Deep Sea Series · Quiz 21 · Adding and Subtracting Signed Numbers

The Guide

Before you begin

This quiz is about adding and subtracting positive and negative numbers. There are four cases to recognize: add a positive, add a negative, subtract a positive, and subtract a negative. And there are three rules for working them out, two for adding and one for subtracting. The page Signed Numbers sets them out one at a time. This quiz shows why the rules are true, using one number line and the two directions you can walk along it. A rule you can see the reason for is a rule you can check yourself.

A number is a place; an operation is a walk

Quiz 20 established that a signed number tells you where you are: −6 degrees, −850 meters, a balance of −45 dollars. Adding and subtracting tell you how you move. The number you start at is a position. The number you add is a journey, and its sign is the direction of travel.

Adding walks you in the direction the sign points. Add a positive and you move right, toward larger. Add a negative and you move left, toward smaller. That is the whole of addition.

Drawn to scale: every tick is one unit. The second journey is longer than the first, so it carries you past zero and out the other side.

Look at what that figure just did. Something was added to 5 and the answer came out smaller than 5. Quiz 11 warned you that multiplying could shrink a number and Quiz 12 that dividing could grow one. Here is the last member of that family: below zero, adding can shrink and subtracting can grow, and for the same reason each time. The operation does what it always did; the numbers have simply acquired a direction.

Two journeys the same way, or two journeys against each other

When the two signs match, both walks go the same way. Their lengths add up and the direction never changes, so −7 + (−5) is twelve units of leftward travel: −12. Owe seven, then owe five more, and you owe twelve. This is Rule 1: when the signs are the same, add the absolute values and keep the sign.

When the signs differ, the walks oppose each other and cancel as far as they can. What survives is the difference of the two distances, pointing whichever way the longer journey pointed. So for 12 + (−5): the distances are 12 and 5, the difference is 7, and the longer journey was the positive one, so the answer is 7. For −12 + 5 the same subtraction gives 7 again, but now the longer journey was the negative one, so the answer is −7.

Different signs, in two steps (Rule 2). First subtract (the smaller absolute value from the larger). Then give the result the sign of the original larger absolute value. A number's absolute value is its distance from zero, which Quiz 20 explained.

Subtracting means turn around

Now the part that usually arrives as a slogan. Subtracting does not introduce a new kind of arithmetic. It takes the journey you were handed and reverses it. Adding 3 walks you three units right; subtracting 3 walks you three units left. Adding −3 walks you three units left; subtracting −3 therefore walks you three units right.

Put as a rule, this is Rule 3: subtracting a number gives the same answer as adding its opposite. So 8 − (−3) is 8 + 3, and 8 − 3 is 8 + (−3). Some teachers call this “keep, change, change.” The Signed Numbers page explains when that short name helps and when it misleads.

Drawn to scale: both arrows are three units long. The same journey, faced the other way, because subtracting reverses whatever it is given.

Why that is not a trick

Money makes it obvious, and it is worth doing once so the rule never has to be trusted on faith. Suppose your balance is −$80: you owe eighty dollars. Now $30 of that debt is written off. A debt is a negative, so what has been removed is −30, and the arithmetic reads −80 − (−30). Your new balance is −$50.

You are thirty dollars better off than you were. Nothing arrived in your hand, but something was taken away from the wrong side of the ledger, and taking away a negative is a gain. That is all two minuses make a plus ever meant. Not a rule about symbols: a fact about debts.

The guard

Ask which way you end up before you work out how far. If the negative journey is the longer one, the answer is negative, whatever the digits turn out to be. And a sanity check that catches most slips: adding a negative should leave you worse off, and subtracting one should leave you better off. If your answer moved the wrong way, the sign is wrong.

Worked Examples

Three to study before you start

Example 1 · Both journeys the same way

Work out −7 + (−5). Both signs are negative, so both walks head left. Seven units left, then five more, is twelve units of leftward travel, and the answer is −12. In words: owe seven dollars, then owe five more, and you owe twelve. Note that nothing was subtracted anywhere, and the number still got smaller, because the direction did that, not the operation.

