns little, and mentioning that division is not closed and subtraction not commutative often earns a mark of its own.
And in word problems, declare your sign convention and give the final answer in words with its unit.
Write out the full expression again after removing each bracket. Four clean lines earn more than one crowded line with the right answer.
Remove brackets in order — bar, round, curly, square — and never start from the outside. If a bar appears, it goes first even when it is buried deep inside.
When a minus sign sits before a bracket, evaluate the bracket completely, then subtract the whole result.
For property questions, always verify with numbers. *Commutative: .* Stating the property without an example earns little, and mentioning that division is not closed and subtraction not commutative often earns a mark of its own.
And in word problems, declare your sign convention and give the final answer in words with its unit.
Did you know
Why is division the one operation integers are not closed under?
Because dividing two whole quantities can leave a piece smaller than one.
Add, subtract or multiply any two integers and the result is always another integer — there is simply nowhere else for it to land. But falls between and , and no integer sits there.
That gap is exactly why the next chapter exists. Rational numbers are invented to fill the spaces between the integers, so that division finally has an answer everywhere except by zero.
Add, subtract or multiply any two integers and the result is always another integer — there is simply nowhere else for it to land. But falls between and , and no integer sits there.
That gap is exactly why the next chapter exists. Rational numbers are invented to fill the spaces between the integers, so that division finally has an answer everywhere except by zero.
Key takeaways
Integers: quick revision
- Like signs give a positive product or quotient, unlike signs a negative one; count the negative factors, since an odd count gives a negative product.
- divided by a non-zero integer is , but division by zero is undefined.
- Integers are closed under addition, subtraction and multiplication but not division; addition and multiplication are commutative and associative, subtraction and division are not.
- is the additive identity, the multiplicative identity, and the additive inverse of .
- The distributive property gives .
- Remove brackets in the order vinculum, ( ), { }, [ ], following BODMAS inside each, and state your sign convention in word problems.
You will remember all of this far better after answering five questions on it than after reading it twice.
- divided by a non-zero integer is , but division by zero is undefined.
- Integers are closed under addition, subtraction and multiplication but not division; addition and multiplication are commutative and associative, subtraction and division are not.
- is the additive identity, the multiplicative identity, and the additive inverse of .
- The distributive property gives .
- Remove brackets in the order vinculum, ( ), { }, [ ], following BODMAS inside each, and state your sign convention in word problems.
You will remember all of this far better after answering five questions on it than after reading it twice.