Negative Numbers in Context

You already use numbers like 6 and 20. Negative numbers let you describe amounts below zero. The number −9 means nine below zero. Say “negative nine.” For a temperature, you can also say “minus nine” or “nine below zero.”

Keep the sign with the number

The minus sign in −9 is part of its name. On its own, −9 tells you a number. You do not need another number to take nine away from.

A positive number is greater than zero. You can write positive nine as 9 or +9. Both mean the same thing. The plus sign is often left out. The minus sign in −9 must stay: 9 and −9 are different numbers.

Zero is neither positive nor negative. Think of it as the meeting place between the two sides. The integers are 0, the positive whole numbers, and their negative partners: …, −3, −2, −1, 0, 1, 2, 3, … . A number such as 1.5 is not an integer.

Ask what zero means

Zero has a job in each story. For elevation, zero means sea level. For a simple money balance, $0 means no money held or owed. On a Celsius thermometer, 0 °C is water’s freezing point.

Imagine a sign that says, “9 feet below sea level.” Write the height as −9 feet. A sign saying “9 feet above sea level” gives a height of 9 feet. The amount is nine feet in both cases. The sign tells you which side of sea level it lies on.

Words help you choose a direction. “Gain 6 points” describes a change of +6 points. “Lose 6 points” describes a change of −6 points. A loss tells you the change; you still need the starting score to find the new score.

Smaller below zero

Below zero, going farther down gives a smaller number. So −10 is less than −2. The digits alone can mislead you: ten is greater than two, but ten below zero is lower than two below zero.

A thermometer is like a number line standing up. Higher readings are warmer. Lower readings are colder. Find zero: negative readings sit below it, and positive readings sit above it.

Check °F or °C

The letters beside the numbers tell you the scale. Fahrenheit (°F) is used for everyday US weather. Celsius (°C) is used in science and for weather in most other countries.

Water freezes at 0 °C, the same temperature as 32 °F. A day at 20 °F is below freezing even though 20 is positive. “Below zero” and “below freezing” do not mean the same thing on a Fahrenheit thermometer.

The two scales use different numbers for the same temperature. For example, 0 °F is about −18 °C. Also, −5 °C is the same temperature as 23 °F. Read the unit before comparing two temperatures. A smaller number on a different scale need not mean colder.

Find the value of one space

Imagine a Fahrenheit thermometer with labels at 0 and 12. There are four equal spaces between those labels. The liquid ends at the third mark below zero.

1. The gap from 0 to 12 is 12 degrees.
2. Four spaces share that gap: 12 ÷ 4 = 3 degrees per space.
3. Count down from zero: −3, −6, −9.

The reading is −9 °F, or “nine degrees below zero.” Count the spaces between labels, not the marks including both ends. Four spaces have five end marks.

You can check from another label. If −12 is one space below the liquid, move up three degrees: −12 to −9. Moving up makes the number greater, even below zero.

Read between the marks

The liquid can stop between marks. Halfway between −6 °F and −4 °F is −5 °F. This value is colder than −4 °F and warmer than −6 °F.

A home freezer should be at 0 °F or colder. A reading of −12 °F is colder than 0 °F; a reading of 20 °F is warmer. All three readings use Fahrenheit, so you can compare their positions directly.

A minus sign can describe a place or an amount you owe. To read it, find the zero level first. Then ask whether the number is above or below that level.

Heights and depths

Elevation is height compared with sea level. Sea level is 0 feet. A hill 60 feet above sea level has elevation 60 feet. A place 60 feet below it has elevation −60 feet.

Badwater Basin in California is 282 feet below sea level: −282 feet. Telescope Peak, in the same national park, is 11,049 feet above sea level. Its elevation is positive.

For an underwater position, you can use the water’s surface as zero. A diver 20 feet below the surface is at −20 feet. A fish at −35 feet is deeper. Both are below zero, but the fish is farther down.

Floor labels

In a building with the ground floor numbered 0, floor 1 is just above ground. Floor −1 is just below ground. Floor −2 is one floor below −1.

Some buildings use a different list: B2, B1, 1, 2, 3. Here, ground level is floor 1 and B1 is directly below it. There is no floor 0 to pass. Read the floor labels before you count an elevator trip.

Money held and money owed

A balance keeps track of money you have or owe. Having $12 gives a balance of $12. Owing $5 gives a balance of −$5. A balance of $0 means neither money held nor money owed.

Suppose a class fund has $6 and buys $10 of supplies. The teacher lends the class the missing money.

