Deep Sea Series · Quiz 20 · Below Zero

The Guide

Before you begin

Six quizzes have been spent in the deep sea without anybody mentioning the obvious. Every depth in every one of them was a negative number. The anglerfish at a thousand meters down was at −1,000. The mud star was near −5,000. The trench floor is at −10,994. You have been reading negative numbers since Quiz 14 and nobody made you say so.

That is the useful way in. A negative number is not a strange kind of number and it is certainly not less than nothing. It is a position, measured from a place somebody chose to call zero.

Zero is a decision

Sea level is not the bottom of anything. It is a line drawn on the side of the ocean because it was convenient, and everything gets measured from it: mountains up, trenches down. Freezing point is the same, a place chosen on a thermometer, with plenty of temperature on both sides of it. A bank balance of zero is not an empty vault, it is the line between having and owing.

Once the zero is chosen, the numbers run away from it in two directions, and the sign tells you which way.

Drawn to scale: every tick is one unit. Further to the right is always larger, on both halves of the line.

Opposites

Two numbers are opposites when they sit the same distance from zero on different sides. The opposite of 3 is −3. The opposite of −12 is 12. Taking the opposite twice brings you home, because the opposite of the opposite of 7 is 7 again. And zero is its own opposite, since it is already there and has nowhere to go.

Which is bigger, and the trap

On a number line, further right is greater, and that rule does not change when you cross zero. So −3 is greater than −8, even though 8 is greater than 3. This is the single most common error in the topic, and it comes from reading size off the digits without looking at the sign.

Say it in words and it stops being strange. Owing three dollars is better than owing eight. Three degrees below freezing is warmer than eight below. Three meters down is nearer the surface than eight meters down. In every case the one with the smaller digit is the larger number, because it is closer to the zero we agreed on.

The rule in one line. With negatives, the bigger the digit, the smaller the number. Colder, deeper, and more owed are all directions away from zero, and away from zero on the left means down in value.

Absolute value: how far, not which way

Sometimes the direction is exactly what you want to ignore. If you ask how deep a dive was, nobody wants the answer −320. They want 320. That is what absolute value does: it reports the distance from zero and throws the sign away.

It is written with a pair of upright bars around the number. |−15| = 15 and |15| = 15. Both numbers sit fifteen units from zero, one on each side, and absolute value cares only about the fifteen. A distance is never negative, so an absolute value is never negative either. Opposites always have the same absolute value, which is really just the definition of opposites said backwards.

The line the whole series has been living on

Here is that line drawn properly, with sea level as zero, the highest point on land above it and the deepest point of the ocean below. Both bars are to scale.

Drawn to scale. |−10,994| is greater than |8,849|, so the deepest point of the ocean is further from sea level than the highest point on land, by 2,145 meters.

Look at how little room the sunlit and twilight water takes. At this scale the whole of it, from the surface down to a thousand meters, is a strip about as thick as the line itself. Almost the entire ocean is below the reach of the sun, which is the fact this series has been quietly built on.

The guard

Two checks. Before comparing, put both numbers on a line in your head and ask which is further right, not which digit is bigger. And after taking an absolute value, check that the answer came out positive, because a distance never has a sign.

Worked Examples

Three to study before you start

Example 1 · Putting them in order

Order these from least to greatest: 3, −8, 0, 8, −3. Do not sort the digits. Place them on a line instead. The negatives go to the left of zero, and the further left, the smaller, so −8 sits furthest left of all. The order is −8, −3, 0, 3, 8. Notice that the two negatives came out in the reverse of the order their digits suggest, which is the whole difficulty of this topic in one observation.

Example 2 · Opposites, including the awkward one

The opposite of −12 is 12: same distance from zero, other side. The opposite of 7 is −7, by the same reasoning running the other way. And the opposite of 0 is 0, which surprises people. Zero is the only number that is its own opposite, because it is standing on the line between the two directions and has no distance to travel. Note also that a minus sign in front of a negative number means the opposite of, so the opposite of −12 can be written −(−12), and that equals 12.

