Negative numbers
Some amounts run both ways from a starting point, below freezing and above it, below sea level and above it. How do you write the ones on the far side of zero?
Where you are 6 in Integers
Why this exists
The whole numbers 0, 1, 2, 3, \ldots answer “how many” and “how far along”. They start at 0 and only go up, which is fine for apples and pages, but many amounts are measured from a starting point that has something on both sides of it.
On a Celsius thermometer, 0 is the freezing point of water, and the temperature can be 6 degrees above it or 6 degrees below it. Heights are measured from sea level, and a hill top is above it while the sea bed is below it. A bank account can hold money, or it can owe money. In each case “6” on its own is not enough: you have to say which side of the starting point you mean.
Negative numbers are the numbers for the other side. They extend the number line past zero to the left, so that “6 below” becomes a single number, -6, that can be placed, compared and later added and subtracted like any other. They also give an answer to a question the whole numbers cannot answer at all, such as 3 - 5, which is the story of why negative numbers were needed.
Definition
A number is positive when it is greater than 0, and negative when it is less than 0. The number 0 is neither positive nor negative.
On a number line, numbers grow to the right. To make room for numbers less than 0, carry the line on to the left of zero, keeping the marks the same distance apart as on the right. The first mark to the left of 0 is -1, the next is -2, and so on. In general, for a positive number n, the point n units to the left of 0 is the number -n, which is read as negative n. So -3 is three units left of zero, and 3 is three units right of it.
A positive number may be written with a plus sign, as +3, but the sign is usually left off: a number written in digits with no sign in front is positive, unless it is 0. The counting numbers, their negatives and 0 together make up the integers, \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots. Negative numbers need not be whole: -2.5 sits halfway between -3 and -2, and \displaystyle -\frac{3}{4} sits three quarters of the way from 0 to -1. Those are the subject of negative fractions and decimals.
Opposites. A number’s opposite is the number just as far from 0 as it is, on the other side of 0. The opposite of 2 is -2 and the opposite of -2 is 2, so 2 and -2 are opposites of each other. The opposite of 0 is 0, since 0 is no distance from itself and has no other side to move to.
The opposite of a number a is written -a and read “the opposite of a”. This gives the minus sign three jobs, and the place where it sits tells you which one it is doing:
| Written | Where the sign sits | Read as |
|---|---|---|
| 9 - 5 | between two numbers | nine minus five (subtraction) |
| -6 | in front of a number | negative six |
| -x | in front of a letter | the opposite of x |
| -(-3) | in front of a bracket | the opposite of negative three |
The two readings of -6 never disagree: “negative six” and “the opposite of six” name the same point, six units left of 0.
Negative numbers in the world
To describe a situation with a signed number, two things are fixed first. The zero is the reference point, the place or amount that counts as nothing either way. The positive direction is the side of zero whose amounts are counted as positive. Once both are fixed, an amount on the positive side of zero is a positive number, an amount on the other side is a negative number, and an amount at the reference point is 0. The size of the number says how far the amount is from the reference point, and its sign says which side it is on.
Amounts in the world come in two kinds.
- A level says where something is, measured from the zero: a temperature, a height, a floor of a building, the balance of a bank account, a day counted from today.
- A change says how far something moved, measured from where it started: a rise or a fall in temperature, a deposit into an account or a withdrawal from it, a profit or a loss. For a change the zero is “no change”, and the positive direction is the one that makes the level go up.
Everyday language marks the two sides with pairs of opposite words. With the usual choice of positive direction, they sort like this:
| The zero | Positive side | Negative side |
|---|---|---|
| sea level | above sea level, a climb | below sea level, a descent |
| 0 on a thermometer | above zero, a rise | below zero, a fall |
| an account that holds nothing and owes nothing | money held, a deposit | money owed, a withdrawal |
| breaking even | a profit | a loss |
| the ground floor | floors above, going up | floors below, going down |
| today | days after today, later | days before today, earlier |
The opposite of a situation swaps each word for its partner and keeps the size. The opposite of a withdrawal of 30 dollars is a deposit of 30 dollars, and the opposite of 6 meters below sea level is 6 meters above it. The number for the opposite situation is the opposite number: -30 becomes 30, and -6 becomes 6.
