1. Keep the first number
The starting amount never changes: in -5 - (-9), the -5 stays exactly as it is. Only the subtraction and the second number are touched.
Subtracting negative fractions
Why Do Negative and Positive Integers Switch Signs?
They switch because subtraction means adding the opposite. The minus between the numbers becomes a plus, and the second number flips across zero — keep the first, change the operation, change the second number.
- keep change change
- add the opposite
- why signs flip
- subtracting integers
Quick answer
Keep, change, change — then add
Every subtraction can be rewritten as an addition of the opposite: 3 - 8 = 3 + (-8) and -5 - (-9) = -5 + 9. The apparent “sign switch” is the second number being replaced by its opposite while the operation flips from take-away to add. The first number never changes.
The rewrite is forced, not chosen: addition and subtraction must undo each other, so subtracting (-9) has to act exactly opposite to adding (-9) — and acting opposite to a leftward move is a rightward move of the same size.
The three moves
Keep-change-change, in order
The mnemonic names the three changes in sequence. Saying them in order prevents the classic half-rewrite, where the operation flips but the number does not (or the reverse).
2. Change the operation
Every subtraction can be rewritten as adding the opposite. The minus sign between the numbers becomes a plus — this is the moment the problem "switches signs".
3. Change the second number’s sign
The opposite of -9 is +9, so the second number flips: -5 - (-9) becomes -5 + 9. What looks like a sign switch is just the opposite of the number you were going to take away.
4. Add with the integer rules
Different signs compete: 9 outgrows 5 and keeps its positive direction, so the answer is +4. Same-sign additions after the flip simply combine.
Worked example
Why 3 - 8 becomes 3 + (-8)
3 - 8 = 3 + (-8) = -5
Even without a visible negative, subtraction means adding the opposite. The switch is invisible when both numbers are positive — 3 + (-8) and 3 - 8 look like the same problem. The rule is doing the same work in every row of this page.
Worked example
The famous double negative: -5 - (-9)
-5 - (-9) = -5 + 9 = 4
Both switches fire: the subtraction becomes addition and -9 becomes +9. The answer 4 sits nine steps right of -5, which is exactly what removing a 9-debt should do.
Worked example
Same-sign flip: 4 - 10
4 - 10 = 4 + (-10) = -6
Adding the opposite lands past zero into negative territory. The sign switch explains why subtracting a bigger positive crosses zero — the addition form 4 + (-10) makes the direction impossible to misread.
Worked example
A fraction keeps the same rule: -1/2 - (-3/4)
-1/2 - (-3/4) = -1/2 + 3/4 = -2/4 + 3/4 = 1/4
Keep-change-change is not an integer-only trick. The fraction version needs one extra step — matching denominators — before the signed numerators combine: -2 + 3 = 1, so the answer is 1/4.
The switch is the opposite, not a trick
Subtracting a number means adding its opposite. That single sentence generates every "sign switch" students see. In 3 - 8, the opposite of 8 is -8, so the problem is 3 + (-8). In -5 - (-9), the opposite of -9 is +9, so the problem is -5 + 9. The signs appear to switch because taking the opposite flips a number across zero — the operation changed, and the second number followed.
The first number never switches. It is the amount you start with, and no rewrite touches it. When students flip both numbers, the problem is no longer the one asked: -5 - (-9) rewritten as 5 + 9 = 14 has kept the size but thrown away both directions. Keep the first, change the operation, change the second — the order is the mnemonic.
Why the two operations must mirror each other
Addition and subtraction are inverse operations: subtracting must undo whatever adding does. Adding -9 to 5 gives -4, so undoing it — subtracting (-9) from -4 — must return to 5. The only way that works is if -4 - (-9) = -4 + 9. The sign switch is forced by the structure of the number line, not by a classroom convention.
This is also why the switch survives contact with fractions. The values -1/2 and +1/2 are opposites just as -9 and +9 are, and subtraction must undo addition for every point on the line, not only for whole numbers. Keep-change-change with a common-denominator step solves every signed fraction subtraction there is.
Common mistakes with the sign switch
- Changing the first number: Keep-change-change keeps the first number. In -5 - (-9), writing 5 + 9 flips the start across zero and produces 14 instead of 4.
- Changing the operation but not the number (or the reverse): Both changes happen together or neither does. -5 + (-9) = -14 and -5 - 9 = -14 both come from half a rewrite. Say the full sentence — "subtracting means adding the opposite" — and both switches fire.
- Switching signs in addition: The switch belongs to subtraction. In -5 + (-9), the plus sign stays and the signs combine: -14. Rewriting an addition problem with keep-change-change invents a subtraction that was never asked for.
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Related fraction topics
Common questions
Integer Sign Switching FAQ
Because subtraction means adding the opposite. The minus sign between two numbers becomes a plus, and the second number takes its opposite across zero: 3 - 8 = 3 + (-8) and -5 - (-9) = -5 + 9. The "switch" is the second number being replaced by its opposite, not the signs changing on their own.