Example 3 · Which is further from sea level

Everest's summit stands at +8,849 meters and Challenger Deep lies at −10,994 meters. Which is further from sea level? As numbers, −10,994 is much the smaller of the two, since it is far to the left on the line. But the question asks about distance, not position, so take absolute values: |8,849| = 8,849 and |−10,994| = 10,994. The trench wins. It is 2,145 meters further from sea level than the mountain is, which means you could sink Everest into the Mariana Trench and still have well over two kilometers of water above the peak.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. When in doubt, draw the line and put the numbers on it.

Put these in order from least to greatest: −8, 3, −3, 0, 8
What is the opposite of −12?
Work out: |−15|
One diver is at −40 m and another is at −25 m. Which statement is true?
- A) The diver at −40 m is deeper, and −40 is the greater number.
- B) The diver at −40 m is deeper, and −25 is the greater number.
- C) The diver at −25 m is deeper, and −25 is the greater number.
- D) They are at the same depth, since 40 and 25 are both positive distances.
Work out: |8| − |−3|
A submersible sits 320 meters below sea level. (a) Write its position as a signed number. (b) Write the opposite of that number, and say what it would mean.

ReadingA reading rest stop, for the stubborn ones.

Five minutes with the Monterey Bay Aquarium's online exhibit, Down to the Deep, which walks the same line this quiz is about, from the surface downward.

No picture at this rest stop. The engraved plate that carried the last six quizzes has been used up, and we would rather leave the space empty than fill it with something we cannot vouch for.

Andre says that −8 is greater than −3, because 8 is greater than 3. What went wrong?
- A) Nothing; −8 is greater than −3.
- B) He compared the digits and ignored the sign. On the line −8 sits further left than −3, so −8 is the smaller number.
- C) He should have taken the absolute value of both first.
- D) Negative numbers cannot be compared to each other.
Challenger Deep is at −10,994 m and the summit of Everest is at +8,849 m. (a) Which is further from sea level? (b) By how many meters?
Explain why, of two dive depths, the deeper one is always written as the smaller number.
A week of morning temperatures, in degrees Fahrenheit: Monday −4, Tuesday 6, Wednesday −11, Thursday 0, Friday −4. (a) Put them in order from coldest to warmest. (b) Which morning was furthest from zero, and how do you know?

Answer Key

Check your work

Open the key: after you've finished all ten

1 −8, −3, 0, 3, 8

Least means furthest left, so the negatives lead and the larger digit goes first among them. If you wrote −3 before −8 you sorted by digit rather than by position, which is question 7.

2 12

Same distance from zero, other side. Twelve units to the left becomes twelve units to the right. It can also be written −(−12), which is the same instruction in symbols.

3 15

Absolute value is distance from zero, and −15 is fifteen units away. Answering −15 keeps the sign that the bars exist to remove; answering 0 confuses distance with position. A distance is never negative.

4 B.

Both halves have to be right, and this is the crux of the quiz. Deeper means further from the surface, which is a question about distance, so −40 wins that. Greater means further right on the line, which is a question about position, so −25 wins that. A) gets the depth right and the comparison backwards. D) throws the signs away entirely, and the two divers are plainly not in the same place.

5 5

Do the bars first, as though each pair were a set of parentheses: |8| is 8 and |−3| is 3, so the sum is 8 − 3. Answering 11 means subtracting a negative, which is Quiz 21's business and is not what the bars asked for; the sign was gone before the subtraction began.

6 (a) −320 m · (b) +320 m, which would be 320 meters above sea level.

Below sea level is negative because sea level is where the zero was put. The opposite is the same distance in the other direction, which for part (b) means up in the air rather than down in the water. Worth noticing that nothing about the ocean made 320 negative. The choice of zero did.

7 B.

Andre reads size off the digits without asking what the sign is doing, which is exactly his habit from Quiz 9, where a bigger denominator looked like a bigger fraction, and from Quiz 17, where 0.5% looked like a half. Here it makes −8 look larger than −3. It is not: owing eight is worse than owing three. C) is a real operation applied to the wrong question, since taking absolute values would answer which is further from zero, and −8 does win that one, which is precisely why it is a tempting wrong answer.