Intuition
Think of 0 as a mirror standing upright on the number line. Each number has a reflection the same distance away on the other side, and that reflection is its opposite. Reflect twice and you are back where you started. The one point that does not move is the one the mirror stands on, 0.
“Less than” keeps the meaning it had for whole numbers: {a < b} means a is to the left of b, and {a > b} means a is to the right of b. Nothing about that changes when the line is carried on past zero, and it settles every comparison with negatives. -5 is to the left of -2, so -5 < -2, even though 5 is more than 2. Seen in the mirror, that is not a surprise. The mirror swaps left and right, so it turns the order round: 2 is left of 5, and their reflections come out the other way, -5 left of -2.
A situation turns into a number once two things are fixed: where zero is, and which direction counts as positive. For money, zero is an account that neither holds nor owes anything and money held is positive, so a balance of -40 dollars means the account owes 40 dollars. A thermometer or a height scale is a number line stood on its end, positive upward, and there “less than” means “lower down”.
In the world, then, “less” takes its meaning from the scale: lower on a height scale, colder on a thermometer, poorer on a bank balance, earlier on a timeline. That is why the words can seem to run backwards on the negative side. An account that owes more is worse off, so its balance is the smaller number. A diver who goes deeper is lower down, so her height is the smaller number. “More below” always means “less”.
Results
Three kinds of number. Every number is exactly one of these: positive, negative, or 0.
The opposite of the opposite. For every number a,
The signs of opposites. If a is positive, then -a is negative. If a is negative, then -a is positive. If a = 0, then -a = 0. So -a is negative exactly when a is positive: a minus sign in front of a letter does not say that the value is negative.
Negatives come first. Every negative number is less than 0, and 0 is less than every positive number. So every negative number is less than every positive number.
Opposites reverse the order. For any two numbers a and b,
For example, 2 < 5, and so -5 < -2.
Negative numbers in the world
More below means less. If two amounts lie below the same zero, d units below and e units below, and d < e, then
So of two debts the larger one is the smaller balance, of two depths below sea level the deeper one is the smaller height, and of two temperatures below zero the colder one is the smaller number.
The opposite situation. Swapping a situation for its opposite, with the same size, changes its number into the opposite number. An amount d units on one side of zero, written a, becomes d units on the other side, which is -a.
Why it is true
Three kinds of number. A point of the number line is either at 0, to the right of 0, or to the left of 0, and it cannot be two of these at once. Right of 0 means greater than 0, which is positive; left of 0 means less than 0, which is negative.
The opposite of the opposite. Say a is d units from 0. Its opposite, -a, is also d units from 0, on the other side. The opposite of -a is then d units from 0 on the other side of that, which is the side a started on, at the same distance. Only one point is d units from 0 on that side, and it is a. If a = 0, all three numbers are 0.
The signs of opposites. Taking the opposite keeps the distance from 0 and changes the side. A positive number is on the right, so its opposite is on the left, which makes it negative; a negative number’s opposite is on the right, so it is positive. The opposite of 0 is 0, by the definition.
Negatives come first. A negative number is to the left of 0, and 0 is to the left of any positive number, so the negative number is to the left of the positive one too.
Opposites reverse the order. Picture turning the number line over, like a page, with the fold through 0. Every number lands on its opposite, because it keeps its distance from 0 and ends up on the other side. Turning the line over swaps left and right. So if a was to the left of b before the turn, then after it, a’s image -a is to the right of b’s image -b. That is -b < -a.
For two positive numbers you can also see it by counting. If 0 < a < b, then -a is a units left of 0 and -b is b units left of 0. Since b is the larger distance, -b is further left, so -b < -a.
More below means less. An amount d units below zero is the number -d, and an amount e units below zero is -e. Both d and e are positive, and d < e, so by the counting argument just given -e is further left than -d, that is, -e < -d. On a scale stood on its end, further left is further down, so the amount further below is the smaller number.
