This page is about signed numbers: positive numbers, negative numbers, and zero, and how to add, subtract, multiply and divide them. It starts from the beginning and assumes only that you can add, subtract, multiply and divide ordinary numbers.
Signed numbers are used on almost every page of the GED math test, because half the numbers in algebra are negative. Most mistakes in algebra are not really algebra mistakes. They are sign mistakes. So this page goes slowly, and it asks you to do one thing again and again, until you do it without thinking. Part 3 explains what that one thing is.
Each part uses only what the parts before it have explained, so read it in order. Keep paper and a pencil beside you, and try each worked example before you read the answer. It is a long page: three or four sittings is a good pace. There is practice in every part, a card with all the rules in one place, practice rounds you can time yourself on, and a check at the end.
Numbers below zero
A number line is a straight line with the numbers marked on it in order, evenly spaced. Zero is in the middle. The numbers to the right of zero are the positive numbers: 1, 2, 3, and so on. The numbers to the left of zero are the negative numbers: −1, −2, −3, and so on.
A negative number is written with a small dash in front of it, called a negative sign: −3 is read “negative three.” Zero is neither positive nor negative.
A positive number can be written with a plus sign in front of it, +3, but it almost never is. A number with no sign in front of it is positive. So 3 means positive 3. This will matter a great deal in Part 3.
Where negative numbers show up
Negative numbers are used whenever there is a starting point, a zero, with amounts on both sides of it.
- Temperature. Zero degrees is a point on the thermometer. −5°F means five degrees below zero.
- Money. A bank balance of zero means you have nothing and owe nothing. A balance of −40 dollars means you owe $40: the account is overdrawn.
- Floors. The street level is zero. Floors above are positive; basement levels below are negative.
- Gains and losses. A football team that loses 6 yards on a play has a change of −6 yards.
In every one of these, the negative number is not “less than nothing” in some strange way. It is an amount on the other side of zero.
Which is larger, opposites, and absolute value
Which of two numbers is larger
On the number line, the number farther to the right is always the larger one. That is true on both sides of zero.
So 5 is larger than 2, and 2 is larger than −3. And −3 is larger than −8, because −3 is farther to the right. This surprises people, because 8 is a bigger number than 3. But think of money: owing $3 is better than owing $8. Or temperature: three degrees below zero is warmer than eight below.
Opposites
Two numbers are opposites when they are the same distance from zero, on different sides. 5 and −5 are opposites. So are −12 and 12. The opposite of zero is zero.
To find the opposite of a number, change its sign. The opposite of 7 is −7. The opposite of −7 is 7.
Absolute value
The absolute value of a number is its distance from zero. A distance is never negative, so an absolute value is never negative. The absolute value is the number without its sign.
It is written with two straight bars around the number: |−6| means “the absolute value of −6.” |−6| = 6 and |6| = 6, because both are 6 steps from zero.
Farther from zero
One of the rules for adding signed numbers asks this question: which of the two numbers is farther from zero? That is the same as asking which one has the larger absolute value. Look only at the numbers without their signs.
Which is farther from zero, −9 or 4? Without their signs, they are 9 and 4, and 9 is larger. So −9 is farther from zero, even though 4 is the larger number. Keep those two ideas apart: larger means farther to the right; farther from zero means a larger absolute value, in either direction.
Practice: the number line and absolute value
Type a number. Type a negative number with the minus key, like -4.
Which is larger, −2 or −7?
−2 is farther to the right on the number line. Owing $2 is better than owing $7.
What is the opposite of −15?
Change the sign: 15. It is the same distance from zero, on the other side.
What is |−11|?
−11 is 11 steps from zero. Absolute value is a distance, so it is positive: 11.
What is |8|?
8 is 8 steps from zero.
Which is farther from zero, −6 or 10?
Without their signs, the numbers are 6 and 10. 10 is farther from zero.
Which is farther from zero, 3 or −12?
Without their signs, the numbers are 3 and 12. −12 is farther from zero, although 3 is the larger number.
