Introduction to Negative Numbers

Introduction

Imagine that the temperature is 3 degrees below zero, an elevator is on a basement floor, or a diver is below sea level. In each case, ordinary counting numbers are not enough. You need negative numbers.

A negative number is a number less than zero. It is written with a minus sign, such as −1, −4, or −20. Negative numbers help you describe positions, temperatures, amounts, and changes that are below a chosen zero point.

In this aiMOOC, you will learn how to:

Recognize negative numbers: Tell positive numbers, negative numbers, and zero apart.
Use a number line: Locate, compare, and order integers.
Find opposites: Connect numbers such as 5 and −5.
Think about distance from zero: Explain why −7 and 7 are equally far from zero.
Use integers in real life: Model temperature, floors, sea level, money, and simple changes.

What Are Negative Numbers?

Numbers greater than zero are positive. Numbers less than zero are negative. Zero sits between the positive and negative numbers and is neither positive nor negative in ordinary school arithmetic.

The numbers ..., −3, −2, −1, 0, 1, 2, 3, ... are called integers. Integers include zero, positive counting numbers, and their negative opposites.

The symbol "−" can do two jobs. In −5, it is a negative sign and tells you that the number is below zero. In 8 − 5, it is a subtraction sign and tells you to subtract.

The Number Line

A number line shows numbers in order. Zero is a useful reference point. Positive numbers are to the right of zero, and negative numbers are to the left.

The farther right a number is, the greater it is. The farther left a number is, the smaller it is. For example, −2 is greater than −6 because −2 is farther to the right.

Try tracing a number line with your finger. Start at 0. Move right to reach 1, 2, and 3. Move left to reach −1, −2, and −3. This picture helps you understand negative numbers as positions, not just as symbols.

Comparing Negative Numbers

Comparing negative numbers can feel surprising at first. On the positive side, 8 is greater than 3. On the negative side, −3 is greater than −8 because −3 is closer to zero and farther to the right.

You can use these comparison signs:

Greater than: −2 > −5 because −2 is farther right.
Less than: −9 < −4 because −9 is farther left.
Equal to: −6 = −6 because both symbols name the same number.

A useful rule is: look at position, not just at the digits. When comparing two numbers on a number line, the number farther to the right is always greater.

Opposites and Absolute Value

Two numbers are opposites if they are the same distance from zero but on different sides. For example, 4 and −4 are opposites. The opposite of −9 is 9. The opposite of 0 is 0.

Absolute value tells you a number's distance from zero, without using direction. The absolute value of −6 is 6, and the absolute value of 6 is also 6.

You may see absolute value written with vertical bars:

|−6| = 6
|6| = 6
|0| = 0

Distance cannot be negative, so an absolute value is never less than zero.

Negative Numbers in Everyday Life

Negative numbers are useful whenever a quantity can go below a reference point.

Temperature

On a Celsius thermometer, temperatures below 0 °C are written with negative numbers. A temperature of −5 °C is colder than −2 °C because −5 is lower on the temperature scale.

If the temperature rises from −4 °C to 1 °C, it has risen 5 degrees. You can count the steps on a number line: −4, −3, −2, −1, 0, 1.

Floors in a Building

Some buildings use negative floor numbers for levels below the ground floor. If ground level is 0, then floor −1 is one level below ground and floor −3 is three levels below ground.

If an elevator is on floor 2 and moves down four floors, it reaches floor −2. A vertical number line can model this movement.

Sea Level and Depth

Sea level can be used as a zero point. A place above sea level can be represented with a positive value, while a place below sea level can be represented with a negative value.

For a simple model, a diver 6 metres below sea level can be described as being at −6 m, while a lookout point 6 metres above sea level can be described as +6 m.

Money and Scores

Negative numbers can also describe amounts below a starting balance or a score below a reference value. For example, if a game gives you a 3-point penalty, the change can be written as −3 points.

When using money examples, be precise about what zero means. A balance of −5 units can represent owing 5 units, while a balance of +5 units can represent having 5 units available.

Moving on the Number Line

A number line can help you understand simple addition and subtraction with integers.

To add a positive number, move to the right. For example, −2 + 3 means start at −2 and move 3 steps right. You land on 1.

To add a negative number, move to the left. For example, 2 + (−1) means start at 2 and move 1 step left. You land on 1.

Here is another example: −2 + (−3). Start at −2 and move 3 steps left. You land on −5.

For Grades 5–6, focus on understanding the direction and meaning of each move. A clear number-line model is more useful than memorizing rules without understanding them.

A Second Number-Line Example

Watch how negative numbers can be located and reasoned about on a number line.

After watching, draw your own number line from −10 to 10. Mark −7, −2, 0, 4, and 9. Then explain which marked number is greatest and which is smallest.

Common Mistakes to Avoid

Mistake 1: Thinking that −8 is greater than −3 because 8 is greater than 3. On a number line, −8 is farther left, so −8 < −3.

Mistake 2: Calling zero a positive number. In ordinary school arithmetic, zero is neither positive nor negative.

Mistake 3: Confusing a negative sign with subtraction. In −6, the sign is part of the number. In 10 − 6, the same symbol tells you to subtract.

Mistake 4: Saying that absolute value means "make the number positive." It is better to think of absolute value as distance from zero. This idea also explains why |0| = 0.

Key Ideas to Remember

Negative numbers: They are less than zero.
Integers: They include negative whole-number values, zero, and positive whole-number values.
Number line: Numbers farther right are greater; numbers farther left are smaller.
Opposites: They are the same distance from zero on opposite sides.
Absolute value: It is the distance from zero.
Real-life models: Temperature, floors, sea level, and balances can all use negative numbers.