The opposite situation. Swapping each word for its partner moves the amount to the other side of the same zero and leaves its size, its distance from zero, alone. The number on the other side of 0 at the same distance is, by the definition, the opposite number. If the situation was at the zero itself, its opposite is the same situation, and the number stays 0.
Worked examples
Writing a situation as a number
Write each as a single number, and say what zero and the positive direction are.
(a) A temperature 12 degrees below zero.
(b) A cave floor 86 meters below sea level.
(c) The balance of a bank account that owes 35 dollars.
(d) A lift stopped two floors below the ground floor, counting the ground floor as floor 0 and floors above it as positive.
Solution(a) Zero is the 0 mark on the thermometer, and warmer is positive. Below zero is the other side, so the temperature is -12 degrees.
(b) Zero is sea level and up is positive. The cave floor is below, so its height is -86 meters.
(c) Zero is an account that holds nothing and owes nothing, and money held is positive. Money owed is on the other side, so the balance is -35 dollars.
(d) On this count, two floors below the ground floor is floor -2. Real buildings label the floors below ground in different ways, which is why the count has to be stated.
In every part the number does two jobs at once: its size says how far from zero, and its sign says which side.
Negative numbers in the world: changes and opposite situations
Write each change as a single number, taking as positive the direction that makes the level go up.
(a) A deposit of 45 dollars into an account.
(b) The temperature falls 3 degrees.
(c) A lift goes down 4 floors.
(d) A shop makes a loss of 120 dollars in one day, counting a profit as positive.
Then give the opposite situation, and its number, for (e) a withdrawal of 30 dollars, (f) a climb of 2.5 meters, and (g) a point 6 meters below sea level.
SolutionParts (a) to (d) are changes, so zero means “no change”, and the sign says which way the level moved.
(a) A deposit makes the balance go up, so it is the positive direction: 45.
(b) A fall makes the temperature go down: -3.
(c) Going down makes the floor number go down: -4.
(d) A loss makes the shop’s money go down, the opposite of a profit: -120.
For the opposite situations, swap each word for its partner and keep the size.
(e) The opposite of a withdrawal of 30 dollars is a deposit of 30 dollars. The withdrawal was -30, and the deposit is its opposite, 30.
(f) The opposite of a climb of 2.5 meters is a descent of 2.5 meters. The climb was 2.5, and the descent is -2.5.
(g) Being 6 meters below sea level is a level, not a change, but the same swap works: the opposite is 6 meters above sea level. The height -6 becomes 6.
Check. In each of (e) to (g) the two numbers have the same size and different signs, as the result on opposite situations says they must.
Negative numbers in the world: a number as a situation
Describe a situation that each number could stand for, saying where zero is and which direction is positive: (a) -8, (b) 12, (c) -12.75, (d) 0.
SolutionEach number has many good answers. What makes an answer right is that the zero and the positive direction are stated and the sign puts the amount on the correct side.
(a) With zero at the 0 mark of a thermometer and warmer as positive, -8 is a temperature 8 degrees below zero. Equally, with sea level as zero and up as positive, -8 is a diver 8 meters below the surface.
(b) With the ground floor as floor 0 and up as positive, 12 is the twelfth floor above the ground floor. With today as 0 and later as positive, 12 is twelve days from now.
(c) The two decimal places suggest money: dollars and cents. With an account that holds and owes nothing as zero and money held as positive, -12.75 is a balance that owes 12.75 dollars. As a change, it could be a price that fell by 12.75 dollars. It could not sensibly be a floor of a building, since floors are counted in whole numbers.
(d) 0 is the reference point itself: a height at sea level, a temperature of exactly 0 degrees, an account that holds nothing and owes nothing. As a change, 0 means no change at all.
Finding opposites
Find the opposite of each: (a) 12, (b) -5, (c) 0. Then (d) simplify -(-15).
Solution(a) 12 is twelve units right of 0. The point twelve units left of 0 is -12, so the opposite of 12 is -12.
(b) -5 is five units left of 0. The point five units right of 0 is 5, so the opposite of -5 is 5.
(c) The opposite of 0 is 0. It is the only number that is its own opposite.
