Adding integers

Adding two whole numbers like $3 + 4$ is easy. Adding integers gets a little trickier because we have to handle negative numbers. But there are just two simple cases, and once you learn them, you'll never go wrong.

Concept

Case 1: Same signs

When both numbers have the same sign, add their sizes and keep the common sign.

$(+3) + (+4) = +7$ . Both positive, sum is $7$ , sign is $+$ .
$(-5) + (-2) = -7$ . Both negative, sum of sizes is $7$ , sign is $-$ .

In short: $(+a) + (+b) = +(a + b)$ and $(-a) + (-b) = -(a + b)$ .

Case 2: Different signs

When the two numbers have different signs, find the difference of their sizes and keep the sign of the number with the bigger size.

$(+8) + (-3)$ : sizes $8$ and $3$ . Difference $5$ . The $8$ is bigger, and it is positive, so the answer is $+5$ .
$(+4) + (-9)$ : sizes $4$ and $9$ . Difference $5$ . The $9$ is bigger, and it is negative, so the answer is $-5$ .

In short: if $a > b > 0$ , then $(+a) + (-b) = +(a - b)$ and $(-a) + (+b) = -(a - b)$ .

Brahmagupta's elegant rules

The Indian mathematician Brahmagupta wrote these rules clearly in $628$ CE in his book Brahmasphutasiddhanta. He called positive numbers "fortunes" and negative numbers "debts". His rules in words:

A fortune plus a fortune is a fortune.
A debt plus a debt is a debt.
A fortune and an equal debt cancel.
A fortune plus a debt: the bigger wins.

The same idea, $1{,}400$ years ago.

Picture on a number line

To add an integer $b$ to $a$ :

Start at $a$ .
If $b > 0$ , move $b$ steps to the right.
If $b < 0$ , move $|b|$ steps to the left.
Where you land is the answer.

This visual rule covers both cases above without separating into rules.

Useful facts

$a + 0 = a$ . (Adding zero changes nothing.)
$a + (-a) = 0$ . (A number and its opposite cancel out.)
$a + b = b + a$ . (Addition is commutative , order doesn't matter.)
$(a + b) + c = a + (b + c)$ . (Addition is associative , grouping doesn't matter.)

Worked examples

Example 1. $(+12) + (+8)$ .

Same sign (both positive). Sum: $12 + 8 = 20$ . Answer: $+20$ .

Example 2. $(-7) + (-9)$ .

Same sign (both negative). Sum: $7 + 9 = 16$ . Answer: $-16$ .

Example 3. $(+15) + (-6)$ .

Different signs. Difference: $15 - 6 = 9$ . The $15$ is bigger (positive), so answer is $+9$ .

Example 4. $(-12) + (+4)$ .

Different signs. Difference: $12 - 4 = 8$ . The $12$ is bigger (negative), so answer is $-8$ .

Example 5. $(-5) + 5$ .

Different signs. Difference: $0$ . Answer: $0$ (the two cancel).

Example 6. Word problem. The temperature at sunrise is $-3°$ C. By noon it has risen $11°$ . What is the noon temperature?

$-3 + 11 = +8$ . So $8°$ C.

Example 7. $(-3) + (+4) + (-7)$ .

Group: $(-3) + (+4) = +1$ . Then $(+1) + (-7) = -6$ .
Answer: $-6$ .

Try it yourself

$(+5) + (+9)$ .
$(-8) + (-6)$ .
$(+12) + (-7)$ .
$(-15) + (+9)$ .
$(-3) + (+3)$ .
$(-4) + (-1) + (+9)$ .
$20 + (-7) + (-6) + 5$ .
The temperature drops $5°$ from $2°$ C. What is the new temperature?
Word problem: A diver is $-12$ m. He rises $8$ m. What is his new depth?
Tricky: $(-100) + (+50) + (-25) + (+30)$ .

Activity

Lift game. Pick a starting floor (e.g., $+2$ ) in Bela's building. Use a stack of cards: each card has a number with a sign ( $+5$ , $-3$ , $+2$ , $-7$ , $-1$ , etc.). Draw a card and add it to your current floor , this means moving up (if positive) or down (if negative) by that many floors. After $5$ cards, where are you? Compare with a friend's path.