Lesson 1.3: Add and Subtract Integers

New Concept

Welcome to the world of integers! You'll master adding and subtracting positive and negative numbers by using key concepts like opposites and absolute value to correctly handle their signs in every calculation.

What’s next

Get ready to apply these rules. You'll work through interactive examples and a series of practice cards to build your skills with integer operations.

Property

Negative numbers are numbers less than 0. The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero. The notation $-a$ is read as “the opposite of $a$.” The whole numbers and their opposites are called the integers. The integers are the numbers $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$.

Examples

• The opposite of $15$ is $-15$, as both are 15 units from zero.
• The opposite of $-9$ is $9$. This can be written as $-(-9) = 9$.
• If $y = -25$, then $-y$ means the opposite of $-25$, which is $-(-25) = 25$.

Explanation

Integers expand our number system to include negative values, which are like mirror images of positive numbers across zero. The 'opposite' of a number is simply its reflection on the other side of the number line.

Property

The absolute value of a number is its distance from $0$ on the number line. The absolute value of a number $n$ is written as $|n|$.

Property of Absolute Value
$|n| \geq 0$ for all numbers. Absolute values are always greater than or equal to zero!

Examples

• The absolute value of $-18$ is its distance from 0, so $|-18| = 18$.
• To simplify $30 - |12 - 4(5)|$, we calculate inside the bars first: $30 - |12 - 20| = 30 - |-8| = 30 - 8 = 22$.
• Compare $|-11|$ and $-|-11|$. We get $11$ and $-11$. So, $|-11| > -|-11|$.

Property

To add integers with the same sign, add their absolute values and keep the common sign.

To add integers with different signs, subtract the smaller absolute value from the larger absolute value. The sum has the sign of the number with the larger absolute value.

Examples

• To add $-15 + (-8)$, the signs are the same. Add $15 + 8 = 23$ and keep the negative sign, so the sum is $-23$.
• To add $-10 + 25$, the signs are different. Subtract $25 - 10 = 15$. Since $25$ has the larger absolute value, the result is positive: $15$.
• To add $12 + (-30)$, the signs are different. Subtract $30 - 12 = 18$. Since $-30$ has the larger absolute value, the result is negative: $-18$.

Property

Subtraction Property
$a - b = a + (-b)$

Subtracting a number is the same as adding its opposite.

Examples

• To solve $10 - 18$, change it to adding the opposite: $10 + (-18)$. The result is $-8$.
• To solve $-5 - (-12)$, change it to adding the opposite: $-5 + 12$. The result is $7$.
• Simplify $20 - (-5 - 2) - 10$. First solve inside the parentheses: $20 - (-7) - 10$. Then change to addition: $20 + 7 + (-10) = 17$.