Example 2 · Two journeys against each other

Work out 12 + (−5), and then −12 + 5. Both have the same two distances, 12 and 5, and in both the walks oppose each other, so both cancel down to a difference of 7. What separates them is which journey was longer. In the first, the twelve was positive, so the survivor points right: 7. In the second, the twelve was negative, so the survivor points left: −7. Same subtraction, opposite signs, and the deciding question was only ever which number sat further from zero.

Example 3 · Taking away a debt

A balance stands at −$80 and $30 of the debt is forgiven. Since a debt is negative, the amount removed is −30, so the calculation is −80 − (−30). Subtracting reverses the journey: the −30 would have walked thirty units left, so subtracting it walks thirty units right. The new balance is −$50. Still in the red, but thirty dollars nearer the surface than before, which is the sense in which taking away a negative is a gain.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. If a question resists, draw the line and walk it.

Work out: −7 + (−5)
Work out: 12 + (−5)
Work out: −6 − 4
Which of these is another way of writing 8 − (−3)?
- A) 8 − 3
- B) 8 + 3
- C) −8 + 3
- D) −(8 + 3)
Work out: −15 − (−15)
A checking account is overdrawn, showing a balance of −$45. A paycheck of $120 is deposited. What is the balance now?

ReadingA reading rest stop, for the stubborn ones.

Five minutes with the Monterey Bay Aquarium's online exhibit, The Midwater, which is the stretch of ocean where the lanternfish in question 10 does its traveling.

Still no picture at this rest stop, and still for the same reason: the 1885 plate that carried Quizzes 14 to 19 has been used up, and we would rather leave the space honest than fill it with something we cannot vouch for.

Denise works out −9 + 4 and gets 5. What went wrong?
- A) Nothing; 5 is correct.
- B) She found the difference of the distances correctly but gave it the wrong sign. The 9 is further from zero than the 4, so the answer keeps the negative: −5.
- C) She should have added the two distances to get 13.
- D) She should have subtracted 4 from 9 and then subtracted again.
At 5 a.m. the temperature was −6°F. By one in the afternoon it had risen 21 degrees. What was the temperature then?
A friend forgives $50 of a debt you owe. Using this example, explain why subtracting a negative leaves you better off.
A lanternfish spends the day at −480 m. At dusk it rises 430 m to feed. At dawn it descends 455 m again. (a) Where is it at night? (b) Where is it the next morning? (c) How far below the surface is that?

Answer Key

Check your work

Open the key: after you've finished all ten

1 −12

Both journeys head left, so their lengths add and the direction is unchanged. Seven left then five more left is twelve left. Answering −2 means treating the second sign as an instruction to turn back, and nothing here asked for a reversal: both numbers were simply negative.

2 7

Opposite directions, so they cancel as far as they can. The distances are 12 and 5, the difference is 7, and the longer walk was the positive one, so the answer points right. Sense check: a debt of five against twelve dollars in hand leaves seven.

3 −10

Subtracting a positive 4 means walking four units left, and you began six units left already, so you finish ten units out. Written as a journey it is −6 + (−4), which is question 1 again in different clothes. Six degrees below freezing, then four degrees colder, is ten below.

4 B.

Subtracting reverses the journey it is given. The −3 would have walked three units left, so subtracting it walks three units right, which is exactly what adding 3 does. Both come to 11. A) gives 5, which is what 8 + (−3) would give, and those two expressions differ by six units, so the confusion is expensive. C) and D) invent a minus sign in front of the 8 that was never there.

5 0

Fifteen units left, then reverse a fifteen-unit leftward journey, which walks fifteen units back to the right. You end where you began, at zero. Any number minus itself is zero, and that stays true on the negative side of the line, which is a quiet but useful confirmation that nothing has changed except direction.

6 $75

Start at −45 and walk 120 units right. The first 45 of those only bring you back to zero, and the remaining 75 are actual money. Written out: the distances are 45 and 120, the difference is 75, and the longer journey was the deposit, so the balance is positive. Worth noticing that $45 of that paycheck was spent before it arrived.

7 B.

Denise's subtraction is correct: 9 − 4 is 5, and 5 is the right distance. What she dropped was the sign, which belongs to whichever number sat further from zero, and that was the −9. The answer is −5. This is her habit from Quiz 3, where she took the smaller digit from the larger to dodge a trade, and from Quiz 14, where she lined up the right-hand edges because that was the easy thing to do. Each time the digits are handled correctly and something structural falls off. The guard catches it: starting nine units below zero and walking only four units back cannot possibly land you above zero.

