Adding and Subtracting Integers
The stronger-sign rule for adding and subtracting integers, why subtracting a negative adds, and how bank balances make it intuitive.
--
Multiplication and division have a tidy sign rule. Addition and subtraction work differently, and applying the multiplication rule here is a common source of errors. What you need instead is the idea of the stronger sign.
Concept
The stronger sign wins
The answer takes the sign of the number with the larger absolute value, and its size is the difference between the two.
Two questions, answered separately: which sign is stronger, and how big is the gap between the two amounts.
Example
The rule in action
Work out .
Compare sizes: 3 is larger than 2, and it carries a minus.
So the answer is negative.
The difference is 
.
⟹ −1
Now : the 7 is larger and negative, and the difference is
, giving
.
Example
When the positive is stronger
These are the familiar cases, and the rule handles them too. It is one method covering every combination, not a special trick for negatives.
Concept
Subtracting a negative
Taking away a negative adds. The two minus signs sitting together turn into a plus, and the problem becomes an ordinary addition.
Rewrite the expression first, then apply the stronger-sign rule to what remains.
Concept
Two negatives added
When both numbers are negative there is no contest between signs — the total simply grows more negative.
Note the contrast with multiplication, where two negatives give a positive. Here they reinforce each other instead.
Example
A bank balance
A positive balance is money you hold; a negative balance is money you owe. An account is overdrawn by 50, then 200 is deposited.
The balance is 
.
Adding the deposit: 
.
200 is larger and positive, so the answer is positive.
The difference is 
.
⟹ a balance of 150
The debt is cleared first and the remainder is what is left over — exactly what the stronger-sign rule computes.
Note
Mistakes to avoid
Adding the two numbers when their signs differ, instead of subtracting.
Taking the sign of the first number rather than the larger one.
Leaving %20%7D)
as a subtraction instead of rewriting it as 
.
Applying the multiplication sign rule: %20%2B%20(-6)%20%7D)
is 
, not 
.
Forgetting that a number without a sign is positive.
Summary
- The result takes the sign of the number with the larger absolute value.
- Its size is the difference between the two numbers.
- Subtracting a negative becomes an addition: 5 − (−2) = 5 + 2 = 7.
- Two negatives added together grow more negative, not positive.
- Bank balances make the whole rule intuitive: debts and deposits cancel out.