1. The class uses its $6, bringing the balance to $0.
2. It still needs 10 − 6 = 4 dollars.
3. The class now owes $4, so its balance is −$4.

The minus sign does not mean the supplies cost a negative price. It shows the money the class owes after buying them.

Scores and changes

A football play that loses 4 yards has a change of −4 yards. In golf, par is the expected number of strokes. A score of −2 is two strokes under par. Golf rewards the lowest score, so a negative score can be a good result.

Draw a straight line and mark zero. Give each equal space the same number value. Your line can keep going in both directions, even beyond the ends you draw.

Put the sides in order

On a horizontal line, numbers get greater to the right. Starting at zero and moving left by ones gives −1, −2, −3. Moving right gives 1, 2, 3.

Read this stretch from left to right: −4, −3, −2, −1, 0, 1, 2, 3, 4. The minus signs stay with the numbers left of zero. The number −1 belongs next to zero; −4 belongs farther left.

On a vertical line, numbers get greater upward. Positive numbers sit above zero. Negative numbers sit below it, just as on a thermometer.

Work out the step

Imagine a line where −24 and 0 are six equal spaces apart. What number belongs at the second mark left of zero?

1. The distance from −24 to 0 is 24 units.
2. Each of the six spaces is worth 24 ÷ 6 = 4 units.
3. From zero, count left: −4, then −8.

The second mark is −8. The labels from −24 to 0 are −24, −20, −16, −12, −8, −4, 0. There are seven marks and six spaces. The step stays four on both sides of zero, so the next mark to the right is 4.

A common slip is to count marks when finding the step. Put your finger in each gap as you count the spaces between the two labels.

Find places between labels

Numbers can sit between the marked values. Halfway between −10 and 0 is −5. You travel five units from either end to reach it.

Halfway between −2 and −1 is −1.5. It belongs just left of −1, not on the positive side. The dot in 1.5 is a decimal point; it helps show half a unit.

You can also estimate a place. On a line labeled −100, 0, and 100, put −50 halfway between −100 and 0. Put −90 much closer to −100. Keep the spaces in proportion: a ten-unit gap must look shorter than a fifty-unit gap.

When you count back by one, zero sits between 1 and −1. Say each landing in order: 4, 3, 2, 1, 0, −1, −2.

Count jumps after the start

Start at 2 and count back four jumps. Your landings are 1, 0, −1, −2. You finish at −2. The starting 2 is where you begin; it is not the first jump.

Skipping zero would make 1 to −1 look like one jump. It takes two: 1 to 0, then 0 to −1.

Counting forward goes the other way. From −3, count −2, −1, 0, 1, 2. Below zero, the digits get smaller as the numbers get greater.

Keep bigger jumps the same size

You can count in twos, threes, fours, or fives. The step must stay the same when you cross zero.

Counting back by five from 10 gives 10, 5, 0, −5, −10.

Now start at 8 and count back by three: 8, 5, 2, −1, −4. This pattern passes zero without landing on it. Here is the jump from 2 to −1:

1. Move back two units from 2 to 0.
2. One unit of the three-unit jump is left.
3. Move that last unit to −1.

Stopping at zero would shorten this jump to two units.

Continue a pattern

Look at 14, 10, 6, 2, … . Each landing is four less than the one before. To find the next number, move four units back from 2. Two reach zero and two more reach −2. Another four-unit jump reaches −6.

For a forward pattern, try −11, −9, −7, −5, −3, −1, 1. Every jump is two units to the right. The jump from −1 to 1 passes zero halfway through.

To check a missing number, look at the jumps on both sides of it. In 6, 3, __, −3, the missing number is 0. Both 3 to 0 and 0 to −3 are three-unit jumps.

The numbers 8 and −8 sit on opposite sides of zero. Both are eight units away. Numbers with matching distances on opposite sides are opposites.

Find the matching place

To find an opposite, cross zero and keep the same distance. The opposite of −13 is 13. The opposite of 13 is −13. Taking the opposite twice brings you back to your starting number.

Zero stays at zero. Its opposite is itself, because its distance from zero is zero.

You can picture opposites with heights. A platform 8 feet above sea level and a hollow 8 feet below sea level have elevations 8 and −8 feet. The signs give different sides; the distances from sea level match.

Read absolute value bars

A number’s absolute value is its distance from zero. Straight bars around a number ask for that distance: |−13| = 13. Also, |13| = 13 and |0| = 0.

Distance has no negative direction. You can be thirteen units left of zero, but the distance you measure is thirteen units. So |−13| is 13, not −13.

