What Are Integers?

Numbers Below Zero

For most of elementary school, numbers start at zero and go up: 0, 1, 2, 3... But what about temperatures below zero? What about owing money? What about elevations below sea level? The real world needs numbers that go in both directions. Integers are the complete set of whole numbers: positive numbers (1, 2, 3...), negative numbers (−1, −2, −3...), and zero. Together, they fill the entire number line from left to right, forever in both directions.

The Number Line

On a number line, zero sits in the center. Positive numbers extend to the right, getting larger. Negative numbers extend to the left, getting smaller. An important concept: −5 is less than −2, even though 5 is bigger than 2 as a digit. Think of it as temperature: −5°C is colder (less) than −2°C. The further left you go on the number line, the smaller the number.

Adding and Subtracting Integers

Adding a positive number moves you right on the number line. Adding a negative number moves you left. So 3 + (−5) means start at 3 and move 5 to the left, landing on −2. Subtracting a negative is the same as adding a positive: 4 − (−3) = 4 + 3 = 7. The phrase "subtracting a negative makes a positive" confuses many students, but on the number line it makes visual sense — reversing a leftward move sends you right.

Integers in Real Life

Integers are everywhere: bank accounts (deposits are positive, withdrawals are negative), elevators (floors above and below ground), golf scores (under par is negative), football (yards gained and lost), and sea level (altitude above and depth below). Understanding integers prepares you for algebra, where variables can represent any integer.

Why This Matters

Integers extend the number line below zero, opening up a whole new world of mathematics. Temperature drops below freezing, a football team loses yards on a play, a bank account goes into overdraft — these everyday situations all require negative numbers. Without integers, children can only describe half of reality. Learning integers prepares students for algebra, where variables can take any value, and for coordinate geometry, where points can be in any quadrant.

Understanding integers also builds abstract thinking. Negative numbers don't represent physical objects you can hold — you can't show someone "negative three apples." Grasping this abstraction is a major cognitive leap that prepares children for higher-level mathematical reasoning.

Where Kids Get Stuck

The most persistent confusion is that students think "bigger number = bigger value" even with negatives. They see −8 and −3 and assume −8 is greater because 8 is bigger than 3. Using a vertical number line (like a thermometer) where "up is warmer" makes it visually clear that −3 is above −8 and therefore greater.

Subtracting a negative number also baffles many students. "5 − (−3) = 8" feels wrong — how can subtracting make a number bigger? The phrase "subtracting a negative is the same as adding" sounds like a rule to memorize, but modeling it on a number line (removing a leftward arrow means you go right) makes the logic visible.

Another stumbling block is multiplication rules for signs. Students can usually accept that positive × negative = negative, but negative × negative = positive feels arbitrary. Patterns help: 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0, so (−1) × (−2) must equal 2 to continue the pattern.

Try This at Home

• Thermometer math — Look up temperatures for different cities (including cold places). Order them from coldest to warmest. Ask: how many degrees warmer is City A than City B?
• Elevator game — Imagine a building with floors from −3 (underground parking) to 10. Start on any floor, then add or subtract to move up or down.
• Football yardage — Track a football team's plays: gained 7, lost 3, gained 12, lost 5. What's the total yardage? Use positive and negative numbers to calculate.
• Integer card game — Use black cards as positive and red cards as negative. Draw two cards, add them together. Highest (or lowest) sum wins the round.

For more ideas, see our guide: Helping Kids With Word Problems.

Last reviewed: May 2026