Class 7 Mathematics Chapter 10: Operations with Integers

Two directions from one place

Some quantities run both ways from a starting point. A temperature can be above or below zero, and a balance can be in credit or in debt.

Numbers that carry a direction like this are the integers. They are the whole numbers together with their negatives.

Writing the direction in

A number with a minus sign in front is negative, so -8 is eight in the downward direction. A plus sign can be written in for clarity, as in +5.

Zero has no direction at all, so it takes neither sign.

A pair that is worth nothing

Take blue counters for positives and red counters for negatives. One blue and one red together are worth nothing, because they pull in opposite directions by the same amount.

A blue with a red is called a zero pair. Adding or removing a zero pair changes nothing.

Adding with counters

To add two integers, put out counters for each and then cancel every zero pair you can see.

Whatever is left is the answer, and its colour gives the sign.

Adding two negatives

Take (-4) + (-6). Put out 4 red counters and then 6 more red counters.

There is no blue among them, so no pair cancels. Ten red counters are left, so the answer is -10.

Adding a positive and a negative

Now take (+5) + (-8). Put out 5 blue and 8 red.

Five zero pairs cancel, using up all the blue. Three red counters are left, so the answer is -3.

Which sign the answer takes

Notice what decided that sign. The reds outnumbered the blues, so red survived.

So when the signs differ, the answer takes the sign of whichever side has more, and its size is the difference between the two.

Subtracting is taking away

Subtracting means removing counters. For 9 - 4, put out 9 blue and take 4 of them away, leaving 5.

That much is familiar. The interesting cases are the ones where what you are asked to remove is not there.

When there is nothing to take

Take (-2) - 5. Put out 2 red counters. Now you must remove 5 blue, and there is no blue at all.

So the counters in front of you are not enough to work with yet.

Add a zero pair first

Here is the move that solves it. Adding zero pairs changes nothing, so add as many as you need.

Add 5 zero pairs to the 2 red. You now have 5 blue and 7 red, which is still worth -2, and the 5 blue can be removed. Seven red are left, so the answer is -7.

Subtracting a negative

Now the case that is usually just memorised. Take 7 - (-3), which asks you to remove 3 red from 7 blue.

There is no red, so add 3 zero pairs. You now have 10 blue and 3 red, and removing the 3 red leaves 10 blue. The answer is 10.

Why that came out larger

Look at what the counters did. To make 3 red available, 3 blue had to come in alongside them.

Removing the red leaves those extra blue behind. That is why subtracting a negative adds, and it is a consequence rather than a rule.

A rule worth stating

So subtracting any integer gives the same answer as adding its opposite. The sum 7 - (-3) becomes 7 + 3, which is 10.

In the same way (-2) - 5 becomes (-2) + (-5), which is -7.

That rule is worth having, because it means there is really only one operation to master. Every subtraction can be turned into an addition before any work begins.

Checking on a number line

A number line gives a second check. Mark zero, with positives to the right and negatives to the left.

Adding a positive moves right and adding a negative moves left. Starting at +5 and moving 8 left lands on -3, which agrees with the counters.

Which check to use when

The counters explain why the rules hold, and the number line is quicker once you trust them.

It is worth being able to do both, because a disagreement between them means a slip somewhere.

What carries forward

A blue counter and a red counter make a zero pair, and adding or removing zero pairs changes nothing. To add, cancel every zero pair and read what is left.

To subtract something that is not there, add as many zero pairs as you need first. That is why subtracting an integer is the same as adding its opposite, and why 7 - (-3) is 10.

Scenario:

A cold store records the temperature as a whole number of degrees, above or below zero. On Monday it starts at +5 and falls by 8 degrees during the day. On Tuesday it starts at -4 and falls by a further 6. On Wednesday a clerk writes 7 - (-3) for the change between two readings and answers 4. On Thursday the store is at -2 and a reading 5 degrees warmer must be subtracted from it. The manager says counters would settle all four, and the clerk replies that counters are only for beginners.

Analysis:

Two of the four are plain additions, one is a sign slip, and one needs zero pairs before anything can be removed.

Activity:

Keep a week of temperature readings, write each daily change as an integer sum, and check every one with counters and then on a number line.

Title:

Counters and a line

Safety:

Nothing hazardous here, though small counters should be kept away from very young children.

What You Notice:

No pairs cancel in the first sum and five in the second, giving -10 and -3. The two subtractions need 5 and 3 zero pairs added and come out at -7 and 10. The number line agrees with all four.

Adding a negative number makes a total smaller, which is why adding does not always make things bigger.

Subtracting a negative is the one case where taking away leaves you with more than you started with.

The counters are not a crutch: they are what turns the add-the-opposite rule from a thing to memorise into a consequence.

A blue with a red is worth nothing.

Subtract by adding the opposite.

