Adding Integers: Definition, Rules, and Examples

What does adding integers mean?

Integers are whole numbers that do not have fractional parts. They include positive numbers, negative numbers, and zero. Adding them together allows us to calculate the final result of combined amounts or opposing directions.

For instance, earning $50$ dollars and then spending $20$ dollars involves adding a positive integer to a negative integer. Representing this mathematically requires specific rules to handle the different signs.

Adding integers with the same sign

When combining two or more numbers that share the same sign, they reinforce each other. The total distance from zero increases in the same direction.

To add integers with the same sign, add their absolute value and keep the original sign.

Adding two positive integers works like standard addition. Both numbers are positive, so the sum remains positive. Adding two negative integers works identically, except the sum is negative. Calculating the absolute value of each number helps isolate the size of the jump from the direction.

Adding integers with different signs

When adding a positive integer to a negative integer, the values oppose each other. A positive amount cancels out an equal negative amount, creating zero pairs.

To add integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the larger absolute value.

For example, to find the sum of $5$ and $-3$, find their absolute values ($5$ and $3$). Subtract the smaller from the larger to get $2$. Because $5$ has a greater absolute value and is positive, the final answer is $2$. This is a primary skill required when adding and subtracting positive and negative numbers.

Use a number line

A number line is an excellent tool for visualizing how values combine. It displays the starting point and clearly shows the direction of the change.

Follow these steps to add integers on a number line:

1. Mark the first integer on the number line.
2. If the second integer is positive, move that many units to the right.
3. If the second integer is negative, move that many units to the left.
4. The position where you stop is the sum.

Additive inverses

Every integer has a corresponding opposite integer located the same distance from zero on the number line.

The sum of an integer and its additive inverse is always zero.

For example, adding $-4$ and $4$ gives $0$. Knowing how to spot opposite numbers and additive inverses makes longer calculations much faster, because opposing values immediately cancel out.

Worked examples

Practicing with specific rules builds accuracy.

Example 1: Adding integers with the same sign

Question: What is the sum of $-14$ and $-23$?

Method:

1. Identify the absolute values of both integers.
2. Because both integers have a negative sign, add their absolute values together.
3. Attach the original negative sign to the result.

Answer: The sum is $-37$.

Check: On a number line, starting at $-14$ and moving $23$ units further left results in $-37$.

Example 2: Adding integers with different signs

Question: Find the sum: $35+(-19)$.

Method:

1. Find the absolute values of both numbers.
2. Since the signs are different, subtract the smaller absolute value from the larger absolute value.
3. Apply the sign of the integer with the larger absolute value.

Answer: The sum is $16$.

Check: Because positive $35$ has a greater magnitude than negative $19$, the answer must be positive.

Example 3: Adding three integers

Question: Evaluate $12+(-20)+(-5)$.

Method:

1. Add the numbers from left to right, or group numbers with the same sign first. Grouping negative integers often saves time.
2. Add $-20$ and $-5$ using the same-sign rule.
3. Add the resulting integer to the remaining positive integer.

Answer: The final result is $-13$.

Check: Grouping the negatives gives $-25$. Then, $12+(-25)$ is found by subtracting $12$ from $25$ and keeping the negative sign. The result $-13$ is correct.

Common mistakes

Applying the wrong mathematical rule is a frequent error when working with negative numbers. A common misconception is believing that two negative numbers always make a positive number. That rule applies only to multiplication and division. When adding two negative numbers, the sum is always negative.

Another common mistake is ignoring the sign of the larger absolute value when adding integers with different signs. Always verify which value is farther from zero before writing the final sign.

Frequently asked questions

What is the identity element for adding integers?

The identity element for addition is zero. Adding zero to any integer leaves the integer unchanged. For example, $-17+0=-17$.

How is subtracting integers related to adding integers?

Every addition problem can be written as a subtraction problem, and vice versa. Adding a negative integer is mathematically equivalent to the subtraction of integers. For example, $8+(-3)$ is identical to $8-3$.

Can the sum of two integers be smaller than both integers?

Yes. When adding two negative integers, the result is located further to the left on the number line. For example, $-4+(-5)=-9$. The sum $-9$ is smaller than both $-4$ and $-5$.