The one habit: look at the second number
Every adding or subtracting problem with two numbers has three pieces: a first number, an operation (adding or subtracting), and a second number.
Before you work anything out, stop and answer two questions:
- Am I adding or subtracting?
- Is the second number positive or negative?
Those two questions, asked every time, are the habit this whole page is built on. Two questions with two answers each make exactly four possibilities. They are called the four cases, and Part 4 goes through each one.
One mark, two jobs
The two questions sound easy. The difficulty is that the same small dash does two different jobs. Between two numbers, it is a minus sign, and it means subtract. Attached to the front of a number, it is a negative sign, and it means that number is below zero. Here is how to tell them apart in each kind of problem.
5 + 3
Nobody has any doubt here. We are adding a positive 3.
5 + (−3)
The plus sign says we are adding. The dash inside the parentheses is attached to the 3, so it is a negative sign. We are adding a negative 3. The parentheses are there to keep the two signs apart, so that the operation sign and the negative sign do not run into each other.
5 − 3
This is the one to slow down on. The dash sits between the two numbers, so it is the operation: we are subtracting. And the 3 has no sign of its own, so it is positive 3. We are subtracting a positive 3.
It is easy to look at 5 − 3, see a minus sign next to the 3, and think of the 3 as negative. It is not. The minus sign belongs to the operation, not to the 3. What is being taken away is a positive 3: three of something that was there, removed.
5 − (−3)
Two dashes. The first sits between the numbers, so it is the operation: subtracting. The second is inside the parentheses, attached to the 3, so the second number is negative 3. We are subtracting a negative 3: taking away something negative. That must turn out differently from taking away something positive, and Part 4 shows how.
When the first number is negative
In −5 − 3, the first dash is attached to the 5, at the very start, so the first number is negative 5. The second dash sits between the numbers, so it is subtracting. The 3 has no sign, so it is positive. This is subtracting a positive. The sign of the first number does not change which case it is; only the operation and the second number decide that.
For every problem, say the case in words before you start: “five, subtract a positive three.” “Negative five, add a negative three.” It feels slow at first. It is the habit that prevents most sign mistakes, and after some practice it takes no time at all.
Practice: name the case
Look at the operation, then at the second number. Do not work out the answer yet.
Which case is 7 + 2?
Adding, and the 2 has no sign of its own, so it is positive: add a positive.
Which case is 7 − 2?
The dash between the numbers means subtracting. The 2 has no sign of its own, so it is positive: subtract a positive.
Which case is 7 + (−2)?
Adding, and the 2 has a negative sign attached inside the parentheses: add a negative.
Which case is 7 − (−2)?
The first dash, between the numbers, means subtracting. The second, inside the parentheses, makes the 2 negative: subtract a negative.
Which case is −7 − 2?
The dash at the very start belongs to the 7, so the first number is negative. The dash between the numbers means subtracting, and the 2 has no sign, so it is positive: subtract a positive.
Which case is −7 + (−2)?
Adding, and the second number is negative 2: add a negative. The first number being negative does not change the case.
Which case is −7 − (−2)?
Subtracting, and the second number is negative 2: subtract a negative.
Which case is −7 + 2?
Adding, and the 2 is positive: add a positive.
The four cases
Here are the four cases, each with one example. All four start at 4. For each one there is a picture on the number line and a picture with money.
On the number line, adding moves you in the direction of the second number’s sign: to the right for a positive, to the left for a negative. Subtracting moves you the opposite way.
1. Add a positive: 4 + 2 = 6
Start at 4. Adding a positive moves you to the right, 2 steps. You land on 6.
With money: you have $4, and you are paid $2. Now you have $6.
Adding a positive makes the number go up.
2. Add a negative: 4 + (−3) = 1
Start at 4. Adding a negative moves you to the left, 3 steps. You land on 1.
With money: you have $4, and a $3 bill arrives. A bill is a negative amount: it is money owed. Adding it leaves you with $1.
Adding a negative makes the number go down.
3. Subtract a positive: 4 − 1 = 3
Start at 4. Subtracting a positive moves you the opposite way from adding it: to the left, 1 step. You land on 3.
