Deep Sea Series · Quiz 22 · Multiplying and Dividing Signed Numbers

The Guide

Before you begin

This time the rule really is short, and you have probably heard a piece of it already: two negatives make a positive. The trouble with that sentence is that it is not a reason, it is a noise, and a rule you cannot check is a rule that will desert you the moment you are tired. So the rule is at the bottom of this Guide rather than the top. Three of the four cases fall straight out of Quiz 21. The fourth needs a real argument, and there is one.

Multiplying is doing the same journey several times

Quiz 21 turned every number into a place and every operation into a walk. Multiplication is that walk repeated. So 3 × (−4) means: take the journey four units to the left, and make it three times over.

Drawn to scale: three equal journeys of four units each, all of them leftward.

That handles three of the four cases at once. Three journeys of four right is 12. Three journeys of four left is −12. And since order never matters in multiplication, (−4) × 3 is the same thing said backwards, so it is −12 as well.

Then the wall

Now try (−3) × (−4) with the same picture and watch it fail. You would have to make a journey negative three times, and there is no such thing. You cannot do something a negative number of times. The model has run out, and this is exactly the moment where most people are handed the slogan instead of an argument.

Rate and time

Here is the argument. Change your reading of the two numbers. Let the first one be a rate: how fast something moves and which way, in meters per minute. Let the second be a time: how many minutes, with the future positive and the past negative. Then their product is the change in position.

Neither of those needs teaching. A sub going down has a negative rate. Four minutes ago is a negative time. And now every case has an obvious meaning, including the fourth.

Drawn to scale. The line is the sub's whole history and future at a steady 2 meters per minute down. Read it forward and she is lower; read it backward and she was higher.

There it is, with no slogan anywhere. A thing that is sinking was higher in the past. Sinking is a negative rate, the past is a negative time, and the answer is a rise, which is positive. Negative times negative is positive because something on its way down used to be further up.

Rate	Time	The question	The answer
+3	+4	rising, four minutes from now	+12, higher
+3	−4	rising, four minutes ago	−12, was lower
−3	+4	sinking, four minutes from now	−12, lower
−3	−4	sinking, four minutes ago	+12, was higher

The rule, now that it has been paid for

Same signs give a positive. Different signs give a negative. Work out the digits as though both numbers were positive, then decide the sign separately. Two decisions, never mixed together.

For a string of numbers, just count the negatives. An even number of them is positive, since they pair off and cancel; an odd number leaves one over, so the answer is negative. (−2)(−3)(−4) has three negatives, which is odd, so the answer is negative: −24.

Dividing asks the same question backwards

Division undoes multiplication, so it cannot have a different rule without contradicting itself. If a sub is 84 meters lower after 7 minutes, her rate was −84 ÷ 7 = −12 meters per minute. And if she is 84 meters lower and sinking at 12 meters per minute, the time is −84 ÷ (−12) = 7 minutes, which is in the future, as a positive time should be. Same signs positive, different signs negative, for exactly the same reason.

One warning, and it matters more than the rule

This rule is for multiplying and dividing. It is not for adding and subtracting. Two negatives added together do not make a positive; they make a bigger negative, because both journeys go the same way. −5 + (−3) is −8, and it always will be. The commonest damage done by the slogan is that people carry it into Quiz 21's territory, where it is simply false.

The guard

Decide the sign first, out loud, before you touch the digits. Same or different? Then multiply or divide as if everything were positive, and put the sign you already decided on the front. Doing it in that order means a sign error and an arithmetic error can never hide behind each other.

Worked Examples

Three to study before you start

Example 1 · All four, in one situation

A submersible moves at 3 meters per minute. Take rising as +3 and sinking as −3, four minutes from now as +4 and four minutes ago as −4. Rising, four minutes on: (+3)(+4) = +12, twelve meters higher. Rising, four minutes ago: (+3)(−4) = −12, she was twelve meters lower. Sinking, four minutes on: (−3)(+4) = −12, twelve lower. Sinking, four minutes ago: (−3)(−4) = +12, she was twelve meters higher. That last one is not a trick. It is what sinking means.