Interactive Tasks

Quiz: Test Your Knowledge

Which number is negative? (Negative four) (!Zero) (!Three) (!Eight)

Which number is smallest? (Negative five) (!Negative two) (!Zero) (!Four)

What is the opposite of 7? (Negative seven) (!Seven) (!Zero) (!Fourteen)

Which temperature is coldest? (Negative six degrees Celsius) (!Negative two degrees Celsius) (!Zero degrees Celsius) (!Four degrees Celsius)

Which statement about zero is correct? (Zero is neither positive nor negative) (!Zero is always positive) (!Zero is always negative) (!Zero is less than every negative number)

What is the distance from −9 to zero? (9 units) (!Negative 9 units) (!Zero units) (!18 units)

An elevator starts on floor 1 and moves down 3 floors. Where does it stop? (Floor negative two) (!Floor negative three) (!Floor two) (!Floor four)

Which integer can represent 3 metres below sea level? (Negative three) (!Three) (!Zero) (!Thirty)

Which number is greatest? (Five) (!Two) (!Negative one) (!Negative six)

What is −5 plus 3? (Negative two) (!Negative eight) (!Two) (!Eight)

Memory Game

Negative number	A value less than zero
Positive number	A value greater than zero
Zero	The reference point between positive and negative values
Opposite	A value the same distance from zero on the other side
Absolute value	A number's distance from zero
Number line	A line that shows numbers in order

Drag and Drop

Match the correct terms.	Topic
Below zero	Negative temperature
Basement level	Negative floor
Farther right	Greater number
Same distance from zero	Opposite numbers
Distance from zero	Absolute value

...

Crossword Puzzle

Integer	What kind of number can be negative, positive, or zero without a fractional part?
Negative	What word describes a number less than zero?
Opposite	What do you call a number the same distance from zero on the other side?
Distance	What does absolute value measure from zero?
Thermometer	What instrument can show temperatures below zero?
Elevator	What can travel to floors below ground level?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Human Number Line: Make a number line from −10 to 10 on the floor with paper labels. Stand on five different integers and say whether each one is positive, negative, or zero.
Weather Watch: Record five temperatures from a weather report or a teacher-provided data set, including at least one value below zero, and arrange them from coldest to warmest.
Integer Card Set: Create 12 cards with positive numbers, negative numbers, and zero. Mix the cards, sort them, and explain your sorting rule to a partner.
Opposite Pairs Poster: Draw a poster showing at least five pairs of opposite numbers and mark how far each pair is from zero.

Standard

Elevator Story: Write a short story about an elevator that starts above ground, travels below ground, and returns upward. Show every stop with an integer.
Sea-Level Diagram: Draw a picture with a mountain, sea level, and a diver. Label at least six heights or depths with integers and explain what zero represents.
Distance Challenge: Create six absolute-value questions using a number line. Trade them with a classmate, solve each other's questions, and check the explanations.
Negative Numbers Interview: Interview a family member, teacher, coach, or worker about where they see values below zero in daily life. Summarize at least three examples and explain the meaning of zero in each one.

Advanced

Temperature Data Project: Collect or use a teacher-provided week of temperatures that cross zero. Make a table and a number-line graph, then explain the greatest rise and greatest fall.
Number-Line Video: Produce a one-minute stop-motion or live-action video that demonstrates how to compare two negative integers using movement on a number line.
Freezer Observation: With adult or teacher supervision, observe temperature readings from a safe thermometer near a freezer or outdoor cold area. Record changes over time and explain any readings below zero.
Integer Board Game: Design a board game in which players move right for positive changes and left for negative changes. Include rules, at least 20 move cards, and a written explanation of how the game models integers.

Learning Assessment

Compare and Explain: Put −7, −2, 0, 4, and −5 in order from least to greatest, then explain your order using positions on a number line.
Choose a Model: A submarine is 8 metres below sea level and a helicopter is 30 metres above sea level. Represent both positions with signed numbers and explain what zero means.
Find the Error: A learner says, "−9 is greater than −4 because 9 is greater than 4." Explain the mistake and correct the comparison.
Distance Reasoning: Give two different integers with absolute value 6 and explain why both answers work.
Multi-Step Elevator Problem: An elevator starts on floor −2, moves up 5 floors, then down 4 floors. Find the final floor and show the movement on a number line.
Create Your Own Context: Invent a real-life situation where −3, 0, and +3 all make sense. Define the zero point and explain what the signs mean.

Evidence of Learning

Strong evidence of learning includes both correct answers and clear explanations. By the end of this aiMOOC, you should be able to demonstrate:

Knowledge: You can define negative numbers, integers, zero, opposites, and absolute value in your own words.
Number-line skill: You can place integers correctly and use left and right positions to compare them.
Reasoning: You can explain why −2 is greater than −8 without relying only on a memorized rule.
Representation: You can model below-zero temperatures, underground floors, and below-sea-level positions with signed numbers.
Products: You can create a labeled number line, diagram, data display, poster, story, or game that uses negative numbers correctly.
Communication: You can describe what zero means in a real-life context and explain the meaning of positive and negative signs.
Transfer: You can recognize a new situation in which values can lie on opposite sides of a reference point and choose suitable integers to represent it.

OERs on the Topic

Linked Learning Areas

Negative numbers connect arithmetic with measurement, data, science, geography, money, and later work with coordinates and algebra. Use the links below to continue learning.

Negative Numbers

aiMOOC Projects