8 15°F

Start at −6 and walk 21 degrees right. Six of those degrees are spent getting to zero and the other fifteen carry you above it. If you answered −27 you added the distances as though both journeys went the same way, but a rise and a temperature below freezing point in opposite directions.

9 Because a debt is a negative amount, so removing $50 of debt is subtracting −50, and that moves your balance $50 in the positive direction.

Any answer of that shape earns the mark. The version to carry: nothing was handed to you, but something was taken off the wrong side of the ledger, and that leaves you exactly as much better off as if it had been. This is the entire content of the phrase two minuses make a plus, and it is a fact about debts rather than a rule about symbols.

10 (a) −50 m · (b) −505 m · (c) 505 m

Rising is a positive journey: −480 + 430 = −50, which is fifty meters below the surface and well up in the lit water. Descending is negative: −50 − 455 = −505. Part (c) asks for a distance rather than a position, so the sign comes off and the answer is 505 meters, which is Quiz 20's absolute value doing the last step. Notice the fish did not return to where it started, since the descent was longer than the rise by twenty-five meters, and unequal journeys are exactly why the two operations need separate accounting.

Reading your results

Questions	The skill they test	If they gave trouble
1, 3	Both journeys the same way	Add the distances, keep the direction. Two negatives added is not a reversal.
2, 6, 8	Journeys against each other	Subtract the smaller distance from the larger, then take the sign of whichever number was further from zero.
4, 5, 9	Subtracting a negative	Subtracting reverses the journey. Think of a debt being written off, and the rule stops needing to be remembered.
7	Where the sign comes from	The signature error of the topic. The distance and the direction are two separate answers, and both have to be produced.
10	Several moves, then a distance	Take one journey at a time and write down where you are after each. Then check whether the question wants a position or a distance.

Eight or more right: the line is yours in both directions, and Quiz 22 is next. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: work through Parts 3 to 8 of Signed Numbers, which take the four cases and the three rules one at a time. Then draw a line from −20 to 20 and walk each problem with your finger, saying the case out loud before you move. The rules describe what your finger does, and the walking shows why they are true. Then come back to these same ten.

Where this goes

Next in the Deep Sea Series: Quiz 22 — Multiplying and Dividing Signed Numbers. Two signs, four combinations, and this time the answer really is a short rule. But it will be earned the same way this one was, by watching what repeated journeys do, rather than handed over as something to take on trust.

The Company · An Interlude

The largest migration on Earth, and it is vertical

Question 10 is not invented. Every evening, across most of the ocean, an enormous number of animals swim upward, and every dawn they swim back down. Lanternfish, shrimp, squid, jellyfish and zooplankton spend the daylight hours a few hundred meters down where the water is dark enough to hide in, and climb at dusk to feed in the richer water near the surface under cover of night. Measured by the sheer weight of animals moving, this diel vertical migration is the largest migration that has ever been measured on this planet, and it happens twice a day, every day, and almost nobody has seen it.

We know about it because of a mistake. During the Second World War, crews using the newly developed sonar kept finding a sea floor that had no business being there: a solid-looking bottom at three to five hundred meters, in water known to be far deeper. It was shallower at night than during the day. Some of the first people to record it reported that they had discovered sunken islands.

What they had actually found was several million small fish. Lanternfish carry gas-filled swim bladders, and a swim bladder reflects sound almost as well as rock does, so a dense enough layer of them returns an echo that looks like ground. The layer got a name, the deep scattering layer, and once people understood that it rose and sank on a daily schedule, the schedule turned out to be the point. The false bottom was a commute.

And the detail worth keeping. The layer sits deeper on nights when the moon is out, because moonlight reaches far enough down to make an animal visible. When a cloud crosses the moon, the layer rises. Several hundred meters of ocean, moving in response to a cloud.

Which is what this quiz is about, done at scale. Every one of those animals is running the same arithmetic your fingers were running on the number line: a position, then a journey, then a new position. Minus four hundred and eighty, plus four hundred and thirty, minus four hundred and fifty-five. They do it without instruments, on a schedule set by the light, and they get it right within about a quarter of an hour of sunrise.