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Negative numbers
Temperatures below zero, money you owe, depths under the water: negative numbers are everywhere, and they show up in math word problems. Learn the number line, the sign rules and one idea about distance, and the wrong-sign answers stop catching you.
A negative number is a number below zero
Numbers to the right of zero are positive. Numbers to the left of zero are negative. A negative number is written with a minus sign: −4 is read “negative four.” Zero is neither positive nor negative.
You already use negative numbers. 4 degrees below zero is −4°F. If you owe the bank $40, your balance is −$40. A diver 20 feet under the water is at −20 feet.
Every negative number has a sign and a size
The sign (− or +) tells you which side of zero the number is on. The size tells you how far it is from zero. −4 and 4 have the same size and opposite signs. Most mistakes with negative numbers are a wrong sign, not a wrong size.
How to solve them
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1
The number line: farther right is greater
On a number line, numbers get bigger as you move right and smaller as you move left. So −2 is greater than −7, because −2 is to the right of −7. In winter, −2°F is warmer than −7°F.
Here is the trap: 7 is bigger than 2, but −7 is less than −2. With negative numbers, the bigger the size, the smaller the number.
In word problems, the sign hides in the words. Learn these:
below zero−5 below zero is −5°Fowe, debt, overdrawn−owing $30 is −30drop, fall, lose, spend−a 6-degree drop is −6rise, gain, deposit+a $50 deposit is +50above sea level+200 feet up is +200Next step: Adding−7 is to the left of −2, so −7 is less than −2. Which is greater?−2 or −7
−2: it is to the right
Which is colder?−15°F or −5°F
−15°F: it is farther below zero
Least to greatest3, −1, −6, 0
−6, −1, 0, 3
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2
Adding: same signs or different signs
Think of money. A positive number is money you have. A negative number is money you owe.
Same signs: add the sizes and keep the sign. You owe $6, then you owe $9 more: now you owe $15.
Different signs: subtract the sizes and keep the sign of the bigger size. You owe $12 and you pay back $4: you still owe $8.
Next step: Subtracting- −6 + (−9)
- −(6 + 9)
- −15
Same signs: add the sizes, keep the minus. - −12 + 4
- −8 + (−4) + 4
- −8 + 0
- −8
Split −12 into −8 and −4. The −4 and the 4 cancel: together they make 0. - −5 + 8
- −5 + 5 + 3
- 0 + 3
- 3
Split 8 into 5 and 3. The −5 and the 5 cancel. -
3
Subtracting: add the opposite
a − b = a + (−b)
The opposite of a number is the same distance from zero, on the other side. The opposite of 8 is −8. Subtracting a number is the same as adding its opposite.
On the number line, subtracting a positive number means walking left. If you walk farther than the number you started on, you go below zero.
Next step: Minus a negative- 3 − 8
- 3 + (−8)
- 3 + (−3) + (−5)
- 0 + (−5)
- −5
It is 3°F and the temperature drops 8 degrees: now it is −5°F. - −4 − 6
- −4 + (−6)
- −10
Both pieces are negative: you go farther below zero. -
4
Subtracting a negative: two minus signs make a plus
a − (−b) = a + b
Why? Taking away a debt makes you richer. If someone cancels a $5 debt you owe, you are $5 better off.
You use this every time you find the difference between a warm temperature and a cold one below zero: warmer minus colder.
Careful: “two minus signs make a plus” is only for subtracting a negative, or multiplying or dividing two negatives. Adding two negatives stays negative: −6 + (−9) = −15.
Next step: Times and divide- 7 − (−5)
- 7 + 5
- 12
The afternoon is 7°F and the night is −5°F: a difference of 12 degrees. - −3 − (−9)
- −3 + 9
- −3 + 3 + 6
- 0 + 6
- 6
First turn the two minus signs into a plus. Then it is an ordinary addition with different signs. -
5
Multiplying and dividing: the signs decide first
For × and ÷ there are only two rules. Same signs give a positive answer. Different signs give a negative answer. Find the sign first, then work with the sizes as usual.
Signs for × and ÷Operation Answer Example + × + positive 4 × 3 = 12 − × − positive (−4) × (−3) = 12 + × − negative 4 × (−3) = −12 − × + negative (−4) × 3 = −12 − ÷ − positive (−20) ÷ (−4) = 5 − ÷ + negative (−20) ÷ 4 = −5 Next step: Absolute value- (−2) × (−3) × (−5)
- 6 × (−5)
- −30
Two minus signs make a plus. The third one makes it negative again. ShortcutCount the minus signs
When you multiply several numbers, count the minus signs. An even number of them gives a positive answer; an odd number gives a negative one.