With money: you have $4, and $1 of it is taken away. You have $3 left. Something positive, something you had, was removed.
Subtracting a positive makes the number go down.
4. Subtract a negative: 4 − (−5) = 9
Start at 4. Adding a negative 5 would move you 5 steps to the left. Subtracting it moves you the opposite way: 5 steps to the right. You land on 9.
With money: you have $9 in your wallet, but you owe a friend $5. The debt counts as −5, so what you are really worth is $4: 9 + (−5) = 4. Now your friend says, “Forget it. You don’t owe me.” The −5 is taken away. Nothing new was put in your hand, but you are now worth the full $9. Taking away the debt made you $5 better off: 4 − (−5) = 9.
Subtracting a negative makes the number go up.
The four cases side by side
| The case | Example | The number |
|---|---|---|
| Add a positive | 4 + 2 = 6 | goes up |
| Add a negative | 4 + (−3) = 1 | goes down |
| Subtract a positive | 4 − 1 = 3 | goes down |
| Subtract a negative | 4 − (−5) = 9 | goes up |
Look at which cases go the same way. Adding a negative and subtracting a positive both go down. Having a bill arrive and having money taken away leave you worse off in the same way. Adding a positive and subtracting a negative both go up. Being paid and having a debt forgiven leave you better off in the same way. Part 7 is built on that observation.
Try it: walk any problem
Choose a first number, adding or subtracting, and a second number, positive or negative. The number line shows the move, and the page names the case.
The three rooms of Negative Mazarine! act out these same four cases: walking forward and backward on a path, clamping floats and weights onto a diving craft and cutting them loose, and dropping hot and cold cubes into a vat and scooping them out. Each room has guided practice and challenges.
Practice: the four cases
Name the case to yourself first. Then work it out. The number line is fine to use.
3 + 5
Add a positive: go up. 3 + 5 = 8.
6 + (−4)
Add a negative: go down 4 from 6. The answer is 2.
9 − 3
Subtract a positive: go down 3 from 9. The answer is 6.
2 − (−6)
Subtract a negative: go up 6 from 2. The answer is 8.
10 + (−1)
Add a negative: go down 1. The answer is 9.
1 − (−1)
Subtract a negative: go up 1. The answer is 2.
Adding numbers with the same sign
The four cases tell you which way the number moves. To get exact answers quickly, and without drawing a number line every time, there are three rules: two for adding and one for subtracting. This part and the next give the two rules for adding. Adding two signed numbers happens in two situations, and each has its own rule. The first is when both numbers have the same sign.
Adding two numbers with the same sign
Add their absolute values. Keep the sign they share.
Work out −3 + (−5).
Name the case: adding a negative. Both numbers are negative, so they have the same sign: Rule 1.
Add the absolute values: 3 + 5 = 8. Keep the sign they share, negative. The answer is −8.
With money: you owe $3, then you owe $5 more. Now you owe $8.
When both numbers are positive, Rule 1 is ordinary adding: 3 + 5 = 8. When both are negative, the answer is negative, and it is farther from zero than either number, because you went down and then down again.
Some people remember “two negatives make a positive” and apply it here, to get 8 instead of −8. That saying is about multiplying, which comes in Part 10. For adding, two negatives make a larger negative: owing and then owing more is never a gain.
Practice: same signs
−4 + (−6)
Both negative. 4 + 6 = 10, keep the negative sign: −10.
−12 + (−3)
Both negative. 12 + 3 = 15: −15.
−1 + (−1)
Both negative. 1 + 1 = 2: −2.
−20 + (−25)
Both negative. 20 + 25 = 45: −45.
Adding numbers with different signs
This is the situation where most mistakes are made. It is worth taking one step at a time, and it is worth learning the rule by heart, word for word.
Adding two numbers with different signs
Two steps:
- First subtract (the smaller absolute value from the larger).
- Then give the result the sign of the original larger absolute value.
Say Rule 2 to yourself until you can say it without looking: at the bus stop, on the train, at night before you fall asleep. Two steps. First subtract, the smaller absolute value from the larger. Then give the result the sign of the original larger absolute value. When you meet a problem with different signs, say the two steps as you do them.
