Textbook
The integers are $\mathbb Z=\{\ldots,-2,-1,0,1,2,\ldots\}$. They include natural numbers, zero, and negative whole numbers. An integer has no nonzero fractional part. Writing $a\in\mathbb Z$ restricts permitted values to this set.
Adding, subtracting, and multiplying integers produces integers. Division need not preserve this property: $3/2$ is not in $\mathbb Z$. Every integer $n$ can be written as $n/1$, so the integers form a subset of the rational numbers. The reverse inclusion does not hold.
Every integer $n$ has an opposite integer $-n$, and their sum is zero. No integer lies strictly between consecutive integers $n$ and $n+1$. This concerns the chosen measurement unit: a quantity represented by an integer in one unit can become fractional in another. Membership in $\mathbb Z$ is checked for the number in the stated model, not inferred from the quantity’s physical name.
Definitions and further reading: Mathematics for Computer Science.
In plain language
Integers describe whole steps in either direction from zero. In an illustrative stock balance, two units followed by three issued units give −1: the sign records a debt, not a negative apple on the shelf. This model has no fractional remainder, so sharing three objects between two people requires another unit or another kind of number. The warehouse keeper need not search for a negative crate; a balance records an obligation rather than a literal inventory.
Example
In an illustrative balance model, subtracting 3 from an initial 2 gives −1. The final step shows the model’s boundary: dividing 3 by 2 leaves the integers.
Integers: Counting on Both Sides of Zero
Starting balance
Two is an integer and positive.
Step 1 of 3: Starting balance
Read all steps
Starting balance
Two is an integer and positive.
- State
- 2current
Subtraction
The balance passes zero and becomes −1, still an integer.
- State
- 2current−3current−1current
Division
One and a half is not an integer.
- State
- 3÷23/2excluded
The subtraction consists of three unit steps: $2\to1\to0\to-1$. Every transition stays within the integers, so the negative result respects the balance model. Division asks a different question: the sought number $q$ must satisfy $2q=3$, which has no integer solution. Rounding to 1 would give another result and leave a remainder; it would not satisfy the original exact equation.
Related concepts: Natural numbers, Rational numbers.