Multiplying is repeated adding

For whole numbers, 3 times 4 means three lots of 4. Adding 4 three times gives 12.

That reading still works when one of the numbers is negative, and it is where we should start.

A positive times a negative

Take 3 times -4. Three lots of -4 means adding -4 three times.

Four below zero, then eight below, then twelve below. So 3 times -4 is -12.

The other order

Now take -3 times 4. Here the repeated adding reading is awkward, because there is no such thing as doing something minus three times.

So we need another way in, and the order itself gives us a clue.

Why the order should not matter

For whole numbers, 3 times 4 and 4 times 3 both give 12. Swapping the two numbers never changes a product.

It would be very odd to lose that for integers. If we want one consistent system, -3 times 4 ought to equal 4 times -3, which is -12.

A pattern that runs past zero

Here is a sharper way to see it. Write out a column of products where the multiplier falls by one each time.

Three times 2 is 6, three times 1 is 3, and three times 0 is 0. Each answer is 3 less than the one above.

Reading the pattern down

So carry on down. The next line is 3 times -1, and the answer should be 3 less than 0, which is -3.

Then 3 times -2 is -6 and 3 times -3 is -9. The pattern decides these for us, and it agrees with the repeated adding.

A second pattern

Now start with the multiplicand negative and let the multiplier fall. Take 2 times -4, which is -8, then 1 times -4, which is -4, then 0 times -4, which is 0.

This time each answer is 4 MORE than the one above.

Where that pattern leads

Carry on down and the next line is -1 times -4. It should be 4 more than 0, so it is 4.

Then -2 times -4 is 8, and -3 times -4 is 12. A negative times a negative has come out positive.

Why that is forced, not chosen

Notice what did the work. We did not decide that two negatives make a positive, and nobody voted on it.

The step in the pattern was already fixed by the lines above zero. Keeping it steady leaves no other value possible.

Counters tell the same story

The counters of the last topic give a second argument. Reading -3 times -4 as removing 3 groups of -4 means removing 12 red counters.

Start from nothing and add 12 zero pairs, giving 12 blue and 12 red. Remove the 12 red and 12 blue are left, so the answer is 12.

The two arguments agree

So two quite different routes give the same answer. The pattern argument says no other value is consistent, and the counters say what the operation actually does.

Agreement between two independent arguments is the strongest reason to trust a result.

The sign rule

Collecting the cases gives a short rule. A positive times a positive is positive, and a negative times a negative is positive too.

A positive times a negative is negative, either way round. So the answer is negative exactly when the two signs differ.

Counting the minus signs

Another way to hold it is to count the minus signs in the product. One minus sign makes the answer negative, and two make it positive.

So -5 times -6 is 30, because two minus signs cancel in the sign, and the sizes multiply as usual.

The sizes never change

Notice that the sign rule is the only new thing here. The size of the answer is always just the two sizes multiplied.

For -5 times -6 that size is 30, and for 3 times -4 it is 12. Only the sign had to be decided.

Checking a product

A quick check is to ask whether the sign looks right before trusting the answer. Count the minus signs, decide the sign, and only then multiply the sizes.

Doing it in that order means a sign slip cannot hide inside a long multiplication.

What carries forward

Multiplying by a negative extends repeated adding, and the pattern running past zero forces every value, including the one where both numbers are negative. The counters agree, since removing groups of red leaves blue behind.

The answer is negative exactly when the two signs differ, and positive when they match. The size is always the two sizes multiplied, so only the sign needed deciding.

Scenario:

A diving log records depths as negative numbers of metres. A diver descends 4 metres in each of 3 stages, which the log writes as 3 times -4. A trainee then has to record the reverse: taking away 3 of those descents, written -3 times -4, and he answers -12 because two minuses look like more of the same. He also writes -5 times -6 as 60, saying two minus signs double the answer. The instructor says the first pattern of products settles both, and the counters settle them again.

Analysis:

One sign has been read as a size, and one rule has been invented. The two independent arguments of this topic settle both.

Activity:

Record five dives as integer products, work out each sign before any multiplying, and check two of them with counters.

Title:

Build the columns

Safety:

Nothing hazardous, though small counters should be kept from very young children.

What You Notice:

The first column steps down by 3 and reaches -9 at 3 times -3. The second steps up by 4 and reaches 12 at -3 times -4. The counters give 12 as well, because removing 12 red from 12 of each colour leaves 12 blue.

Nobody chose that two negatives give a positive: the step in the pattern was already fixed by the lines above zero.

Two minus signs change only the sign of a product, never its size, so -5 times -6 is 30 rather than 60.

Multiplying by a negative crosses zero, which is why multiplying does not always make a number bigger.

Signs differ, answer negative.

Settle the sign, then multiply the sizes.