(d) Read -(-15) as “the opposite of negative fifteen”. Negative fifteen is fifteen units left of 0, and its opposite is fifteen units right:
This is the result -(-a) = a with a = 15.
The opposite of a letter
Evaluate -x when (a) x = 9, (b) x = -9, (c) x = 0. Is -x a negative number?
Solution-x means the opposite of x, so find the opposite of whatever number x is.
(a) When x = 9, -x is the opposite of 9, which is -9.
(b) When x = -9, -x is the opposite of -9:
(c) When x = 0, -x is the opposite of 0, which is 0.
So -x is negative in part (a), positive in part (b), and neither in part (c). Whether -x is negative depends on x: it is negative exactly when x is positive. The minus sign in -x says “opposite”, not “negative”.
Comparing two numbers
Write < or > between each pair: (a) -5 \;\square\; {-2}, (b) -1 \;\square\; 3, (c) 0 \;\square\; {-7}, (d) -12 \;\square\; {-15}.
SolutionThe number further left is the smaller one.
(a) -5 is five units left of 0 and -2 is two units left, so -5 is further left: -5 < -2.
(b) -1 is negative and 3 is positive, and every negative number is less than every positive number: -1 < 3.
(c) -7 is negative, so it is less than 0. Written with 0 first: 0 > -7.
(d) -15 is further from 0 on the left than -12 is, so -15 is further left and -12 > -15. The same answer comes from reversing the order: 12 < 15, so -15 < -12.
Putting a list in order
Write these numbers from least to greatest: 4, -6, 0, -1, 3, -10.
SolutionPlace them on a number line. Least to greatest is left to right.
The negatives come first, then 0, then the positives. Among the negatives, the one furthest from 0 is furthest left, so they go -10, then -6, then -1. Among the positives, 3 comes before 4. In order:
Read from right to left, the same line gives the list from greatest to least: 4, 3, 0, -1, -6, -10.
Negative numbers in the world: above and below sea level
In a made-up harbor, a gull flies 6 meters above the water, a buoy floats on the surface, a fish swims 4 meters below the surface, a diver is 6 meters below it, and an anchor rests on the sea bed 14 meters below it. Take the surface as sea level.
(a) Write each height as a number.
(b) Which is highest, and which is lowest?
(c) Apart from the buoy, which is closest to sea level?
(d) Which two are the same distance from sea level?
(e) Compare the heights of the diver and the anchor with <.
(f) The gull flies directly over the diver. How far apart are they?
Solution(a) Sea level is zero and up is positive. The gull is 6, the buoy is 0, the fish is -4, the diver is -6 and the anchor is -14, all in meters. Standing the number line on its end puts them in their real places.
(b) Higher means greater on this scale. The greatest of 6, 0, -4, -6, -14 is 6, so the gull is highest, and the least is -14, so the anchor is lowest.
(c) Closest to sea level asks how far each is from zero, whichever side it is on: the gull 6, the fish 4, the diver 6, the anchor 14. The fish, 4 meters below, is closest. It is not the highest; “closest to zero” and “greatest” are different questions.
(d) The gull and the diver are both 6 meters from sea level, one above and one below. Their heights, 6 and -6, are opposites.
(e) The anchor is further below sea level, so by “more below means less”, -14 < -6.
(f) The gull is 6 meters above the surface and the diver 6 meters below it, so the line between them crosses the surface, and the distance is 6 + 6 = 12 meters.
Negative numbers in the world: when more means less
(a) Five made-up bank balances, in dollars, are: Ivo -850, Lena -230, Omar -1200, Pia -90, Ravi -980. List the people from the least debt to the most debt, and then their balances from greatest to least.
(b) An account owes more than 30 dollars. Write an inequality for its balance b, in dollars, and give one possible balance.
(c) In one town it is colder than -4 degrees. Write an inequality for its temperature t, and give one possible temperature.
