Read the signs of both addends
Each number carries a direction: positive points right, negative points left. Write the problem as signed numbers before computing — 5 + (-8), not "5 minus something".
Subtracting negative fractions
Addition of Positive and Negative Integers
Two rules cover every signed addition: same signs combine and keep their sign, different signs compete and the larger size decides. Every subtraction rewrites into this addition, so these rules are the engine behind the whole topic.
- same signs add
- different signs compete
- sign of the larger
- number line addition
Quick answer
Same signs add; different signs compete
-4 + (-3) = -7: both moves point left, so the sizes add and the answer stays negative. -4 + 9 = 5: the moves disagree, nine outgrows four, and the answer keeps the winner’s positive sign. Opposites such as 4 + (-4) cancel to exactly 0.
A number line makes the rules visible. Start at the first number and walk the second number’s size in its direction — right for positive, left for negative. Where you land is the sum.
Sign rules
Four cases, two rules
Read the two signs, pick the row, and the whole method is decided before any arithmetic. Notice the different-sign rows share one rule — only which size wins changes.
Sign combinationRuleExamples
Positive + positiveAdd the sizes and keep the positive sign.4 + 3 = 7
Negative + negativeAdd the sizes and keep the negative sign.-4 + (-3) = -7
Different signsSubtract the smaller size from the larger; the winner’s sign wins.-4 + 9 = 5 and 4 + (-9) = -5
OppositesEqual sizes with different signs cancel to zero.4 + (-4) = 0
The method
Four steps: read, sort, compete, confirm
Sort the case: same signs or different signs
Same-sign problems combine the sizes and keep the shared sign. Different-sign problems are a contest between two opposite moves.
For different signs, subtract the sizes
The larger size decides the answer’s sign; the answer’s size is the difference. In -4 + 9, nine outgrows four, so the answer is +5. In 4 + (-9), it is -5.
Confirm with a number line walk
Start at the first number and walk the second number’s size in its direction. -4 + 9 starts left of zero and walks nine right, landing on 5.
Worked example
Same negative signs: -6 + (-3)
-6 + (-3) = -9
Both moves point left, so the sizes add and the sign stays negative: nine left of zero. Two debits combine into a bigger debit — there is no competition because nothing points right.
Worked example
Different signs, positive wins: -6 + 10
-6 + 10 = 10 - 6 = 4
Ten steps right against six steps left leaves four steps right. The larger size (10) takes the answer’s sign, and the answer’s size is the difference 10 - 6 = 4.
Worked example
Different signs, negative wins: 6 + (-10)
6 + (-10) = -(10 - 6) = -4
The same walk in mirror image: six right cannot cover ten left, so the landing point sits four left of zero. Same sizes as the previous example — only the winner’s sign changed.
Worked example
Fractions play by the same rules: -3/4 + 1/2
-3/4 + 1/2 = -3/4 + 2/4 = -1/4
Match the denominators first: halves become quarters. Then the different-sign rule runs on the numerators — 3 outgrows 2 and keeps the negative direction, so the answer is -1/4.
Why same signs add but different signs subtract
Addition is a walk on the number line. A positive addend is a walk to the right, a negative addend a walk to the left. When both walks point the same way, the distances simply stack — that is why -4 + (-3) travels seven left, not one. When the walks point in opposite directions, they partially undo each other, and what remains is the difference between the two sizes, facing the direction of the longer walk.
This picture explains the case students find strangest: adding a negative makes the total smaller. In 6 + (-10), the sum is -4 — smaller than 6 — because a leftward walk of ten more than undoes the six steps right. “Adding” does not mean growing; it means applying the second number’s signed move.
How this powers subtracting negative fractions
Every subtraction on this topic rewrites as one of these additions. -3/4 - (-1/4) becomes -3/4 + 1/4 — a different-sign addition where 3 outgrows 1 and the negative direction wins: -1/2. 1 - (-3/4) becomes 1 + 3/4 — a same-sign addition stacking to 1 3/4. Once the rewrite is made, the rules on this page finish every problem.
Fractions add one familiar step: the sizes can only compete after the denominators match. Rewrite both fractions over the least common denominator first — -2/3 + 1/6 becomes -4/6 + 1/6 — and then the signed numerators follow the same two rules as integers: sizes compete, larger wins.
See the minus-a-minus rewrite in actionPractice routine
Predict the sign before computing
For each problem, say which rule applies and which sign should win before doing any arithmetic. Then compute and check.
TryWhat to noticeAnswer
-8 + (-5)Same signs: add the sizes, keep the sign.-13
-8 + 5Different signs: 8 wins, so the answer is negative.-3
8 + (-5)Mirror of the last problem: the positive size wins.3
-7 + 7Exact opposites cancel.0
-2/3 + 1/6Match denominators (sixths), then run the different-sign rule.-1/2
Keep learning
Related fraction topics
Common questions
Adding Positive and Negative Integers FAQ
Same signs: add the sizes and keep the sign — -4 + (-3) = -7. Different signs: subtract the smaller size from the larger and keep the sign of the larger — -4 + 9 = 5, while 4 + (-9) = -5. Opposites always cancel to zero.