The Number Line: Integers and Rational Numbers

The Temperature Problem

It is January in Amarillo, Texas. The weather forecast says tonight's low will be -8°F. Tomorrow's high will be 15°F. Your friend in Houston texts you: "It's 62 degrees here. How much colder is it where you are?"

To answer that question — and hundreds of real questions like it — you need numbers that go below zero. You need a way to measure distance from zero. And you need a system for comparing numbers that sit on opposite sides of it.

Welcome to the rational number line. It is the foundation underneath every math topic you will study this year.

Numbers with Direction

You already know the counting numbers: 1, 2, 3, 4, 5... They go on forever in one direction. But the real world does not only go in one direction.

• A football team gains 7 yards, then loses 3 yards.
• A bank account has $200 deposited, then $50 withdrawn.
• An elevator goes up 4 floors, then down 6 floors.

Each of these situations has two directions. To handle that, we extend the number line to the left of zero. The numbers to the right of zero are positive. The numbers to the left are negative. Zero itself is neither positive nor negative — it is the dividing line between the two directions.

The integers are the set of all positive whole numbers, negative whole numbers, and zero:

... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...

Opposites: Same Distance, Different Direction

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

• The opposite of 5 is -5.
• The opposite of -12 is 12.
• The opposite of 0 is... 0. (Zero is its own opposite.)

On the number line, opposites are mirror images across zero. If you fold the number line at zero, 5 and -5 would land on top of each other.

Here is a useful fact: adding a number and its opposite always gives you zero. Try it: 5 + (-5) = 0. This will matter a lot when we start solving equations later this year.

Absolute Value: How Far from Zero?

Sometimes you do not care about direction — you just want to know distance. How far is -8 from zero? How far is 8 from zero? Both are 8 units away.

The absolute value of a number is its distance from zero, regardless of direction. We write it with vertical bars:

|5| = 5

|-5| = 5

|0| = 0

|-137| = 137

Absolute value is always zero or positive. You cannot have a negative distance — distance is just "how far," not "which way."

Back to Amarillo: The temperature is -8°F. How cold is that compared to zero? |−8| = 8. It is 8 degrees below zero. The absolute value tells you the intensity without worrying about the direction.

❓

Concept Check

A submarine is at a depth of -450 feet (below sea level). A drone is flying at an altitude of 380 feet (above sea level). Which one is farther from sea level?

▸

The submarine is farther from sea level. |-450| = 450 feet from sea level, while |380| = 380 feet from sea level. Even though the submarine's position is written with a negative number, its distance from sea level is greater.

Beyond Integers: Rational Numbers

Integers are whole numbers — they land exactly on the tick marks of the number line. But what about the spaces between them?

You already know about fractions and decimals: 1/2, 0.75, -2.3, 3/4. These numbers fill in the gaps between integers. Together with the integers, they form the rational numbers — any number that can be written as a fraction a/b, where a and b are integers and b is not zero.

Every integer is also a rational number (just put it over 1: 5 = 5/1). Every terminating decimal is rational (0.75 = 3/4). Every repeating decimal is rational (0.333... = 1/3).

Here is a list of rational numbers, from negative to positive:

-2.5, -3/2, -1, -1/4, 0, 0.6, 7/8, 1, 1.75, 2

Plotting Rational Numbers on the Number Line

To place a rational number on the number line:

1. Identify which two integers it falls between.
2. Divide that segment into equal parts based on the denominator.
3. Count from the lower integer.

Example: Where does -3/4 go?

It is between -1 and 0. Divide that segment into 4 equal parts. Starting from -1, count 1 part to the right. That puts you at -3/4. (Or from 0, count 3 parts to the left.)

Example: Where does 1.6 go?

It is between 1 and 2. Since 0.6 = 6/10, it is 6/10 of the way from 1 to 2.

Comparing and Ordering Rational Numbers

Here is the single most important rule for comparing numbers on the number line:

Numbers to the right are greater. Numbers to the left are less.

This sounds obvious until negative numbers get involved:

• Is -3 greater than or less than -1?
• Is -0.5 greater than or less than -2?

-3 < -1 because -3 is farther to the left. Think of it as temperature: -3°F is colder (less) than -1°F.

-0.5 > -2 because -0.5 is farther to the right. -0.5 is closer to zero — it is "less negative."

Comparing Fractions and Decimals

When fractions have different denominators, convert them to the same denominator or to decimals:

Compare -2/3 and -5/8.

Convert to decimals: -2/3 ≈ -0.667 and -5/8 = -0.625.

Since -0.625 is to the right of -0.667, we have -5/8 > -2/3.

Or find a common denominator: -2/3 = -16/24 and -5/8 = -15/24. Since -15/24 > -16/24, again -5/8 > -2/3.

Ordering from Least to Greatest

Put these in order from least to greatest: 1/2, -3, 0, -1.5, 2, -2/3

Step 1: Convert to comparable form.

• 1/2 = 0.5
• -3 = -3.0
• 0 = 0
• -1.5 = -1.5
• 2 = 2.0
• -2/3 ≈ -0.667

Step 2: Order from left to right on the number line.

-3, -1.5, -2/3, 0, 1/2, 2

Or with inequality symbols: -3 < -1.5 < -2/3 < 0 < 1/2 < 2

❓

Concept Check

Order these rational numbers from least to greatest: 3/4, -1, -1/2, 0.8, -0.75, 0

▸

Convert to decimals: 3/4 = 0.75, -1 = -1.0, -1/2 = -0.5, 0.8 = 0.8, -0.75 = -0.75, 0 = 0. From least to greatest: -1, -0.75, -1/2, 0, 3/4, 0.8. Written with original forms: -1, -0.75, -1/2, 0, 3/4, 0.8.

Real-World Connections

Rational numbers are not an abstract concept that lives only in math class. They show up everywhere:

• Banking: Your account balance can be positive (you have money) or negative (you owe money — overdraft).
• Elevation: The Dead Sea is about -1,412 feet (below sea level). Mount Everest is 29,032 feet.
• Sports stats: A golfer's score of -5 (five under par) is better than +2 (two over par).
• Cooking: A recipe calls for 3/4 cup of flour — that is a rational number between 0 and 1.

The number line is not just a classroom tool. It is a map of how quantities work in the real world — with direction, distance, and everything in between.

Looking Ahead

Now that you know what rational numbers are and how they sit on the number line, the next step is learning how to do things with them — add, subtract, multiply, and divide. In Unit 2, you will discover why a negative times a negative is positive, and why subtracting a negative is the same as adding. The number line you built here is the foundation for all of it.

❓

Concept Check

What is the absolute value of -17.5? Explain what that number represents in real-world terms if -17.5 represents a temperature in degrees Fahrenheit.

▸

|-17.5| = 17.5. In real-world terms, this means the temperature is 17.5 degrees below zero Fahrenheit. The absolute value tells you how far below zero the temperature is, without the direction — just the intensity of the cold.