Think of it with money. A positive number is money you have; a negative number is money you owe. Adding them together means paying the debt with the money you have. What is left over is the difference between them. Whether it is money left or money still owed depends on which was bigger.
Rule 2, one step at a time
- Check: are the signs different? One number is positive and one is negative. If so, Rule 2 applies.
- First subtract. Take the two numbers without their signs, their absolute values, and subtract the smaller from the larger. That gives the size of the answer.
- Then give the result the sign of the original larger absolute value. Look back at the problem as it was written. Find the number whose absolute value was larger, and give your answer its sign.
When the positive number is farther from zero
Work out 9 + (−4).
The signs are different, so Rule 2.
First subtract. Without their signs the numbers are 9 and 4. 9 − 4 = 5.
Then give the result the sign of the original larger absolute value. The larger absolute value is 9, and in the original problem the 9 is positive. The answer is 5.
With money: you have $9 and a $4 bill. Pay it, and $5 is left.
When the negative number is farther from zero
Work out 4 + (−9).
The signs are different, so Rule 2.
First subtract. Without their signs the numbers are 4 and 9. 9 − 4 = 5.
Then give the result the sign of the original larger absolute value. The larger absolute value is 9, and in the original problem it is −9, negative. The answer is −5.
With money: you have $4 and a $9 bill. Pay all $4, and you still owe $5.
Look at those two examples together. The numbers 9 and 4 appear in both, and the subtraction is 9 − 4 = 5 both times. Only the sign of the answer is different. The subtraction gives the size of the answer but not its sign, because it was done with the signs taken off. That is why the second step sends you back to the original problem: the sign is there, and only there. The most common mistake in this whole topic is to stop after the subtraction and write 5 for 4 + (−9). The rule is not finished until the second step is done.
The order does not matter
Work out −9 + 4.
This is the same as 4 + (−9) with the numbers in the other order. Adding can be done in either order: $4 and a $9 bill leave you in the same place as a $9 bill and $4.
The signs are different. First subtract: 9 − 4 = 5. Then the sign of the original larger absolute value: that is −9, so the answer is −5.
When they are the same distance from zero
Work out −6 + 6.
The two numbers are opposites: each is 6 from zero. First subtract: 6 − 6 = 0. There is no larger absolute value, and zero has no sign, so the second step has nothing to do. The answer is 0.
A number and its opposite always add to zero. Such a pair is called a zero pair. A $6 debt paid with $6 leaves nothing either way.
Practice, one step at a time
Each problem below takes Rule 2 one step at a time: first the check, then the subtraction, then the sign. Answer each step, and the next one opens.
Practice: different signs
Now without the help. Say the two steps to yourself as you do them.
8 + (−3)
First subtract: 8 − 3 = 5. The original larger absolute value is 8, positive. The answer is 5.
2 + (−7)
First subtract: 7 − 2 = 5. The original larger absolute value is −7, negative. The answer is −5.
−11 + 4
First subtract: 11 − 4 = 7. The original larger absolute value is −11, negative. −7.
−3 + 10
First subtract: 10 − 3 = 7. The original larger absolute value is 10, positive. 7.
−25 + 25
Opposites, a zero pair: 0.
−45 + 120
First subtract: 120 − 45 = 75. The original larger absolute value is 120, positive. An overdraft of $45 and a $120 deposit leave $75.
Subtracting means adding the opposite
Now the third rule, the one for subtracting. Part 4 showed that subtracting a positive does the same thing as adding a negative: both go down. And subtracting a negative does the same thing as adding a positive: both go up. Here are the pairs side by side:
4 − 1 = 3 and 4 + (−1) = 3
4 − (−5) = 9 and 4 + 5 = 9
In each pair, the subtraction and the addition give the same answer. And in each pair, the number being added is the opposite of the number being subtracted: 1 and −1, −5 and 5.
