What Are Integers?
Positive, negative, and zero — the complete number line that extends in both directions forever.
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.
The concept of negative numbers was controversial for centuries. Ancient Greek mathematicians rejected them as absurd — how could you have less than nothing? Indian mathematicians were the first to use negatives systematically, around 600 CE. European mathematicians didn't fully accept negative numbers until the 1600s. The great French mathematician Blaise Pascal once declared that subtracting 4 from zero was "utter nonsense." Today, negative numbers are so routine that we check them on our phone's weather app every morning.
Last reviewed: April 2026