Finding an opposite and finding an absolute value have different jobs. The opposite of 13 is −13, a new place. The absolute value of 13 is 13, a distance. For −13, both questions happen to give 13.

A lower balance can mean more owed

Imagine two balances: −$18 and −$7. The first is lower on the number line. Yet it shows more money owed: $18 compared with $7.

A balance below −$30, such as −$35, means more than $30 owed. The lower balance has a greater distance from zero.

Use two distances

Place −8 and 5 on a line. The first number is eight units from zero. The second is five units from zero. The gap crosses zero, so the full distance is 8 + 5 = 13 units.

Picture two numbers on a number line. The number farther right is greater. On a vertical line, the higher number is greater. Use this rule even when both numbers are negative.

Compare below zero

Which is greater: −14 or −6? Both sit left of zero. The number −6 is closer to zero, so it is farther right. That makes −6 greater than −14.

Do not compare just 14 and 6. Their minus signs matter. Fourteen below zero is lower than six below zero.

Any positive number is greater than zero. Zero is greater than every negative number. So 2 is greater than −90, and 0 is greater than −1. Zero is not the smallest number.

Write < or >

The symbol < means “is less than.” The symbol > means “is greater than.” The open side faces the greater number. The point faces the smaller number.

Write −14 < −6 and read it aloud: “Negative fourteen is less than negative six.” You can turn the statement around: −6 > −14. It says the same comparison in the other order.

Build the statement in two steps. First decide which number is greater. Then place the symbol with its open side toward that number. This helps you keep the symbol from turning the right idea into a wrong sentence.

Say what the comparison means

In a temperature story, greater means warmer. At −6 °F, it is warmer than at −14 °F. Both days are cold, but one reading is higher.

For elevation, greater means higher. A shelf at −6 feet is above a ledge at −14 feet when both use the same zero level.

For a balance, greater can mean owing less. A balance of −$6 is greater than −$14. You would need fewer dollars to bring −$6 back to zero.

For golf scores, the lowest score wins. A score of −5 beats −2 because −5 is the smaller number. “Greater number” does not always mean “better result.”

Read a comparison in words

“The temperature is below −5 °F” means less than −5 °F. A reading of −8 °F fits; −3 °F does not. The word “below” compares with −5, not with zero.

Ascending order means least to greatest. Follow a number line from left to right. Descending order means greatest to least. Follow it from right to left.

Start with three groups

Put these numbers in ascending order: 7, −9, 0, −2, 15, −13.

1. Gather the negatives: −9, −2, −13. They all come before zero.
2. Order them from farthest left: −13, −9, −2.
3. Put zero next, then the positives from least to greatest.

The finished list is −13, −9, −2, 0, 7, 15. Each number is greater than the one before.

A common slip is to begin the negative group with −2. That number is closest to zero, so it belongs last among these negatives. The number −13 is the smallest of the whole list.

For descending order, reverse the finished list: 15, 7, 0, −2, −9, −13. Check the requested direction before writing your answer.

Order real readings

Three ponds have temperatures −8 °C, 1 °C, and −3 °C. From coldest to warmest, they go −8 °C, −3 °C, 1 °C. The coldest reading is the smallest.

In golf, the smallest score is best. For scores −6, 1, and −2, best to worst is −6, −2, 1. Read what the order means in the story, as well as which numbers are greatest.

Include numbers between marks

The number −1.5 lies halfway between −2 and −1. In ascending order, these numbers go −2, −1.5, −1. It is tempting to put −1.5 after −1 because 1.5 is greater than 1. On the negative side, −1.5 lies farther left.

List integers between two ends

Which integers lie between −3 and 3? Begin just after −3 and stop just before 3: −2, −1, 0, 1, 2. There are five integers.

“Between” leaves out the two ends here. Including −3 and 3 would answer a different question. Keep zero in your list when it lies inside the gap. If you need a count, list the integers first so you can see each one.

The difference between two numbers tells you how far apart they are. Here, a difference is a distance. It can be zero or positive, but never negative. Use words such as “warmer” or “lower” to give a direction.

Split a gap at zero

At night it is −5 °F. In the afternoon it is 8 °F. How much warmer is the afternoon? Both parts of the trip count:

1. From −5 to 0 is five units.
2. From 0 to 8 is eight units.
3. Together, the change is 5 + 8 = 13 degrees.

The afternoon is 13 degrees warmer. Its final reading is 8 °F; thirteen degrees tells you how much it changed.

