Integers — Definition, Examples & Properties

Integers

All positive and negative whole numbers (including zero). That is, the set {... , –3, –2, –1, 0, 1, 2, 3, ...}. Integers are indicated by either or J.

Nested diagram showing number sets: natural numbers ⊂ whole numbers ⊂ integers ℤ ⊂ rationals ℚ ⊂ algebraic ⊂ reals ℝ, plus...

Key Formula

\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}

Where:

$\mathbb{Z}$ = The standard symbol for the set of all integers, from the German word 'Zahlen' meaning 'numbers'
$0$ = Zero, which is an integer that is neither positive nor negative
$\ldots$ = Indicates the set continues infinitely in both the positive and negative directions

Worked Example

Problem: Determine which of the following numbers are integers:

-7, \; 3.5, \; 0, \; \frac{2}{3}, \; 12, \; -100

Step 1: Recall the definition: integers are whole numbers that can be positive, negative, or zero. They have no fractional or decimal part.

\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}

Step 2: Check each number.

-7

is a negative whole number, so it is an integer.

3.5

has a decimal part, so it is NOT an integer.

-7 \in \mathbb{Z}, \quad 3.5 \notin \mathbb{Z}

Step 3:

0

is included in the set of integers by definition.

\frac{2}{3}

is a fraction that does not simplify to a whole number, so it is NOT an integer.

0 \in \mathbb{Z}, \quad \tfrac{2}{3} \notin \mathbb{Z}

Step 4:

12

is a positive whole number, so it is an integer.

-100

is a negative whole number, so it is also an integer.

12 \in \mathbb{Z}, \quad -100 \in \mathbb{Z}

Answer: The integers in the list are

-7

0

12

, and

-100

. The non-integers are

3.5

and

\frac{2}{3}

Another Example

This example focuses on closure properties — which operations on integers always produce integers — rather than simply identifying integers from a list.

Problem: Perform the following operations with integers and determine whether each result is also an integer: (a)

-8 + 5

, (b)

-3 \times 4

, (c)

7 \div 2

Step 1: Compute

-8 + 5

. Adding a positive number to a negative number: since

8 > 5

, the result is negative with magnitude

8 - 5 = 3

-8 + 5 = -3

Step 2:

-3

is a whole number, so the result is an integer. This illustrates that integers are closed under addition — adding any two integers always produces an integer.

-3 \in \mathbb{Z} \quad \checkmark

Step 3: Compute

-3 \times 4

. A negative times a positive gives a negative result.

-3 \times 4 = -12

Step 4:

-12

is an integer. Integers are also closed under multiplication.

-12 \in \mathbb{Z} \quad \checkmark

Step 5: Compute

7 \div 2 = 3.5

. This is not a whole number, so the result is NOT an integer. Integers are not closed under division.

7 \div 2 = 3.5 \notin \mathbb{Z}

Answer: (a)

-3

is an integer. (b)

-12

is an integer. (c)

3.5

is not an integer. Integers are closed under addition, subtraction, and multiplication, but not under division.

Frequently Asked Questions

Is zero an integer?

Yes, zero is an integer. The set of integers includes all positive whole numbers, all negative whole numbers, and zero. Zero is the integer that separates the positive integers from the negative integers on the number line.

What is the difference between integers and whole numbers?

Whole numbers are the non-negative counting numbers including zero:

\{0, 1, 2, 3, \ldots\}

. Integers include all whole numbers plus their negative counterparts:

\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}

. Every whole number is an integer, but not every integer is a whole number — for instance,

-5

is an integer but not a whole number.

Are fractions and decimals integers?

A fraction or decimal is an integer only if it simplifies to a whole number. For example,

\frac{6}{3} = 2

is an integer, and

4.0

is an integer because it equals

4

. However,

\frac{1}{2}

and

3.7

are not integers because they have non-zero fractional parts.

Integers vs. Rational Numbers

	Integers	Rational Numbers
Definition	All positive and negative whole numbers and zero: $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$	All numbers that can be written as a fraction $\frac{a}{b}$ where $a, b$ are integers and $b \neq 0$
Symbol	$\mathbb{Z}$	$\mathbb{Q}$
Examples	$-5, \; 0, \; 42$	$\frac{1}{3}, \; -2.75, \; 7$
Includes fractions?	No	Yes
Relationship	Every integer is a rational number (e.g., $3 = \frac{3}{1}$ )	Not every rational number is an integer (e.g., $\frac{1}{2}$ )
Closed under division?	No ( $7 \div 2 = 3.5 \notin \mathbb{Z}$ )	Yes, except division by zero

Why It Matters

Integers appear throughout algebra whenever you work with signed numbers, solve equations like

x + 5 = 2

(where

x = -3

), or plot points on a coordinate plane. Understanding the number hierarchy — naturals

\subset

wholes

\subset

integers

\subset

rationals

\subset

reals — is essential for knowing what kinds of solutions an equation can have. In real life, integers model quantities like temperature below zero, floors of a building (including basement levels), and financial debts.

Common Mistakes

Mistake: Thinking that negative numbers cannot be integers.

Correction: The set of integers explicitly includes all negative whole numbers. Numbers like

-1

-50

, and

-1000

are all integers.

Mistake: Assuming that any result of dividing two integers is also an integer.

Correction: Integers are not closed under division. For example,

5 \div 3 = 1.\overline{6}

, which is a rational number but not an integer. Only when the divisor divides the dividend evenly is the quotient an integer.

Related Terms

Natural Numbers — Positive integers (sometimes including zero)
Whole Numbers — Non-negative integers: $\{0, 1, 2, \ldots\}$
Positive Number — Integers greater than zero
Negative Number — Integers less than zero
Rational Numbers — Superset of integers including fractions
Real Numbers — Superset containing all integers and irrationals
Algebraic Numbers — Larger set; every integer is algebraic
Gaussian Integer — Complex numbers with integer real and imaginary parts