Lesson 59: Adding Integers on the Number Line

New Concept

Welcome to Course 1! This course expands our number system beyond whole numbers to include integers (positive and negative numbers) and the rules of algebra.

What’s next

We begin this journey by exploring how integers work. Next, you'll get a visual breakdown of how to add them using a number line.

Property

Integers include all the whole numbers and also the opposites of the positive integers (their negatives).

Examples

• The numbers $-10$, $0$, and $42$ are all proud members of the integer family.
• Numbers like $3\frac{1}{2}$ and $-1.5$ are not integers because they are not whole numbers.
• A temperature of $-5$ degrees, a depth of $-100$ feet, and a profit of $500$ dollars are all described with integers.

Explanation

Think of integers as all the whole-number steps you can take on a number line, both forward (positive) and backward (negative), including standing still at zero. No fractions or decimals are allowed in this exclusive club! It’s the set of whole numbers and their grumpy, negative twins, plus the very neutral zero who gets along with everyone.

Property

The absolute value of a number is its distance from zero on a number line.

Examples

• The distance of $-8$ from zero is 8, so $|-8| = 8$.
• A submarine at $-200$ feet is the same distance from sea level as a helicopter at $+200$ feet, so $|-200| = |200| = 200$.
• Always simplify inside the bars first: $|5 - 12| = |-7| = 7$.

Explanation

Absolute value is like asking, “How many jumps from zero are you?” It doesn’t care about direction (left or right), only distance! That's why the absolute value is always positive or zero—it represents pure, unfiltered distance. So, the absolute value of a number is its happy, positive twin, no matter how negative it started out.

Property

The sum of two opposites is always zero.

Examples

• Taking 8 steps to the right and 8 steps to the left gets you back to the start: $(+8) + (-8) = 0$.
• If you borrow 20 dollars ($-20$) and then earn 20 dollars ($+20$) to repay the debt, your balance is zero: $(-20) + (+20) = 0$.
• On a number line, an arrow from $0$ to $+15$ and an arrow from $+15$ back to $0$ show that $(+15) + (-15) = 0$.

Explanation

Adding two opposite numbers is like taking five steps forward and then taking five steps backward—you end up exactly where you started: at zero! They perfectly cancel each other out, like a superhero and their arch-nemesis in a final battle where both disappear in a flash. Poof! Nothing is left but a perfect, balanced zero.