Positive & Negative Rational Number Operations — Print
Negative numbers are directions, not just "less than nothing". Subtraction is adding the opposite, and sign rules come from patterns you can see.
What the practice questions cover
These questions use 3 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
At eight this morning it was 8 degrees. Now it is 2 degrees below zero. How much has it dropped?
You cannot say "8 minus 2 is 6." The thermometer went past zero, and below zero is real.
Count it on the scale instead. From +8 down to 0 is eight degrees. From 0 down to -2 is two more.
Ten degrees altogether. Written down, that is 8 - (-2) = 10.
Subtracting a negative added. Not by decree — because going down past zero covers extra ground.
Every sign rule in this unit is that number line in disguise. When one is hard to remember, put it back on the thermometer and read it off.
The number line is the whole idea
Put positives to the right of zero and negatives to the left. Adding a positive walks right. Adding a negative walks left.
-3 + 5: start at -3, walk 5 to the right, land on 2.
4 + (-9): start at 4, walk 9 left, land on -5.
Subtraction is adding the opposite
This one rule replaces all the confusing cases: a - b = a + (-b).
5 - 8 = 5 + (-8) = -3. And -2 - (-6) = -2 + 6 = 4.
Two minus signs next to each other turn into a plus, because the opposite of going left is going right.
Keep–Change–Change
Keep the first number. Change the - to +. Change the sign of the second number. Now you only ever add.
Why a negative times a negative is positive
Look at a pattern instead of memorizing:
3 × (-2) = -6, ;2 × (-2) = -4, ;1 × (-2) = -2, ;0 × (-2) = 0.
Each time the left factor drops by 1, the answer rises by 2. Keep going: -1 × (-2) must be 2, and -2 × (-2) must be 4. The pattern forces it.
Sign rules
Same signs → positive. Different signs → negative. This holds for both × and ÷.
Counting the negatives
In a long product, count how many negative factors there are. An even count gives a positive answer; an odd count gives a negative one.
(-1)(-2)(-3)(-4) = 24 — four negatives, even, positive.
(-2)5 = -32 — five negatives, odd, negative.
The trap: -32 versus (-3)2
(-3)2 = (-3)(-3) = 9. The parentheses say the negative is part of the base.
-32 = -(32) = -9. Without parentheses, the exponent grabs only the 3, and the minus sign waits outside.
Worked examples
A chain of signs
Compute -8 - (-3) + (-5).
- Rewrite every subtraction as addition. Keep–Change–Change.
- -8 + 3 + (-5) -(-3) became +3.
- -8 + 3 = -5 Start at -8, walk 3 right.
- -5 + (-5) = -10 Walk another 5 left.
-10
Negatives with fractions
Compute -34 ÷ 25.
- Handle the sign first: one negative, so the answer is negative. Different signs give a negative.
- Now the size: 34 × 52 = 158 Keep, change, flip.
- Put the sign back: -158 Or -178.
-158
Order of operations with signs
Compute -22 + (-2)3 - (-4).
- -22 = -(2 × 2) = -4 No parentheses, so the exponent takes only the 2.
- (-2)3 = -8 Three negatives — odd count — so negative.
- -(-4) = +4 Subtracting a negative adds.
- -4 + (-8) + 4 = -8 Combine left to right.
-8
Words to know
- Opposite
- Same distance from zero, other side. The opposite of -7 is 7.
- Absolute value
- |x| — distance from zero, always positive or zero.
- Additive inverse
- The number you add to get 0. -7 + 7 = 0.
- Integer
- A whole number, positive, negative, or zero.
Mistakes to watch for
Reading -32 as 9. Without parentheses the exponent attaches only to the 3, so it is -(32) = -9. You get 9 only from (-3)2, where the brackets make -3 the base.
Treating "two negatives make a positive" as a rule about addition. It is a rule about multiplication. -3 + (-5) = -8: two debts make a bigger debt, not a credit.
Losing the sign halfway through a long calculation. Decide the sign first, by counting the negative factors, then do the arithmetic with plain numbers. An even count is positive, an odd count negative.
Ordering negatives by their size. -0.8 is less than -0.75, because it sits further left on the number line. The bigger the digits, the smaller the negative.
Negative numbers were used in China by about 200 BC, written with black counting rods for positive and red rods for negative.
European mathematicians resisted them for another 1,500 years. As late as the 1700s some called them "absurd" or "fictitious" and refused to accept a negative answer.
Modern accounting reversed the colours — being "in the red" now means negative.
Problem 1
Compute -7 + 4.
Worked answer
-3
-7 + 4 = -3
Problem 2
Compute 5 + (-12).
Worked answer
-7
Start at 5, walk 12 left
5 - 12 = -7
Problem 3
Compute 3 - (-8).
Worked answer
11
3 - (-8) = 3 + 8
= 11
Problem 4
Compute (-6) × 7.
Worked answer
-42
6 × 7 = 42
Different signs, so -42.
Problem 5
Compute (-48) ÷ (-6).
Worked answer
8
48 ÷ 6 = 8
Same signs, so +8.
Problem 6
Compute -9 - (-4) + (-6).
Worked answer
-11
-9 + 4 + (-6)
-9 + 4 = -5
-5 + (-6) = -11
Problem 7
Compute (-3)2.
Worked answer
9
(-3)(-3) = 9
Same signs, positive.
Problem 8
Compute -32.
Worked answer
-9
-32 = -(32) = -(9) = -9
The minus sign waits outside the exponent.
Problem 9
Compute (-2)(-3)(-5).
Worked answer
-30
2 × 3 × 5 = 30
Three negatives — an odd count — so the answer is -30.
Problem 10
Compute -23 × 94.
Worked answer
-3/2
One negative, so the answer is negative
23 × 94 = 1812 = 32
Answer: -32
Problem 11
Compute -35 ÷ (-910).
Worked answer
2/3
Same signs, so the answer is positive
35 × 109 = 3045 = 23
Problem 12
The temperature is -4°C at dawn and rises 11° by noon, then falls 6° by night. What is the night temperature?
Worked answer
1
-4 + 11 = 7
7 - 6 = 1
It is 1°C at night.
Problem 13
Compute -23 + (-4)2 - (-5).
Worked answer
13
-23 = -(8) = -8 — no parentheses, so the sign stays outside
(-4)2 = 16 — two negatives, positive
-(-5) = +5
-8 + 16 + 5 = 13
Problem 14
Compute -24 + 9-3 - 7.
Worked answer
-2
Top: -24 + 9 = -15
-15-3 = 5 — same signs, positive
5 - 7 = -2
Problem 15
A submarine at -180 m rises at 15 m per minute. What is its depth after 7 minutes?
Worked answer
-75
It rises 15 × 7 = 105 m
-180 + 105 = -75
It is 75 m below the surface.
Problem 16
What is the value of (-1)99 + (-1)100?
Worked answer
0
(-1)99 = -1 — odd count of negatives
(-1)100 = 1 — even count
-1 + 1 = 0