ExampleA fuel tank loses 15 gallons a day for 4 days
- Losing is negative: each day is −15 gallons.
- 4 × (−15): different signs, so the answer is negative.
- The size: 4 × 15 = 60. The change is −60 gallons.
Answer −60 gallons
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6
Absolute value: the distance from zero
The absolute value of a number is its distance from zero. It is written between two bars: |−6| = 6 and |6| = 6. A distance is never negative, so an absolute value is never negative.
The distance between two numbers is the bigger one minus the smaller one. When the smaller one is below zero, that is a subtraction of a negative, so it becomes an addition.
Next step: Three examples|−6| = 6: −6 is 6 steps from zero. ExampleA diver is 35 feet under the water. A pelican flies 20 feet above the water. How far apart are they?
- Under the water is negative: the diver is at −35 ft. The pelican is at +20 ft.
- Bigger minus smaller: 20 − (−35).
- Two minus signs make a plus: 20 + 35 = 55.
Answer 55 feet
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7
Three worked examples, from easy to test level
Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.
In every example, decide the sign before you work out the size. Say it out loud if it helps: “different signs, so subtract, and the bigger one is negative.”
Example · EasyWhat is −9 + 4?
First try
−9 + 4 = −13- The signs are different: −9 is negative and 4 is positive. Adding the sizes is the rule for same signs only.
- Different signs: subtract the sizes. 9 − 4 = 5.
- Keep the sign of the bigger size. The bigger size is 9, and −9 is negative.
- So the answer is −5. Check with money: you owe $9, you pay back $4, and you still owe $5.
Answer −5
Example · MediumWhat is −6 − (−14)?
First try
−6 − 14 = −20- The second minus sign matters. Subtracting a negative is adding: −6 − (−14) becomes −6 + 14.
- Different signs: subtract the sizes. 14 − 6 = 8.
- The bigger size, 14, is positive, so the answer is positive: 8.
Answer 8
Example · Test levelAt 6 a.m. the temperature at a base was −11°F. By noon it had risen 18 degrees. By midnight it had dropped 15 degrees from the noon temperature. What was the temperature at midnight?
First try
−11 + 18 = 7, then 15 − 7 = 8°F- Rising is adding: −11 + 18. Different signs: 18 − 11 = 7, and 18 is the bigger size, so at noon it was 7°F.
- Dropping is subtracting: 7 − 15. Do not flip it to 15 − 7. The drop is bigger than 7, so the temperature goes below zero.
- Split 15 into 7 and 8: 7 − 7 = 0, and 8 more degrees down is −8.
- At midnight it was −8°F.
Answer −8°F
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Adding the sizes when the signs are different
−7 + 3: one number is negative and one is positive.
Wrong−7 + 3 = −10
Right−7 + 3 = −4
Dropping the second minus
5 − (−4): you are subtracting a negative.
Wrong5 − 4 = 1
Right5 + 4 = 9
Two negatives multiplied
(−6) × (−5): the signs are the same.
Wrong(−6) × (−5) = −30
Right(−6) × (−5) = 30
Bigger size, smaller number
Which is less, −12 or −3?
Wrong−3, because 3 is smaller than 12
Right−12: it is farther left
15 questions in three levels
Warm-up first, then core, then test level. Pick an answer, then read why.
Negative numbers are not a separate test. You need them in math questions with temperatures, bank balances, depths and changes, and when you solve equations.
There is no calculator, so getting the sign right is your job. A wrong sign gives a choice with the right size and the wrong sign, so if you see both a number and its opposite in the choices, slow down and check your sign.
A tip: decide the sign first and write it down, then work out the size. On the computer test you cannot go back to a question, so check the sign before you choose.
What is −7 + 12?
Show the answer
C. The signs are different, so subtract the sizes: 12 − 7 = 5. The bigger size is 12, and it is positive, so the answer is positive. −19 and 19 add the sizes, which is the rule for same signs only, and −5 has the right size with the wrong sign. The answer is 5.
What is 4 − 11?
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B. Start at 4 and walk 11 steps left. The first 4 steps take you to 0, and the other 7 take you to −7. 7 flips the subtraction to 11 − 4 and loses the sign, 15 adds 4 and 11, and −15 adds them with a minus. The answer is −7.
What is (−9) × 4?
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A. Different signs give a negative answer. Then the size: 9 × 4 = 36. 36 forgets the sign, and −13 and 13 add or subtract the numbers instead of multiplying them. The answer is −36.
Which of these numbers is greater than −5?
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D. Greater means farther right on the number line. −2 is to the right of −5, closer to zero. −12, −9 and −6 look bigger because 12, 9 and 6 are bigger than 5, but all three are to the left of −5. The answer is −2.