Example 2 · Sign first, then digits

Work out (−7) × (−3). Sign first: both are negative, so the signs are the same, so the answer is positive. Say it before going on. Now the digits, as though neither sign existed: seven threes are 21. Put the two decisions together and the answer is 21. Deciding in that order is worth the discipline, because if you work out the digits first the sign becomes an afterthought, and an afterthought is what gets dropped.

Example 3 · Dividing is the same question turned around

A sub is 84 meters lower than she was, seven minutes after starting. What was her rate? That is −84 ÷ 7. Different signs, so negative; 84 divided by 7 is 12; the rate is −12 meters per minute, which is a descent, as it should be. Now the other way: she is 84 meters lower and she has been sinking at 12 meters per minute, so how long? That is −84 ÷ (−12). Same signs, so positive; 84 divided by 12 is 7; the answer is +7 minutes, a positive time, meaning it is ahead of the start rather than behind it.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. For every calculation, write the sign down before you work out the digits.

Work out: (−4) × 6
Work out: (−7) × (−3)
Work out: −36 ÷ 9
A submersible is sinking steadily at 5 meters per minute. Where was she 3 minutes ago, compared with where she is now?
- A) 15 meters lower, because sinking always means going down.
- B) 15 meters higher, because something on its way down was further up before.
- C) 8 meters higher, since 5 and 3 make 8.
- D) At the same depth, since the past does not count.
Work out: −45 ÷ (−5)
Work out: (−2)(−3)(−4)

▶ VideoA video rest stop, for the stubborn ones.

Six minutes with the Monterey Bay Aquarium Research Institute on how and why deep-sea animals make their own light, filmed from their own remotely operated vehicles.

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

Marcus works out −5 + (−3) and answers +8, because two negatives make a positive. What went wrong?
- A) Nothing; +8 is correct.
- B) He used the rule for multiplying on a problem about adding. Two negatives added both go the same way, so the answer is −8. Had it been (−5)(−3), his rule would have applied and given 15.
- C) He should have subtracted 3 from 5 and answered 2.
- D) The rule only works when the two numbers are equal.
A debt of $45 is shared equally among 3 people. Write the calculation as a division of signed numbers, and say what each person owes.
Using something that is sinking, explain in your own words why a negative multiplied by a negative comes out positive.
A submersible descends steadily at 12 meters per minute. (a) How far will she have moved in 7 minutes? (b) Where was she 4 minutes ago, compared with now? Write both as multiplications of signed numbers.

Answer Key

Check your work

Open the key: after you've finished all ten

1 −24

Different signs, so the answer is negative; four sixes are 24. As journeys it is six trips of four units left, which lands you twenty-four units out.

2 21

Same signs, so positive; seven threes are 21. Read it as rate and time: sinking seven meters a minute, three minutes ago, puts her twenty-one meters higher than she is now.

3 −4

Different signs, so negative; 36 divided by 9 is 4. Division carries the same rule as multiplication because it is the same relationship asked backwards: −4 times 9 is −36, which closes the loop.

4 B.

The rate is −5 and the time is −3, so the change is (−5)(−3) = +15: fifteen meters higher. A) is the answer of someone who has decided that "sinking" settles the question without noticing that the time is in the past. C) adds where the situation multiplies. D) is worth a moment, because it sounds almost sensible, and the reply is that arithmetic runs backwards perfectly well: if you know the rate, you know where she was as surely as where she will be.

5 9

Same signs, so positive; 45 divided by 5 is 9. Notice that dividing a negative by a negative gives a positive for the same reason multiplying two negatives does, and not as a separate fact to be learned.

6 −24

Count the negatives: three of them, which is odd, so the answer is negative. Then 2 times 3 times 4 is 24. Working left to right gives the same thing: (−2)(−3) is 6, and 6 times (−4) is −24. Two negatives pair off and cancel; a third has nothing to pair with.

7 B.

Marcus has the right rule in the wrong room, which is his habit: in Quiz 12 he flipped a fraction that was not the one needing flipping, and in Quiz 16 he divided the two numbers the wrong way round. Each time the tool was real and the job was different. Here, adding two negatives sends both journeys the same way, so they pile up: −8. The rule he reached for is genuine, and had the problem been (−5)(−3) it would have given 15. This is the single most expensive confusion in the whole topic, and it exists because the slogan is taught without the room it belongs to.

8 −45 ÷ 3 = −15, so each person owes $15.