Solution(a) Each balance is negative, so each person owes the size of the number: Ivo 850, Lena 230, Omar 1200, Pia 90, Ravi 980. From least debt to most: Pia, Lena, Ivo, Ravi, Omar. A smaller debt is a greater balance, so the balances from greatest to least come in the same order:
(b) Owing more than 30 dollars puts the balance further below zero than -30, and “more below means less”, so
For example, an account that owes 45 dollars has b = -45, and -45 < -30. A balance of -20 would not do: it owes only 20 dollars.
(c) Colder means lower on the thermometer, which means less, so
For example, t = -9 works, since -9 is further below zero than -4. A temperature of -1 degree is warmer than -4, and so is 3 degrees.
In each part a word that sounds like “more” (more debt, colder, deeper) turns into “less than” for the number. The words describe the size of the amount below zero; the inequality compares positions on the line.
Where the opposites of a range of numbers lie
A number x lies strictly between -4 and -1, that is, -4 < x < -1. Between which two numbers does -x lie?
SolutionTake the two comparisons one at a time, and reverse each with the rule “if a < b, then {-b < -a}”.
From -4 < x, reversing gives -x < -(-4), and -(-4) = 4, so
From x < -1, reversing gives -(-1) < -x, and -(-1) = 1, so
Together, 1 < -x < 4. The range from -4 to -1 is reflected in the mirror at 0 onto the range from 1 to 4, and its ends swap: the left end -4 goes to the right end 4.
Check. Try x = -2.5, which lies between -4 and -1. Then -x = 2.5, which lies between 1 and 4.
Boundaries
A sign means nothing until zero and a direction are fixed. “Negative” means “on the other side of the chosen zero”. Change the zero and the sign can change: a ledge 5 meters below the top of a 30 meter cliff is at -5 if heights are measured from the cliff top, but at 25 if they are measured from the beach at its foot. Which direction is positive is a choice too. A diver 20 meters down is at -20 if up is positive and at 20 if down is positive; both are right, as long as the choice is stated.
Some amounts have no other side. A count of people, a length or a weight is measured from nothing, and there is nothing below nothing. Negative numbers describe amounts measured from a starting point with room on both sides, such as a temperature, a height or a balance, and changes that can go either way, such as a rise or a fall.
A scale may take only some of the numbers. Floors of a building are counted in whole numbers, so floor -2.5 means nothing, while a bank balance of -2.50 dollars is perfectly real. Before reading a number as a situation, check that the scale allows it.
Greater means further right, not further from zero. -10 < -2, although -10 is further from 0 than -2 is. How far a number is from zero, whatever its side, is a different measure, its absolute value, and it does not decide which number is greater. In the world this is the difference between “lowest” and “furthest from sea level”, or between “the smallest balance” and “the biggest debt”.
No end on either side. There is no smallest negative number: for any negative number, the number one unit further left is smaller and still negative. Among the integers, -1 is the greatest negative one. Among all numbers there is no greatest negative number, because for any negative number a, the point halfway between a and 0 is still negative and is greater than a.
“Not negative” includes zero. A number that is not negative is either positive or 0. Such a number is called non-negative, and “not positive” likewise means negative or 0.
Where this goes next
With negative numbers in place, the number line runs both ways from zero, and the next lessons use it in that shape. Integers on a number line practices placing and reading them, and comparing integers turns “further right is greater” into a routine for any pair or list. Temperatures below zero and temperature change takes the thermometer of Examples 1 and 2 further, and works out how far a temperature has risen or fallen when it crosses zero.
The distance from a number to zero, which opposites share and which Example 8 used for “closest to sea level”, becomes absolute value. The fact that a number and its opposite are balanced about zero is the start of opposites and zero pairs: a number and its opposite add to 0, just as a deposit and a withdrawal of the same size leave a balance where it was. That idea is what makes adding and subtracting integers work, and it leads in turn to multiplying and dividing integers.
Two such lines crossing at right angles at their zeros make the coordinate plane, where every point has a pair of signed numbers.
Exercises
Numbers for situations
Write each as a single number.
- A temperature 15 degrees below zero.
- A point 240 meters below sea level, with up as positive.
- The balance of an account that owes 60 dollars.
- A lift seven floors above the ground floor, where the ground floor is floor 0 and up is positive.