What is −24 ÷ (−3)?
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C. Same signs give a positive answer. Then the size: 24 ÷ 3 = 8. −8 has the wrong sign, −21 subtracts instead of dividing, and 72 multiplies 24 by 3. The answer is 8.
What is −8 − (−17)?
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C. Subtracting a negative is adding: −8 − (−17) becomes −8 + 17. Different signs: 17 − 8 = 9, and the bigger size, 17, is positive. −25 drops the second minus and does −8 − 17, −9 has the wrong sign, and 25 adds the sizes. The answer is 9.
What is (−4) × (−7) × (−2)?
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A. Count the minus signs: there are three, an odd number, so the answer is negative. The size: 4 × 7 = 28, and 28 × 2 = 56. 56 forgets the third minus sign, 28 stops after the first two numbers, and −13 adds the numbers instead of multiplying. The answer is −56.
At dawn the temperature was −13°F. By 2 p.m. it was 9°F. How many degrees did the temperature rise?
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D. The rise is warmer minus colder: 9 − (−13). Two minus signs make a plus: 9 + 13 = 22. On the number line, that is 13 degrees up to 0, then 9 more. 4 degrees does 13 − 9 and forgets that the start is below zero, −4 degrees adds −13 and 9, and −22 degrees has the right size with the wrong sign: a rise is positive. It rose 22 degrees.
Jaylen’s checking account is overdrawn: the balance is −$35. He deposits $120. What is the new balance?
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C. A deposit is adding: −35 + 120. Different signs: 120 − 35 = 85, and the bigger size, 120, is positive. The first $35 of the deposit pays off what he owed, and the other $85 stays in the account. −$155 subtracts the deposit, −$85 has the wrong sign, and $155 treats the $35 he owed as money he had. The new balance is $85.
How far apart are −9 and 14 on the number line?
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D. Distance is bigger minus smaller: 14 − (−9) = 14 + 9 = 23. Or count: 9 steps from −9 up to 0, then 14 more. 5 does 14 − 9 and forgets that −9 is on the other side of zero, and 9 and 14 are each only one number’s distance from zero. They are 23 apart.
At 6 p.m. the temperature was 4°F. It dropped 3 degrees every hour until midnight. What was the temperature at midnight?
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B. From 6 p.m. to midnight is 6 hours. The total drop: 6 × 3 = 18 degrees, a change of −18. Then 4 − 18: the first 4 degrees take it to 0, and the other 14 take it to −14. −18°F forgets the starting 4°F, −11°F counts only 5 hours, and 22°F adds the drop instead of subtracting it. At midnight it was −14°F.
Ana’s account is at −$40 after a purchase. The bank charges a $25 overdraft fee. Then she deposits $200. What is her balance now?
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A. The fee is more debt: −40 − 25 is the same as −40 + (−25). Same signs: add the sizes and keep the minus: −65. Then the deposit: −65 + 200. Different signs: 200 − 65 = 135, and 200 is the bigger size, so the balance is positive. $160 forgets the fee, $185 adds the fee instead of taking it away, and $265 adds everything, as if the debt were money she had. Her balance is $135.
A hiking trail starts 120 feet below sea level and ends 340 feet above sea level. How many feet does the trail climb?
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D. Below sea level is negative: the trail starts at −120 ft and ends at +340 ft. The climb is end minus start: 340 − (−120) = 340 + 120 = 460. 220 ft does 340 − 120 and forgets that the start is below sea level, 340 ft counts only the part above sea level, and −460 ft subtracts the wrong way round: a climb is positive. The trail climbs 460 ft.
If a = −6 and b = 4, what is the value of ab − a? (ab means a × b.)
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B. First ab: (−6) × 4 = −24, because different signs give a negative. Then subtract a, which is −6: −24 − (−6) becomes −24 + 6. Different signs: 24 − 6 = 18, and the bigger size, 24, is negative, so the result is −18. −30 drops the second minus (−24 − 6), 30 forgets that ab is negative (24 + 6), and 18 ignores every sign (6 × 4 − 6). The value is −18.
A unit’s fuel log shows five changes, in gallons: +45, −30, −25, +18, −12. What is the total change?
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B. Group the signs. Gains: 45 + 18 = 63. Losses: 30 + 25 + 12 = 67, so −67. Then 63 − 67: different signs, 67 − 63 = 4, and the bigger size, 67, is negative. 4 gallons has the wrong sign, 130 gallons adds every number as a gain, and −130 gallons adds every number as a loss. The total change is −4 gallons.
12 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 12 cards known.
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