If you’ve ever counted apples or checked a thermometer below freezing, you’ve already used integers without realizing it. These whole numbers — positive, negative, and zero — form the foundation of arithmetic and are surprisingly easy to identify once you know what to look for, so here’s what an integer really is with clear examples and a simple identification method.

Definition: Whole number, can be positive, negative, or zero ·
Excludes: Fractions and decimals ·
Set symbol: Z ·
Zero: Is an integer ·
Count: Infinite

What Is an Integer?

Five essential facts about integers, one pattern: they are all about wholeness — no fractions, no decimals.

What is an integer in maths?

Definition of integer

• An integer is a whole number — it can be positive (like 5), negative (like -3), or zero.
• Integers do not include fractions or decimals. (Mathnasium (math education center))
• The set of all integers is denoted by Z, from the German word Zahlen meaning “numbers”.

What is an integer for kids?

An integer is a whole number that can be positive, negative, or zero. It’s like counting your toys — you can have 5 toys (positive), or owe someone 3 toys (-3), or have none at all (0). There are no parts of toys, just whole numbers.

This formal definition means that numbers such as 0, 1, -100, and 2024 all qualify, while 0.5, -3.2, and 7/8 do not. The implication: integers are the simplest complete number system you can count with — no halves, no pieces.

Examples of integers and non-integers

• Integers: 1, 10, -3, 0, 2024 (Study.com (educational resource))
• Non-integers: 0.5, -2.7, π, √2 (DreamBox Learning (math platform))
• A number like 3 1/2 is not an integer because it includes a fraction. (Study.com)

The pattern: any number that can be written without a fractional or decimal component is an integer. Numbers that require a fraction or decimal — even simple ones like 1.5 — fall outside the set.

The number line and integers

• Integers are commonly introduced using a number line with zero in the middle, positive numbers to the right, and negative numbers to the left. (Study.com)
• The integer set is countably infinite — there is no largest or smallest integer.

The catch: while integers stretch infinitely in both directions, every integer is a specific, fixed point on the number line — there are no gaps between them for fractions.

The concept of integers forms the bedrock of arithmetic, setting the stage for more complex number systems.

Is zero an integer?

Zero’s place in the integer set

• Zero is an integer. (DreamBox Learning (math platform))
• Zero is neither positive nor negative — it is the neutral element. (Mathnasium (math education))
• It is included in the definition of integers: “the number zero (0), a positive natural number, or the negation of a positive natural number.”

This may seem trivial, but zero was historically controversial — ancient number systems often lacked it. Today, zero is recognised as the additive identity: add zero to any integer and the number stays the same.

Why zero is considered an integer

• Zero satisfies all integer criteria: it is a whole number with no fractional or decimal part. (Twinkl (teaching resources))
• In the set of integers, zero separates positive from negative numbers. (DreamBox Learning)

The upshot: zero is not just an integer — it’s the pivot point of the entire integer number line.

Historical perspective

The concept of zero as a number emerged independently in India, Babylon, and the Maya civilisation. By the 7th century, Indian mathematicians treated zero as a full integer, a view that spread through Arabic mathematics and eventually to Europe.

What this means: zero’s inclusion in the integer set is not a recent invention — it has been a settled fact for over a millennium.

How do I identify an integer?

1. Check if the number is whole
   • Does the number contain a fraction (like 3/4) or a decimal (like 2.5)? If yes, it is not an integer. (Mathnasium (math education))
   • Numbers with zero after a decimal point (like 5.0) are integers because they represent a whole number. (Study.com (educational resource))
2. Look for a fractional or decimal part
   • Any number with a decimal point and non-zero digits after it is not an integer (e.g., -7.5). (Study.com)
   • A number like -7 is an integer because it is a negative whole number. (DreamBox Learning (math platform))
3. Verify the number is a real number without a remainder
   • An integer can be viewed as a real number with no fractional component.
   • If the number can be divided by 1 without leaving a remainder, it is an integer. (Twinkl (teaching resources))

The trade-off: this step-by-step method works for any real number, but in programming contexts, integers have size limits that don’t exist in pure mathematics.

What is an example of an integer?

Positive integers

• Positive integers are numbers greater than zero: 1, 2, 3, … (DreamBox Learning (math platform))
• Also called natural numbers, they are used for counting objects.

Negative integers

• Negative integers are numbers less than zero: -1, -2, -3, … (DreamBox Learning)
• They represent losses, debts, or temperatures below zero. (Mathnasium (math education))

Zero

Zero is the only integer that is neither positive nor negative. It represents the absence of quantity and serves as the boundary between positive and negative numbers.

Common non-integers

• 0.5, -2.7, π (3.14159…), √2 (1.414…), and fractions like 1/3 are not integers. (DreamBox Learning)
• Any number that cannot be written without a decimal or fraction part is a non-integer. (Study.com (educational resource))

The pattern: real-world examples like the number of students in a class (integer) versus the temperature in Celsius (can be integer but often includes decimals) show when integers are natural and when they are not.

What is an integer and rational number?

Relationship between integers and rational numbers

• Every integer can be written as a fraction with denominator 1 (e.g., 5 = 5/1). Therefore, all integers are rational numbers.
• Not all rational numbers are integers — fractions like 1/2 and decimals like 0.75 are rational but not integers. (Twinkl (teaching resources))

How integers are a subset of rationals

• The set of integers (Z) is a proper subset of the set of rational numbers (Q). (Mathnasium (math education))
• A rational number is any number that can be expressed as a ratio of two integers. Integers meet this condition because they can be written as integer/1.

Examples of rational numbers that are not integers

• 1/2, 3/4, 0.75, 2.5, and -1.3 are rational but not integers. (Study.com (educational resource))
• Repeating decimals like 0.333… (1/3) are also rational but not integers. (DreamBox Learning (math platform))

The implication: when someone says a number is “rational,” it could be an integer or a fraction — the label alone doesn’t guarantee wholeness.

Understanding this subset relationship clarifies why integers are a fundamental building block in mathematics.

Confirmed facts and what’s unclear

“An integer is the number zero (0), a positive natural number (1, 2, 3, …), or the negation of a positive natural number (−1, −2, −3, …).”

— Wikipedia (open encyclopedia)

“An integer is a whole number that can be positive, negative, or zero. Integers do not include fractions or decimals, just whole numbers.”

— Mathnasium (math education center)

For a student tackling algebra, the ability to instantly recognise an integer — and know why 3.14 doesn’t count — can mean the difference between confusion and confidence. The concept is simple, but its implications stretch from counting change to writing computer code.

Understanding integers is essential before exploring factors and their role in mathematics, since every factor of a number must be an integer itself.

Frequently asked questions