A second way to see it: a pattern
Watch what happens as the number being subtracted goes down by 1 each time:
4 − 3 = 1
4 − 2 = 2
4 − 1 = 3
4 − 0 = 4
4 − (−1) = 5
4 − (−2) = 6
4 − (−3) = 7
Each time the number being subtracted goes down by 1, the answer goes up by 1. When we reach subtracting a negative, the pattern carries straight on: 4 − (−1) is 5, and 4 − (−2) is 6. Subtracting a negative gives a larger answer, just as the money example said.
Subtracting
Subtracting a number gives the same answer as adding its opposite. Rewrite the subtraction as an addition, then use Rule 1 or Rule 2.
Rewrite 8 − 3 as an addition, and work it out.
Name the case first: subtracting, and the second number, 3, is positive. Subtract a positive.
The opposite of 3 is −3. So 8 − 3 is the same as 8 + (−3), which is 5.
You did not need to rewrite this one to get 5. But it is worth seeing that the rewrite gives the right answer on a problem you already know, so that you can trust it on the harder ones.
Rewrite −2 − (−8) as an addition, and work it out.
Name the case: subtracting, and the second number is −8. Subtract a negative.
The opposite of −8 is 8. So −2 − (−8) is the same as −2 + 8.
Start at −2 and go up 8: two steps to reach zero, six more to reach 6. The answer is 6.
“Keep, change, change”
Many teachers and books give Rule 3 a short name: keep, change, change. It means: keep the first number as it is; change the subtraction to addition; change the sign of the second number to its opposite.
−2change
− to +change
−8 to 8
The short name is useful once you understand what it stands for. Used without that understanding, it leads to mistakes. Three are common:
- Using it on an addition. 5 + (−3) has no subtraction in it, so there is nothing to change. Keep, change, change is only for subtracting.
- Changing the first number. In −2 − (−8), the −2 stays −2. Only the number right after the minus sign changes.
- Forgetting that a plain number is positive. In 5 − 3, the second number is positive 3, so its opposite is −3: 5 + (−3). The rule works on subtracting a positive just as it does on subtracting a negative.
Taking something away has the opposite effect of adding it: taking away money you had leaves you worse off, like getting a bill; taking away a debt leaves you better off, like getting paid.
Practice: rewrite, then work it out
Rewrite each subtraction as an addition on your paper first. Then type the answer.
7 − (−3)
Subtract a negative. 7 − (−3) becomes 7 + 3 = 10.
6 − 2
Subtract a positive. 6 − 2 becomes 6 + (−2) = 4.
−1 − (−4)
Subtract a negative. −1 − (−4) becomes −1 + 4. From −1, go up 4: the answer is 3.
0 − (−9)
Subtract a negative. 0 − (−9) becomes 0 + 9 = 9.
0 − 9
Subtract a positive. 0 − 9 becomes 0 + (−9) = −9.
Every kind of subtraction
Now all three rules work together. For any subtraction, the steps are always the same:
- Name the case. Subtracting a positive, or subtracting a negative?
- Rewrite it as adding the opposite (Rule 3). Keep the first number, change the subtraction to addition, change the sign of the second number.
- Add, with Rule 1 if the signs are now the same, or Rule 2 if they are different.
Here are all the kinds of subtraction you will meet. The first number and the second number can each be positive or negative, and either one can be farther from zero. It looks like a lot of cases. But every row is handled by the same three steps.
| Problem | The case | Rewritten | Rule | Answer |
|---|---|---|---|---|
| 9 − 4 | subtract a positive | 9 + (−4) | 2: 9 − 4, sign of 9 | 5 |
| 4 − 9 | subtract a positive | 4 + (−9) | 2: 9 − 4, sign of −9 | −5 |
| −4 − 9 | subtract a positive | −4 + (−9) | 1: both negative | −13 |
| 4 − (−9) | subtract a negative | 4 + 9 | 1: both positive | 13 |
| −9 − (−4) | subtract a negative | −9 + 4 | 2: 9 − 4, sign of −9 | −5 |
| −4 − (−9) | subtract a negative | −4 + 9 | 2: 9 − 4, sign of 9 | 5 |
| −6 − (−6) | subtract a negative | −6 + 6 | 2: a zero pair | 0 |
The one to watch: a smaller number minus a larger one
The second row, 4 − 9, deserves its own attention, because algebra produces it constantly and it is easy to answer 5 without thinking. Name the case: subtracting a positive 9. Rewrite: 4 + (−9). The signs are different, and −9 is farther from zero, so the answer is negative: −5.