8 (a) Challenger Deep · (b) 2,145 meters

The question asks about distance from sea level, so compare absolute values: 10,994 against 8,849. The trench is further by 2,145 meters. As positions the two numbers are nowhere near each other, since one is far left on the line and the other far right, and the gap between them as positions is 19,843 meters. Read the question carefully: distance from zero and distance from each other are different quantities.

9 Because depth is measured downward from sea level, and every meter further down is another step to the left on the number line, which makes the number smaller.

Any answer of that shape earns the mark. The version worth remembering: deeper is further from zero, and further from zero on the negative side means smaller in value, even while the distance is getting larger. Depth and value move in opposite directions, and that is the only thing that makes this topic feel strange.

10 (a) −11, −4, −4, 0, 6 · (b) Wednesday, because |−11| = 11 is the largest distance from zero.

Coldest to warmest is the same as least to greatest, and the two Fridays and Mondays at −4 sit together. Part (b) is the quiz's whole distinction in one question: Wednesday is the smallest number and also the furthest from zero, and Tuesday at 6 is the largest number but only six units out. Furthest from zero is a question for the bars.

Reading your results

Questions	The skill they test	If they gave trouble
1, 7, 10	Ordering across zero	Draw the line every time. Further right is greater, and the bigger digit means the smaller number once you are left of zero.
2, 6	Opposites	Same distance, other direction. Zero is its own opposite.
3, 5	Absolute value	Distance from zero, sign discarded. Work out what is inside the bars first, and the result is never negative.
4, 8, 9	Distance against position	Deeper, colder and more owed are all further from zero and smaller in value at the same time. Ask which the question wants.

Eight or more right: the line runs both ways for you now, and Quiz 21 puts it to work. 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. Draw a number line from −10 to 10 on a scrap of paper and keep it beside you. For a week, every time you meet a negative number anywhere, on a forecast, a statement, an elevator panel, point to where it sits. Almost everything in this quiz becomes obvious once the line is in front of you instead of in your head. Then come back to these same ten.

Where this goes

Next in the Deep Sea Series: Quiz 21 — Adding and Subtracting Signed Numbers. You can now say where a number sits. Next is what happens when you move: a rise of 21 degrees from −6, a payment against an overdraft, a climb from the seafloor. The line stays exactly where it is; you walk along it.

The Company · An Interlude

Who decided where zero is

There is no creature at the top of this quiz, because the plate that carried the last six has been used up, and we would rather show you a drawing we made than a photograph we have no right to. So this interlude is about the line instead, and it turns out the line has a history.

Sea level sounds like a fact about the ocean. It is closer to an agreement. The sea is never at one height: it rises and falls twice a day, it stands higher in some places than others, and it is climbing slowly everywhere. What gets called mean sea level is an average, worked out from years of readings at tide gauges, and it comes out slightly different depending on where you take the readings from. The two ends of the Panama Canal do not sit at the same height. Neither do the two ends of a long enough coastline.

So the zero on the drawing above is a human decision, maintained by measurement, and revisited when the measuring improves. In December 2020 China and Nepal announced a joint resurvey of Everest and put the summit at 8,848.86 meters, a little higher than the figure most of us learned. The mountain had not grown much. The instruments had gotten better, and so had the agreement about where to measure from.

Challenger Deep is the same story pointing downward. The figure of 10,994 meters comes from a sonar survey in 2010 and carries an uncertainty of a few tens of meters, and later dives have come back with numbers slightly different again. Nobody has laid a tape measure down there. What exists is a series of careful estimates, all of them counted from a zero that was itself estimated.

Which is the point of this quiz. A negative number never meant less than nothing. It means on the other side of the place we agreed to start counting. Move the agreement and the signs move with it. A basement is −1 from the lobby and +2 from the sub-cellar, and the floor does not shift an inch.

Worth keeping, when the next form or statement or forecast puts a minus sign in front of something. The sign is not a verdict about the number. It is a report on which side of somebody's zero the number is standing.