The Number Pattern That Explains Why Two Negatives Make a Positive
Learn the sign rules for multiplying and dividing integers, see the number pattern that explains them, apply the order of operations, and use the distributive property to calculate faster.
Can multiplying two negative numbers really give a positive?
Yes — and you can watch it happen in a number pattern rather than take it on trust. Multiplication by a negative reverses direction, so reversing twice brings you back to where you started.
This page covers everything in the CBSE Class 7 Mathematics chapter's second half: the sign rules for multiplication, the matching rules for division, evaluating expressions with several operations, and the properties that let you calculate faster.
This page covers everything in the CBSE Class 7 Mathematics chapter's second half: the sign rules for multiplication, the matching rules for division, evaluating expressions with several operations, and the properties that let you calculate faster.
How do the sign rules for multiplying integers work?
There are only two rules. Like signs give a positive product, and unlike signs give a negative one.
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Multiply the values as usual, then attach the sign the rules demand.
The pattern that explains the last line is worth writing out. Keep one factor at and step the other down by one each time:
Each answer has risen by 3. Continuing the same steady rise:
The pattern forces the positive answers; anything else would break the steady increase of 3.
For several factors, count the negatives. An even number of negative factors gives a positive product and an odd number gives a negative one, so — three negatives, hence negative.
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Multiply the values as usual, then attach the sign the rules demand.
The pattern that explains the last line is worth writing out. Keep one factor at and step the other down by one each time:
Each answer has risen by 3. Continuing the same steady rise:
The pattern forces the positive answers; anything else would break the steady increase of 3.
For several factors, count the negatives. An even number of negative factors gives a positive product and an odd number gives a negative one, so — three negatives, hence negative.
How do the sign rules for dividing integers work?
Exactly the same way, because every division is a multiplication fact read backwards.
Since , it must be that . The signs cannot behave differently in the two directions.
So like signs give a positive quotient and unlike signs give a negative one:
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A worked context: a water tank level drops by 24 cm over 6 equal hours, so the change per hour is
a fall of 4 cm each hour.
Two results about zero are examined often. Dividing zero by any non-zero integer gives zero, since . But dividing by zero is undefined — there is no integer that multiplied by 0 gives 7, so names nothing at all.
Since , it must be that . The signs cannot behave differently in the two directions.
So like signs give a positive quotient and unlike signs give a negative one:
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A worked context: a water tank level drops by 24 cm over 6 equal hours, so the change per hour is
a fall of 4 cm each hour.
Two results about zero are examined often. Dividing zero by any non-zero integer gives zero, since . But dividing by zero is undefined — there is no integer that multiplied by 0 gives 7, so names nothing at all.
How do you evaluate an expression with several operations?
Follow the order of operations: brackets, then division and multiplication left to right, then addition and subtraction left to right.
Worked example.
Multiplication and division first: and . Then
With brackets, clear them first:
A temperature example. A cold store cools at per hour for 4 hours starting from :
The boundary case to watch is a minus sign in front of a bracket. In the bracket gives 3, so the answer is ; multiplying the minus into only part of the bracket is the usual error.
Worked example.
Multiplication and division first: and . Then
With brackets, clear them first:
A temperature example. A cold store cools at per hour for 4 hours starting from :
The boundary case to watch is a minus sign in front of a bracket. In the bracket gives 3, so the answer is ; multiplying the minus into only part of the bracket is the usual error.
How do the commutative, associative and distributive properties help?
They let you rearrange a calculation into an easier one without changing its value.
Commutative — order does not matter: .
Associative — grouping does not matter: . Use it to pair the factors that multiply neatly.
Distributive — a multiplier spreads across a sum or difference:
This is the one that saves real time. To find , rewrite 98 as :
Run the other way, it collects a common factor: .
One restriction matters. Subtraction and division are not commutative — and differ, and so do and — so these rearrangements apply to addition and multiplication only.
Commutative — order does not matter: .
Associative — grouping does not matter: . Use it to pair the factors that multiply neatly.
Distributive — a multiplier spreads across a sum or difference:
This is the one that saves real time. To find , rewrite 98 as :
Run the other way, it collects a common factor: .
One restriction matters. Subtraction and division are not commutative — and differ, and so do and — so these rearrangements apply to addition and multiplication only.
Exam tip
Exam tip: deciding the sign before you calculate
Work out the sign first, write it down, and only then multiply or divide the values. Splitting the job in two stops the sign being forgotten in the middle of the arithmetic.
For a product of several factors, count the negatives rather than tracking them one pair at a time: even count means positive, odd count means negative. So is positive 24 before you compute anything.
In a mixed expression, do not let a negative sign jump the queue. In the multiplication happens first, giving , not .
And write brackets around every negative factor as you go, as rather than . Two operation signs side by side are what invites the slip.
For a product of several factors, count the negatives rather than tracking them one pair at a time: even count means positive, odd count means negative. So is positive 24 before you compute anything.
In a mixed expression, do not let a negative sign jump the queue. In the multiplication happens first, giving , not .
And write brackets around every negative factor as you go, as rather than . Two operation signs side by side are what invites the slip.
Did you know
Why does an odd number of negative factors give a negative answer?
Because each negative factor flips the sign, and flipping an odd number of times cannot return you to the start.
Begin at positive. One negative factor takes you to negative, a second brings you back to positive, a third to negative again. Only an even number of flips lands where it began.
It is the same reason a switch pressed an odd number of times ends up in the opposite state — so you can read off the sign of as negative without multiplying a thing.
Begin at positive. One negative factor takes you to negative, a second brings you back to positive, a third to negative again. Only an even number of flips lands where it began.
It is the same reason a switch pressed an odd number of times ends up in the opposite state — so you can read off the sign of as negative without multiplying a thing.
Key takeaways
Multiplying and dividing integers: quick revision
- Like signs give a positive product or quotient; unlike signs give a negative one.
- The number pattern forces the rule: as a factor steps down past zero, the products keep rising, so .
- In a product of several factors, an even number of negatives gives a positive and an odd number gives a negative.
- Division follows multiplication: divided by a non-zero integer is , but division by is undefined.
- Order of operations: brackets, then division and multiplication left to right, then addition and subtraction.
- Commutative and associative properties apply to addition and multiplication only; the distributive property turns into .
You will remember all of this far better after answering five questions on it than after reading it twice.
- The number pattern forces the rule: as a factor steps down past zero, the products keep rising, so .
- In a product of several factors, an even number of negatives gives a positive and an odd number gives a negative.
- Division follows multiplication: divided by a non-zero integer is , but division by is undefined.
- Order of operations: brackets, then division and multiplication left to right, then addition and subtraction.
- Commutative and associative properties apply to addition and multiplication only; the distributive property turns into .
You will remember all of this far better after answering five questions on it than after reading it twice.