The debt is negative because it is money owed rather than held, and the 3 is positive because it is a count of people. Different signs, so the answer is negative, and a negative fifteen dollars is a debt of fifteen dollars. Notice the answer needed translating back into English at the end: the arithmetic gives −15, and what a person actually says is "I owe fifteen."

9 Because sinking is a negative rate and the past is a negative time, and anything on its way down must have been higher up before. So the change works out positive.

Any answer of that shape earns the mark. What is not wanted is "two negatives make a positive," because that repeats the rule instead of giving a reason. The version worth carrying: run the film backwards on something falling and you watch it rise.

10 (a) (−12)(+7) = −84, so 84 meters lower · (b) (−12)(−4) = +48, so she was 48 meters higher.

Descending makes the rate −12 both times, and only the time changes sign. In part (a) the time is ahead, so the signs differ and the answer is negative. In part (b) the time is behind, so the signs match and the answer is positive. One rate, two directions in time, and the arithmetic tells you her whole dive: 48 meters above where she is now, four minutes back, and 84 below it, seven minutes on.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2, 5	The sign rule	Same signs positive, different signs negative. Decide the sign out loud before touching the digits.
3, 8	Dividing	Same rule as multiplying, because division is the same relationship asked backwards.
4, 9, 10	Why the rule is true	Rate and time. A negative rate is going down, a negative time is the past, and something going down was higher before.
6	Strings of negatives	Count them. Even means positive, odd means negative, because they pair off.
7	Which room the rule belongs to	The signature error of the topic. Two negatives added make a bigger negative, always.

Eight or more right: signed arithmetic is complete, and Quiz 23 opens new country. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. Practice one habit only: for a week, every time you meet a signed multiplication or division, say same or different out loud before doing anything else, then work the digits as if both numbers were positive. Almost every lost mark in this topic is a sign decided while distracted by digits. Then come back to these same ten.

Where this goes

Next in the Deep Sea Series: Quiz 23 — Ratios and Unit Rates. This quiz has already used a rate in every question without calling it one. Meters per minute, dollars per hour, miles per gallon: the next quiz is about what the word per is doing, and why it is the most useful preposition in arithmetic.

The Company · An Interlude

Where rate times time gets very strange

Every question in this quiz multiplied a rate by a time. Most of the rates were brisk, because a submersible moving twelve meters a minute is easy to picture. The deep sea also runs some of the slowest processes on the planet, and down there the same multiplication produces numbers that are hard to hold in the head.

Scattered across the abyssal plains, at depths of three to six thousand meters, lie manganese nodules: black lumps, roughly potato-shaped, formed by metal precipitating out of seawater onto a scrap of shell or bone. They grow. The rate is a few millimeters per million years. One estimate puts it between 2 and 15 millimeters in a million years, and a survey team working the Clarion-Clipperton Zone described their accretion as less than half a centimeter every million years, and among the slowest of all geological processes.

Do the multiplication and the size becomes an age. A nodule fifteen centimeters across can be fifteen million years old. It has been sitting on the floor of the Pacific, growing at a rate that would take longer than recorded history to add the thickness of a sheet of paper, since before there were people to notice.

And they were found by the Challenger. The same British expedition of 1872 to 1876 that dredged up half the deep-sea animals anyone knew about in the nineteenth century also brought up the first manganese nodules. Filhol's French voyages on the Travailleur and the Talisman, which supplied the engravings at the head of Quizzes 14 through 19, were sailing in the wake of that one. For a long time nobody knew what the nodules were for, and they were filed as a curiosity.

They are not filed as a curiosity now. Those lumps hold nickel, copper and cobalt, and there are contracts to scrape them off the seafloor across areas the size of a small country. Which puts an arithmetic problem in front of anyone who wants to think about it honestly: something that took fifteen million years to make can be collected in an afternoon, and the rate at which it comes back is the same rate it grew at in the first place. Multiplication does not care which direction you read it in. That is the whole point of this quiz, and it is not always a comfortable one.

There is one more thing worth saying about negative time, since this quiz introduced it. Down there the past is not an abstraction to be calculated: it is stacked up in order, and you can read it. Sediment settles slowly and steadily, so the deeper you drill, the older the layer. A core pulled from the abyssal floor is a negative time axis you can hold in your hands, and every centimeter of it is a multiplication somebody has already done.