Division undoes multiplication

For whole numbers, 12 divided by 3 is 4 because 4 times 3 is 12. The division asks which number multiplies 3 to give 12.

That reading is the key to the whole topic. It means no new sign rule is needed.

Reading a division as a missing factor

So write any division as a multiplication with a gap. The question "what is -12 divided by 3?" becomes "what times 3 gives -12?"

Once the question is in that form, the work of the last topic answers it.

A negative divided by a positive

Take -12 divided by 3. We want the number that multiplies 3 to give -12.

The size must be 4, since 4 times 3 is 12. The answer must be negative, because a positive times a positive could never give -12. So it is -4.

A positive divided by a negative

Now take 12 divided by -3. We want the number that multiplies -3 to give 12.

Again the size is 4. This time the answer must be negative, because 4 times -3 would give -12 and we want +12. So it is -4 once more.

Both of them negative

Now take -12 divided by -3. We want the number that multiplies -3 to give -12.

The size is 4 again, and this time the answer is positive, since 4 times -3 is -12 exactly. So the answer is 4.

The same rule as before

Collect those three cases and the pattern is familiar. The answer is negative when the signs differ and positive when they match.

That is the sign rule of multiplication, word for word.

Why it has to be the same rule

This is not a coincidence, and it is worth seeing why. Every division is a multiplication question in disguise.

So whatever rule governs the signs of products must govern the signs of quotients too. One rule covers both operations.

Checking by multiplying back

That reading also gives a free check. Multiply your answer by the divisor and the dividend should come back.

For -12 divided by -3 we answered 4. Multiplying 4 by -3 gives -12, which is where we started, so the answer stands.

Two more worked cases

Take -20 divided by 4. The size is 5, the signs differ, so the answer is -5. Checking, -5 times 4 is -20.

Take 18 divided by -6. The size is 3, the signs differ, so the answer is -3. Checking, -3 times -6 is 18.

Division is not always exact

Multiplying two integers always gives an integer, but dividing them may not. Take -12 divided by 5.

There is no integer that multiplies 5 to give -12, so this division has no integer answer at all.

Order matters in division

Multiplication does not care about order, but division does. We found -12 divided by 3 to be -4.

Turn it round and 3 divided by -12 is not even a whole number. So the two numbers in a division cannot be swapped.

Zero as the dividend

Take 0 divided by -5. We want the number that multiplies -5 to give 0, and that is 0.

So zero divided by any non-zero integer is zero, whatever the sign of the divisor.

Dividing by zero

Now try the other way round, with 0 as the divisor. What times 0 gives -12?

Nothing does, because anything times 0 is 0. So dividing by zero has no answer, and it is not something to attempt.

The four operations together

The chapter now covers all four operations on integers. Adding and subtracting were settled by zero pairs, and multiplying and dividing share one sign rule.

Subtracting turns into adding the opposite, and dividing turns into a multiplication question.

A habit worth keeping

In all four, settle the sign before the size. Then multiply or divide the sizes as usual, and check by reversing the operation.

Three short steps, and a sign slip has nowhere to hide.

The habit matters most in a long calculation. A wrong sign there produces an answer of exactly the right size, which is why it survives a glance and only the reverse check catches it.

What carries forward

A division is a multiplication question with a gap, which is why integers need no second sign rule: a quotient is negative when the signs differ and positive when they match. Multiplying the answer by the divisor must return the dividend.

Dividing two integers may give no integer at all, the order cannot be swapped, zero divided by anything non-zero is zero, and dividing by zero has no answer.

Scenario:

A lift log records floors as integers, with basements negative. A technician finds the lift has dropped 12 floors in 3 equal stages and writes -12 divided by 3. A trainee answers 4, saying a division cannot produce a negative. A second entry reads -12 divided by -3 and the trainee answers -4, saying two minus signs must give a minus. A third entry reads -12 divided by 5 and the trainee rounds it to -2. A fourth reads -12 divided by 0 and the trainee writes 0.

Analysis:

Four entries and four different errors: a missing sign, a sign rule reversed, a rounding that hides a real answer, and a division that has no answer at all.

Activity:

Take ten integer divisions from a log or a table, work each one by the missing-factor reading, and check every answer by multiplying back.

Title:

Divide, then multiply back

Safety:

Nothing hazardous here.

What You Notice:

The five divisions give -4, -4, 4, -5 and -3, and each multiplies back to its dividend. The sixth has no integer answer, since no integer times 5 reaches -12. Zero divided by -5 is 0, while -12 divided by 0 has no answer at all.

Integers need no second sign rule at all, because every division is a multiplication question with a gap in it.

Division cares about the order of its two numbers, while multiplication never did.

Zero divided by a non-zero integer is zero, but dividing by zero has no answer, so the two sides of a division treat zero quite differently.

A division is a product with a gap.

Settle the sign, then divide the sizes.