Subtracting 5 from 8 would give 3. That measures the gap from positive 5 to 8. Starting at −5 adds a whole stretch below zero, so three units cannot cover this trip.

Check by counting on thirteen from −5. Five jumps reach zero; the remaining eight reach 8.

When both numbers are below zero

Find the distance from −11 to −4. You travel toward zero but stop at −4. The trip does not reach zero or cross it.

The distances from zero are eleven and four. Subtract them: 11 − 4 = 7 units. You can also count seven jumps: −10, −9, −8, −7, −6, −5, −4.

Adding 11 and 4 would count two trips to zero. That would be too far for this gap, because both ends sit on the same side.

For two positive numbers, subtract the smaller from the greater. From 4 to 9 is 9 − 4 = 5 units. If one end is zero, use the other number’s distance from zero: from −7 to 0 is seven units.

Count spaces, not end marks

From −1 to 2, the marked numbers are −1, 0, 1, 2. There are four marks but only three spaces. The difference is three units. Start counting after the first mark so it does not become an extra jump.

A temperature rise means warmer: move up a thermometer or right on a number line. A fall means colder: move down or left. Keep the same scale, °F or °C, for the whole calculation.

Rise through zero

It is −7 °F, then the temperature rises nine degrees.

1. Seven degrees of the rise take you to 0 °F.
2. There are 9 − 7 = 2 degrees left.
3. Move up those two degrees to 2 °F.

The change was nine degrees. The final temperature is 2 °F. They answer different questions.

Fall through zero

A temperature of 6 °C falls ten degrees. The first six degrees reach zero. Four degrees remain, so the new reading is −4 °C.

Count jumps after the start. If you count the starting 6 as the first jump, you will stop one degree too soon.

Not every change crosses zero. A fall of three degrees from −4 °F ends at −7 °F. A rise of four degrees from −10 °F ends at −6 °F. A rise gives a greater reading even when it stays negative.

Find how much it changed

From −12 °F to −3 °F is a rise of nine degrees. Both readings are negative, so use the nine-unit gap between them. From 4 °F to −6 °F is a fall of 4 + 6 = 10 degrees.

For several readings, compare one pair at a time:

The changes are a fall of five, a fall of seven, and a rise of ten degrees. The biggest change is the last one, even though the final reading is only 1 °F.

Work backward to a start

At dawn, it is −3 °F after a fall of eight degrees. What was the temperature before the fall?

Undo the fall by rising eight degrees from −3 °F. Three degrees reach zero, leaving five. The starting reading was 5 °F.

Check in the story’s direction: from 5 °F, falling five degrees reaches zero. Falling three more reaches −3 °F. That uses all eight degrees.

For heights, floors, and money, use the same number line ideas. Decide what zero means, then follow each change. Write the first result before starting the second change.

Heights across sea level

Badwater Basin is at −282 feet. Telescope Peak is at 11,049 feet. Their elevation difference crosses sea level: 282 + 11,049 = 11,331 feet. This is a difference in height, not the length of a walking route.

Now picture a diver at −18 feet, with the water’s surface at zero. The diver swims up eleven feet and reaches −7 feet. A bird is 25 feet above the water. Their vertical gap is 7 + 25 = 32 feet. The diver’s new depth is needed for that second calculation.

Follow an elevator

In a building with ground floor 0, an elevator starts at floor −3. It goes up eight floors, then down six floors.

1. Going up three floors reaches 0. Five floors remain, so it reaches floor 5.
2. Start the next trip at 5. Down five floors reaches 0; one more reaches floor −1.

The elevator has traveled fourteen floors in all: eight up and six down. Its final floor is −1. “How far did it travel?” and “Where did it stop?” need different answers.

A building without floor 0 needs careful counting. If the labels go B1, 1, 2, 3, the trip from B1 to 3 has three floor gaps. Count B1 to 1, 1 to 2, and 2 to 3. Adding an unseen floor 0 would make the answer one too large.

Earn money, then spend

Leah owes $9. Her balance is −$9. She earns $14, then spends $8.

First, nine dollars pays the debt. Five dollars remain, so the balance becomes $5. Next, spending $8 uses that five and leaves three dollars owed. The final balance is −$3.

Check the meaning: earning made the balance greater. Spending made it smaller. The minus sign tells you Leah owes money again.

Choose the answer the question needs

A final temperature, floor, elevation, or balance can be negative. A distance such as “how far apart?” cannot be negative. If a story asks how much is owed, give the debt amount: a balance of −$3 means $3 owed.