- Say whether each number is positive, negative or neither: -8, 0, 3, -0.5, -(-4).
Negative numbers in the world
In 6 to 9, write each change as a single number, taking as positive the direction that makes the level go up.
- A withdrawal of 80 dollars from an account.
- A rise of 4 degrees in temperature.
- A loss of 250 dollars for a shop over one week, counting a profit as positive.
- A price that falls by 0.45 dollars.
- Write sea level as a height, with sea level as zero.
- Give the opposite situation of a deposit of 120 dollars, and its number.
- Give the opposite situation of a fall of 7 meters, and its number.
- Give the opposite situation of being 3 floors below the ground floor, and its number, with the ground floor as floor 0 and up as positive.
- With today as 0 and later days as positive, write as numbers: 3 days ago, 5 days from now, and today.
- A cave floor is 30 meters below the top of a hill, and the hill top is 100 meters above sea level. With up as positive, give the height of the cave floor measured (a) from the hill top, (b) from sea level.
- Describe a situation for each number, saying where zero is and which direction is positive: -20, 6.5, 0.
Opposites
- Find the opposite of 9.
- Find the opposite of -14.
- Find the opposite of 0.
- Simplify -(-11).
- Simplify -(-(-2)).
- Evaluate -x when x = 6, when x = -6 and when x = 0.
- Evaluate -(-y) when y = -7.
- Read each aloud, and say what job the minus sign is doing: 8 - 3, -8, -y, -(-8).
Comparing and ordering
Write < or > between each pair.
- -7 \;\square\; {-3}
- -9 \;\square\; 1
- 0 \;\square\; {-3}
- -20 \;\square\; {-18}
- Write from least to greatest: 2, -3, -8, 0, 5, -1.
- Write from greatest to least: -4, 6, -11, 1, -2.
Comparing in the world
- One morning it is -12 degrees in one town and -9 degrees in another. Which town is colder? Write the comparison with <.
- A lake surface is at -28 meters and a valley floor is at -86 meters, both measured from sea level with up as positive. Which is lower? Write the comparison with <.
- One account has a balance of -45 dollars and another has -70 dollars. Which balance is less, and which account owes more?
On a made-up stretch of coast, heights in meters are measured from sea level with up as positive: a bench on a cliff top at 14, a boat deck at 2, a seal at -3, a reef at -14 and a wreck at -17.
- Which is lowest, and which is closest to sea level?
- Which two are the same distance from sea level? If one is directly above the other, how far apart are they?
- Compare the heights of the seal and the reef with <.
The table gives the change in five made-up share prices over one day, in dollars.
| Share | Change in price (dollars) |
|---|---|
| A | 0.42 |
| B | -0.57 |
| C | 0.08 |
| D | -0.21 |
| E | -0.42 |
- Which prices fell, and which rose?
- Which price changed the most, and which changed the least? By how much did each change?
- Which two prices changed by the same amount?
- An account owes more than 40 dollars. Write an inequality for its balance b, in dollars, and give one possible balance.
- Write these balances in order from the most debt to the least debt: -75, -310, -40, -1000, -290.
Reasoning
- Is -x always a negative number? Explain, with an example.
- A number p has -p > 0. What can you say about p? Explain.
- Two different numbers are opposites of each other. Explain why one of them must be positive and the other negative.
- One report says the temperature “fell to -3 degrees” and another says it “fell by 3 degrees”. Do they say the same thing? Explain.
- Explain why, of two accounts that each owe money, the one that owes more always has the smaller balance.
Answers
- -15. Below zero is the negative side.
- -240. Below sea level is the negative side when up is positive.
- -60 dollars. Money owed is on the other side of zero from money held.
- 7. Above the ground floor is the positive side.
- -8 is negative, 0 is neither, 3 is positive, -0.5 is negative, and -(-4) is positive, since -(-4) = 4.
- -80. A withdrawal makes the balance go down.
- 4. A rise makes the temperature go up.
- -250. A loss is the opposite of a profit.
- -0.45. A fall makes the price go down.
- 0. Sea level is the reference point itself.