With money: you have $4 and you spend $9, paying the rest on credit. You owe $5.
After you have an answer, ask whether it moved the right way. Subtracting a positive should give an answer smaller than the first number. Subtracting a negative should give an answer larger than the first number. If your answer moved the wrong way, the sign is wrong.
Practice: every kind of subtraction
Name the case, rewrite, then add.
3 − 8
Subtract a positive: 3 + (−8). First subtract: 8 − 3 = 5. The original larger absolute value is −8, negative. The answer is −5.
−5 − 6
Subtract a positive: −5 + (−6). Both negative: 5 + 6 = 11. −11.
10 − (−2)
Subtract a negative: 10 + 2 = 12.
−7 − (−3)
Subtract a negative: −7 + 3. First subtract: 7 − 3 = 4. The original larger absolute value is −7, negative. The answer is −4.
−3 − (−7)
Subtract a negative: −3 + 7. First subtract: 7 − 3 = 4. The original larger absolute value is 7, positive. The answer is 4.
12 − 20
Subtract a positive: 12 + (−20). First subtract: 20 − 12 = 8. The original larger absolute value is −20, negative. −8.
−8 − (−8)
Subtract a negative: −8 + 8, a zero pair: 0.
The temperature at 6 a.m. is −4°F. By noon it is 11°F. How many degrees did it rise? (Work out 11 − (−4).)
Subtract a negative: 11 + 4 = 15 degrees. It rose 4 degrees to reach zero, and 11 more after that.
Three or more numbers
Many problems have more than two numbers: a bank balance over a week, the temperature through a day, a football team’s plays. There are two good ways to work them. Both begin the same way.
First: rewrite every subtraction
Go through the problem and rewrite each subtraction as adding the opposite, one at a time. After that, everything is adding, and adding can be done in any order.
Way 1: the positives together, the negatives together
Work out −4 + 9 − 12 + 3.
Rewrite the one subtraction: − 12 is subtracting a positive 12, which becomes + (−12). The problem is now −4 + 9 + (−12) + 3.
Add the positives: 9 + 3 = 12. Add the negatives: −4 + (−12) = −16 (Rule 1).
Combine the two totals: 12 + (−16). Different signs, so Rule 2: first subtract, 16 − 12 = 4; then the sign of the original larger absolute value, −16. The answer is −4.
With money: $12 came in and $16 went out, so you are $4 short.
Way 2: left to right, two at a time
Work out the same problem, −4 + 9 + (−12) + 3, from left to right.
−4 + 9 = 5. Then 5 + (−12) = −7. Then −7 + 3 = −4.
The same answer, −4. Way 2 is good when the numbers are small. Way 1 is good when there are many numbers, because it needs only one step with different signs, at the end.
A number and its opposite add to zero, so they can be crossed out before you begin. In 7 + (−3) + 3 + (−10), the −3 and the 3 cancel, leaving 7 + (−10) = −3.
A bank account starts the week at $35. Rent of $50 is taken out, a paycheck of $120 goes in, and a $40 bill is paid. What is the balance?
Money out is negative and money in is positive: 35 + (−50) + 120 + (−40).
Positives: 35 + 120 = 155. Negatives: −50 + (−40) = −90. Combine: 155 + (−90) = 65. The balance is $65.
Practice: three or more numbers
Rewrite any subtraction first.
5 + (−8) + 2
Positives: 5 + 2 = 7. Negatives: −8. 7 + (−8) = −1.
−3 − 4 + 10
Rewrite: −3 + (−4) + 10. Negatives: −7. Positives: 10. 10 + (−7) = 3.
