Understanding Integers

🌡️ Understanding Integers

Part 1 of 5 — Concept Introduction

Have you ever seen a temperature like $-10°$ on a freezing winter morning? Or heard that a submarine dove to $-200$ feet below sea level? Those numbers with a little "minus" in front are part of a special set of numbers called integers.

Integers are the whole numbers, their opposites, and zero:

$..., -3, -2, -1, 0, 1, 2, 3, ...$

The three little dots ($...$) mean the list keeps going forever in both directions!

The Three Kinds of Integers

💡 Important: Zero is special — it is neither positive nor negative. It is the dividing line between the two!

Notice that fractions and decimals like $\frac{1}{2}$ or $2.5$ are not integers. Integers are only whole numbers and their opposites.

The Number Line 📏

The easiest way to picture integers is on a number line:

$\longleftarrow \;\; -5 \;\; -4 \;\; -3 \;\; -2 \;\; -1 \;\; 0 \;\; 1 \;\; 2 \;\; 3 \;\; 4 \;\; 5 \;\; \longrightarrow$

The number line follows two simple rules:

• As you move right, the numbers get bigger (increase).
• As you move left, the numbers get smaller (decrease).

Opposites ↔️

Every integer has an opposite — the number that is the same distance from zero but on the other side.

• The opposite of $3$ is $-3$.
• The opposite of $-7$ is $7$.
• The opposite of $0$ is just $0$ itself.

Think of opposites as a mirror image, with zero acting as the mirror!

Why Integers Matter 🌍

Negative numbers help us describe things that go below a starting point. Here are some real places you see them:

• Temperature: $-10°$ means 10 degrees below zero (brrr! ❄️).
• Elevation: $-50$ feet means 50 feet below sea level.
• Money: $-20$, i.e. owing $20, means you owe $20 (a debt).

In each case, the positive version means the opposite direction: $+10°$ is above zero, $+50$ feet is above sea level, and $+20$, i.e. having $20, means you have $20.

So integers let us talk about both directions — up and down, above and below, having and owing — all with one set of numbers.

Concept Check 🎯

Let's make sure the basics are clear before moving on.

📝 Worked Examples

Part 2 of 5 — Absolute Value Step by Step

The absolute value of a number is its distance from zero on the number line. Distance is never negative, so absolute value is always positive (or zero).

We write absolute value with two straight bars: $|\;\;|$.

Example 1: Find $|-5|$

• Picture $-5$ on the number line. It sits 5 steps to the left of zero.
• Distance is just how far, not which direction.
• So $|-5| = 5$. ✅

Example 2: Find $|5|$

• The number $5$ sits 5 steps to the right of zero.
• That is also a distance of $5$.
• So $|5| = 5$. ✅

💡 Big idea: A number and its opposite have the same absolute value, because they are the same distance from zero. Both $-5$ and $5$ are $5$ units away.

Example 3: Find $|0|$

Zero is $0$ steps away from itself, so $|0| = 0$.

Putting It Together

Let's read this number line and answer a few quick questions.

$\longleftarrow \;\; -4 \;\; -3 \;\; -2 \;\; -1 \;\; 0 \;\; 1 \;\; 2 \;\; 3 \;\; 4 \;\; \longrightarrow$

• What is the opposite of $-3$? Go the same distance on the other side of zero → $3$.
• What is $|-4|$? The number $-4$ is $4$ units from zero → $4$.
• What is $|2|$? The number $2$ is $2$ units from zero → $2$.

Now it is your turn to try a few!

Your Turn 🧮

Use what you learned about opposites and absolute value. Type a single number for each answer.

1. What is $|-8|$?

2. What is the opposite of $6$? (Include the sign.)

3. What is $|7|$?

🧭 Guided Practice

Part 3 of 5 — Comparing Integers

Remember the golden rule: on a number line, the number farther to the right is greater, and the number farther to the left is less.

Fill In the Comparison 🔍

Choose the correct word to complete each sentence about the number line.

🌍 Application & Word Problems

Part 4 of 5 — Integers in the Real World

Integers show up all around you. The trick to word problems is deciding which direction is positive and which is negative.

Example 🏔️

A hiker starts at sea level ($0$ feet). She climbs to $+300$ feet, then a diver goes down to $-150$ feet in a nearby lake.

• The hiker's elevation $+300$ is above sea level.
• The diver's elevation $-150$ is below sea level.
• Since $+300$ is to the right of $-150$ on a number line, the hiker is higher: $300 > -150$.

Once you label the directions, comparing integers tells you who is higher, colder, or richer!

Word Problem Practice 🧮

On a freezing morning, a thermometer reads 8 degrees below zero. By noon it has warmed up, and a bank account shows a debt (money owed) of $15.

Type a single number for each answer, including a minus sign when needed.

1. Write the morning temperature as an integer.

2. Write the bank account debt of $15 as an integer.

3. What is the absolute value of that temperature, $|-8|$? (This tells how many degrees below zero it is.)

Word Problem Check 📋

🏆 Review & Challenge

Part 5 of 5 — Putting It All Together

You have learned a lot about integers! Here is a quick summary of the big ideas:

💡 Remember: When comparing two negative numbers, the one closer to zero is the greater number. For example, $-2 > -8$.

Now take on the challenge questions below to show what you know!

Challenge Questions 🎯

These mix together everything from this lesson. Take your time!