Integer¶

Definition¶

The integers are the positive and negative whole numbers together with zero:

Every natural number is an integer, so $\mathbb{N}\subseteq\mathbb{Z}\text{.}$ Math I: Calculus takes $\mathbb{Z}$ and its usual arithmetic and order as given.

Interpretation¶

Integers extend the natural numbers by introducing additive inverses $-1,-2,-3,\ldots$. They make subtraction unrestricted: for example, the equation $x+5=3$ has no solution in $\mathbb{N}$ but has the integer solution $x=-2$$-2$.

On the number line the integers are equally spaced points, one unit apart, continuing without end in both directions. The bracket marks $\mathbb{N}=\{0,1,2,\dots\}$$\{0,1,2,\dots \}$, the integers from $0$ upwards; the points to the left of $0$ are the negatives that $\mathbb{Z}$ adds. There is no integer strictly between two neighbouring integers.

Division remains restricted, since $1\div2$ is not an integer; removing this limitation is the step to the rational numbers. A common slip concerns the convention for $\mathbb{N}$: here $0\in\mathbb{N}$$\mathbb{N}$, so the positive integers are $\mathbb{N}^{+}=\{1,2,3,\ldots\}$$\{1,2,3,\dots \}$, not all of $\mathbb{N}$.

Key Results¶

• Every integer is exactly one of the following: zero, a positive natural number, or the negation of a positive natural number.
• The set $\mathbb{Z}$ is closed under addition, subtraction, and multiplication, but not under division.
• Every integer $x$ has a unique additive inverse $-x$ satisfying $x+(-x)=0\text{.}$
• For integers $x$ and $y\text{,}$ we have $x<y$ exactly when $y=x+n$ for some positive natural number $n\text{.}$

Examples¶

• $-7,\,0,$ and $23$ are integers.
• $1/2$ is not an integer.
• The difference $3-5=-2$ is an integer.

Prerequisites¶

• Natural Number

Related Concepts¶

• Subtraction and Difference
• Addition and Sum

Sources¶

• Math I: Calculus · Lecture 2

Symbols: element of, integers, natural numbers, subset

Last updated: 20 October 2026 03:02