6 − (−2) − 9
Rewrite: 6 + 2 + (−9). Positives: 8. 8 + (−9) = −1.
−10 + 4 + (−4) + 15
The 4 and −4 are a zero pair. −10 + 15 = 5.
A diver starts at −12 meters, rises 5 meters, then goes down 9 meters. Where is she?
−12 + 5 + (−9). Negatives: −21. Positives: 5. 5 + (−21) = −16 meters.
A team gains 7 yards, loses 3, loses 6, and gains 1. What is the total change?
7 + (−3) + (−6) + 1. Positives: 8. Negatives: −9. 8 + (−9) = −1: a loss of 1 yard.
Multiplying and dividing
Multiplying and dividing signed numbers have their own rules, and they are different from the adding rules. Work out the size of the answer as if both numbers were positive. Then decide the sign.
Multiplying and dividing two numbers
Same signs: the answer is positive. Different signs: the answer is negative.
| Signs | Multiplying | Dividing |
|---|---|---|
| positive and positive | 3 × 4 = 12 | 12 ÷ 4 = 3 |
| positive and negative | 3 × (−4) = −12 | 12 ÷ (−4) = −3 |
| negative and positive | (−3) × 4 = −12 | −12 ÷ 4 = −3 |
| negative and negative | (−3) × (−4) = 12 | −12 ÷ (−4) = 3 |
Why a negative times a negative is positive
Multiplying 3 × (−4) means three groups of −4: three debts of $4 are a debt of $12, so the answer is −12. For a negative times a negative, a pattern shows what must happen. Watch the answers as the first number goes down by 1:
3 × (−4) = −12
2 × (−4) = −8
1 × (−4) = −4
0 × (−4) = 0
(−1) × (−4) = 4
(−2) × (−4) = 8
Each time the first number goes down by 1, the answer goes up by 4. The pattern does not stop at zero: (−1) × (−4) is 4, and (−2) × (−4) is 8. A negative times a negative is positive.
Dividing follows the same rule, because dividing undoes multiplying: since (−3) × (−4) = 12, it must be that 12 ÷ (−4) = −3.
More than two numbers
Count the negative signs. An even number of negatives gives a positive answer; an odd number gives a negative answer. (−2) × (−3) × (−5) has three negatives, an odd number, so it is negative: 2 × 3 × 5 = 30, and the answer is −30.
Adding: −3 + (−5) = −8. Two negatives added make a larger negative.
Multiplying: (−3) × (−5) = 15. Two negatives multiplied make a positive.
Before you use a sign rule, check whether you are adding or multiplying.
Practice: multiplying and dividing
(−6) × 7
Different signs: negative. 6 × 7 = 42. −42.
(−8) × (−5)
Same signs: positive. 40.
−54 ÷ 9
Different signs: negative. 54 ÷ 9 = 6. −6.
−36 ÷ (−4)
Same signs: positive. 9.
(−1) × (−1) × (−1)
Three negatives, an odd number: negative. −1.
2 × (−3) × (−5)
Two negatives, an even number: positive. 2 × 3 × 5 = 30.
All the rules in one place, and practice for speed
Here is everything on this page on one card. It prints on its own with the button under it, so you can keep it beside you while you practice. The aim is to need it less and less, until you do not need it at all.
Signed numbers: the rules
First, every time: Am I adding or subtracting? Is the second number positive or negative? (A number with no sign is positive.)
4 + 2 = 6
goes up
4 + (−3) = 1
goes down
4 − 1 = 3
goes down
4 − (−5) = 9
goes up
Adding two numbers with the same sign
Add their absolute values. Keep the sign they share.
Adding two numbers with different signs
Two steps:
- First subtract (the smaller absolute value from the larger).
- Then give the result the sign of the original larger absolute value.
Subtracting
Subtracting a number gives the same answer as adding its opposite. Rewrite the subtraction as an addition, then use Rule 1 or Rule 2.
Multiplying and dividing
Same signs: positive. Different signs: negative. For more than two numbers, count the negatives: even is positive, odd is negative.