- A withdrawal of 120 dollars, which is -120. The deposit was 120, and the opposite situation has the opposite number.
- A rise of 7 meters, which is 7. The fall was -7.
- 3 floors above the ground floor, which is 3. Being 3 floors below is -3, and its opposite is 3.
- -3, 5 and 0. Days ago are earlier, the negative side; today is the zero.
- (a) -30 meters: the cave floor is 30 meters below the hill top, which is now the zero. (b) 70 meters: it is 100 - 30 = 70 meters above sea level. The same place has a negative height from one zero and a positive height from another.
- Many answers. For example: -20 is a balance that owes 20 dollars, with an empty account as zero and money held as positive; 6.5 is a temperature 6.5 degrees above zero, with warmer as positive; 0 is a height at sea level, or no change at all. Any answer that states the zero and the positive direction and puts the amount on the matching side is right.
- -9. It is nine units from 0 on the other side.
- 14. -14 is fourteen units left of 0, so its opposite is fourteen units right.
- 0. Zero is its own opposite.
- 11. The opposite of -11 is 11.
- -2. Work from the inside: -(-2) = 2, and the opposite of 2 is -2.
- -6, 6 and 0. When x = 6, -x is the opposite of 6; when x = -6, -x = -(-6) = 6; when x = 0, -x = 0.
- -7. -(-y) = y for every y, so the value is y itself. Directly: -y = -(-7) = 7, and the opposite of 7 is -7.
- “Eight minus three”: subtraction. “Negative eight”: part of the name of a negative number. “The opposite of y”: the opposite. “The opposite of negative eight”: the outer sign is the opposite, the inner sign is part of the number -8; the value is 8.
- -7 < -3. -7 is further left of 0 than -3.
- -9 < 1. A negative number is less than a positive one.
- 0 > -3. Every negative number is less than 0.
- -20 < -18. 18 < 20, and opposites reverse the order.
- -8, -3, -1, 0, 2, 5. Negatives first, the one furthest from 0 first among them.
- 6, 1, -2, -4, -11. Positives first, then the negatives starting with the one nearest 0.
- The town at -12 degrees is colder, since -12 < -9: it is further below zero.
- The valley floor is lower, since -86 < -28.
- -70 is less, since -70 < -45, and that account owes more: 70 dollars against 45.
- The wreck is lowest, at -17, the least of the five heights. The boat deck is closest to sea level, 2 meters from it; the seal is 3 meters from it and the others further.
- The bench and the reef, each 14 meters from sea level, one above and one below. Directly above one another they are 14 + 14 = 28 meters apart.
- -14 < -3. The reef is further below sea level than the seal.
- B, D and E fell, since their changes are negative; A and C rose.
- B changed the most, falling by 0.57 dollars; C changed the least, rising by 0.08 dollars. The size of a change is its distance from 0, whichever way it went.
- A and E, each by 0.42 dollars, A up and E down. Their changes, 0.42 and -0.42, are opposites.
- b < -40. Owing more than 40 dollars puts the balance further below zero than -40. For example, b = -55, an account owing 55 dollars.
- -1000, -310, -290, -75, -40. The most debt is the least balance, so this is the list from least to greatest.
- No. -x is the opposite of x. If x = -5, then -x = -(-5) = 5, which is positive, and if x = 0 then -x = 0, which is not negative either. -x is negative only when x is positive.
- p is negative. -p > 0 says the opposite of p is positive. If p were positive its opposite would be negative, and if p were 0 its opposite would be 0, so p can only be negative.
- Neither number can be 0: the opposite of 0 is 0 itself, so a pair of different opposites cannot include it. So each is on one side of 0, and opposites lie on different sides, one on the right, which is positive, and one on the left, which is negative.
- No. “Fell to -3 degrees” gives a level: the temperature ended 3 degrees below zero. “Fell by 3 degrees” gives a change, -3, and says nothing about where the temperature ended: from 10 degrees it would end at 7, still above zero.
- An account that owes d dollars has balance -d. If one owes d and the other e, with d < e, then opposites reverse the order, so -e < -d: the account owing more has the smaller balance.