Keep, change, change is Rule 3, for subtracting only: keep the first number, change − to +, change the sign of the second number. Two negatives make a positive when multiplying, not when adding.
On test day
Part 1 of the GED math test has no calculator, and signed numbers are all through it. On Part 2, the TI-30XS calculator has two different keys that look alike: the subtract key, and a key marked (−) for making a number negative. To enter 4 − (−5), press 4, subtract, then (−), then 5. Using the subtract key where the negative key belongs gives an error message.
Practice rounds
Accuracy comes first, then speed. Choose a kind of problem and do a round of ten. The page times the round and shows the right answer and the case for any you miss. When you can get ten out of ten, try to do it a little faster the next time.
The words on this page, in one place
Positive number: a number greater than zero, to the right of zero on the number line. A number with no sign is positive.
Negative number: a number less than zero, to the left of zero, written with a negative sign: −3.
Signed numbers: positive numbers, negative numbers, and zero, taken together.
Negative sign: the dash attached to the front of a number, meaning it is below zero.
Minus sign: the same dash placed between two numbers, meaning subtract.
Opposites: two numbers the same distance from zero on different sides, such as 5 and −5.
Absolute value: a number’s distance from zero; the number without its sign. Written with bars: |−6| = 6.
Farther from zero: having the larger absolute value, on either side of zero.
The four cases: add a positive, add a negative, subtract a positive, subtract a negative.
Zero pair: a number and its opposite, which add to zero.
Keep, change, change: a short name for Rule 3: keep the first number, change subtraction to addition, change the sign of the second number.
Eighteen questions on the whole page
Answer each one, then press Check. Each answer comes with its reasoning. Type a negative number with the minus key, like -7.
- 1.
Which is larger, −9 or −4?
−4 is farther to the right on the number line.
- 2.
What is |−14|?
The distance from zero, 14.
- 3.
Which case is 8 − 5?
The dash between the numbers means subtracting, and the 5 has no sign of its own, so it is positive: subtract a positive.
- 4.
Which case is −8 − (−5)?
Subtracting, and the second number is −5: subtract a negative. The first number being negative does not change the case.
- 5.
−6 + (−7)
Same signs, Rule 1: 6 + 7 = 13, keep the negative sign. −13.
- 6.
10 + (−4)
Different signs, Rule 2. First subtract: 10 − 4 = 6. The original larger absolute value is 10, positive. 6.
- 7.
3 + (−11)
Different signs, Rule 2. First subtract: 11 − 3 = 8. The original larger absolute value is −11, negative. −8.
- 8.
−15 + 9
Different signs, Rule 2. First subtract: 15 − 9 = 6. The original larger absolute value is −15, negative. −6.
- 9.
5 − 12
Subtract a positive: 5 + (−12). First subtract: 12 − 5 = 7. The original larger absolute value is −12, negative. −7.
- 10.
−2 − 6
Subtract a positive: −2 + (−6). Both negative: −8.
- 11.
7 − (−8)
Subtract a negative: 7 + 8 = 15.
- 12.
−10 − (−3)
Subtract a negative: −10 + 3. First subtract: 10 − 3 = 7. The original larger absolute value is −10, negative. −7.
- 13.
−3 − (−10)
Subtract a negative: −3 + 10. First subtract: 10 − 3 = 7. The original larger absolute value is 10, positive. 7.
- 14.
−5 + 8 − 6 + (−2)
Rewrite: −5 + 8 + (−6) + (−2). Positives: 8. Negatives: −13. 8 + (−13) = −5.
- 15.
(−4) × (−9)
Same signs: positive. 36.
- 16.
−63 ÷ 7
Different signs: negative. −9.
- 17.
Which of these equals 6 − (−2)?
Rule 3: keep 6, change subtraction to addition, change −2 to 2: 6 + 2. Only the second number changes sign.
- 18.
An account is overdrawn by $25 (a balance of −25 dollars). A deposit of $60 is made, then a $50 bill is paid. What is the balance?
−25 + 60 + (−50). Positives: 60. Negatives: −75. 60 + (−75) = −15. The account is overdrawn by $15.