What is an Integer? Definition, Examples & Key Facts

Symbol: ℤ ·
Infinite Set: Yes, extends infinitely in both positive and negative directions ·
Zero Included: Yes, 0 is an integer ·
Example: -5, 0, 42 ·
Smallest Positive Integer: 1 ·
Largest Negative Integer: -1

The key facts below summarise the core properties of integers.

How do I define an integer?

What is the formal definition of an integer?

An integer is a whole number that has no decimal or fractional part. The set of integers includes zero, all positive natural numbers (1, 2, 3, …), and their negative counterparts (-1, -2, -3, …). As Math is Fun (educational math reference) explains, integers are “the counting numbers, zero, and the negatives of the counting numbers.” The formal notation is {…, -3, -2, -1, 0, 1, 2, 3, …}, denoted by the symbol ℤ, derived from the German word Zahlen meaning “numbers.”

What are examples of integers?

• Positive integers: 1, 2, 3, 100, 1,000,000 (Study.com (educational resource))
• Negative integers: -1, -12, -500 (Mathnasium (math learning center))
• Zero: 0 is an integer (Math is Fun (educational math reference))

What are non-examples of integers?

• 1.5 (decimal) (Study.com (educational resource))
• 3.14 (decimal)
• ½ (fraction)
• -0.7 (decimal)

The implication: any number that can be written with a fractional component—whether a decimal point, a fraction bar, or a percent sign—fails the integer test. Only whole numbers qualify.

Is 1.5 an integer?

Why are decimals not integers?

A number with a non-zero decimal part is not an integer. The decimal point signals a fractional portion that breaks the “wholeness” requirement. For example, 1.5 represents one and a half—not a complete, undivided count. As Study.com (educational resource) puts it, “Numbers like 3 1/2 and -7.5 are not integers because they contain fractions or decimals.”

What about 0.0 and 1.0?

Zero written as 0.0 is still the integer 0—the fractional part is zero, so it qualifies. Similarly, 1.0 equals the integer 1. The key distinction: if the decimal part equals zero, the number is an integer; if the decimal part is anything else, it is not.

How to check if a number is an integer

1. Look at the number. Does it contain a decimal point or a fraction bar?
2. If it has a decimal point, check the digits after it. If any non-zero digit exists, it is not an integer.
3. If there is no decimal point (or only zeros after it) and no fraction bar, the number is an integer.

Math is Fun (educational math reference) emphasizes that integers “do not include fractions or decimals.”

The catch: many people mistakenly think 0.5 is an integer because it’s small and common. But any non-zero fractional component disqualifies it. Zero is an integer; 0.5 is not.

How to explain integers to kids?

Simple definition for kids using number lines

Integers are like whole steps on a number line. Start at zero. Steps to the right are positive (1, 2, 3). Steps to the left are negative (-1, -2, -3). You cannot take half a step. Each step is a complete, indivisible unit. Kiddle (kid-friendly encyclopedia) describes integers as “positive numbers, negative numbers, and zero,” noting they “do not have decimals or fractions.”

Hands-on activities: counting steps and money

• Temperature: A thermometer shows 0°C for freezing. Above zero is positive (warm), below is negative (cold).
• Elevator floors: Ground is 0, going up is positive floors, going down to a basement is negative floors.
• Bank account: Money you have is positive, debt you owe is negative.
• Game scores: Points earned are positive integers; points lost (or penalties) are negative integers.

Common misconceptions about negative numbers

Many children think negative numbers are “not real” or “just imaginary.” Historically, Wikipedia (free encyclopedia) notes that mathematicians debated whether negative numbers were “real” for centuries. Today, we know they represent debt, below-zero temperatures, and backward steps—all very real.

What this means: the number line is the best tool. Show a child that -3 is three steps to the left of zero, no different conceptually than +3 is three steps to the right.

What is the difference between a number and an integer?

The table below contrasts integers with other types of numbers.

Integer vs whole number

All whole numbers (0, 1, 2, 3, …) are integers, but integers also include negative numbers (-1, -2, -3, …). The term “whole number” is sometimes used to mean non-negative integers only, so integers are a broader set.

Integer vs rational number

Every integer is a rational number because it can be written as a fraction with denominator 1 (e.g., 5 = 5/1). But not every rational number is an integer. Wikipedia (free encyclopedia) explains: “The set of integers is a subset of rational numbers.” Fractions like 3/4 are rational but not integer.

Real numbers include integers and non-integers

Real numbers cover everything: integers, fractions, decimals, irrationals (like π and √2). Integers are a small but fundamental subset of the real numbers.

Why this matters: when a programmer defines a variable as an integer type, they get whole-number operations—fast, exact, and memory-efficient—but lose the ability to store fractional values. That trade-off is central to both mathematics and computer science.

What is an integer in Python?

Integer data type in programming

In Python, an integer (int) is a built-in data type that stores whole numbers. As Mimo (coding education platform) explains, integers “are commonly used for counting, indexing, and calculations in code.” You can assign an integer to a variable: score = 42 or temperature = -5.

How Python handles arbitrary-precision integers

Unlike many languages, Python integers can grow arbitrarily large—there is no fixed upper limit. A Python integer can hold 2¹⁰⁰⁰ without overflow. Kiddle (kid-friendly encyclopedia) notes that “many languages have an int or integer type for whole numbers,” but Python’s int is special because it dynamically expands as needed.

Integer limits in languages like C vs Python

• Python: No fixed limit; memory is the only constraint.
• C: 32-bit integer max = 2,147,483,647; min = -2,147,483,648.
• JavaScript: Safe integer range up to 9,007,199,254,740,991 (Number.MAX_SAFE_INTEGER).
• Java: 32-bit int and 64-bit long with fixed ranges.

The implication: a Python developer never worries about integer overflow, making the language ideal for large-number calculations (cryptography, scientific computing). A C developer must carefully choose the right integer size or risk bugs.

Expert perspectives on integers

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

Mathnasium (math learning center)

“The set of integers consists of zero, the positive natural numbers, and the additive inverses of the positive natural numbers (the negative integers).”

Wikipedia (free encyclopedia)

These definitions reinforce the same core idea from different angles: integers are the whole-number backbone of mathematics.

Putting it all together: why integers matter

From counting apples to writing Python loops, integers are the backbone of exact, discrete measurement. They give us a reliable way to count, compare, and compute without the ambiguity of fractions. For the math student, the programmer, or the curious kid, understanding integers is the first step into a world where numbers behave predictably—and that’s powerful.

The choice is clear: when you need an exact count, use an integer. When you need precision beyond the whole step, reach for a decimal or a fraction. Each has its purpose, and knowing the difference is the foundation of mathematical literacy.

For more practice with integers in an educational context, check out Text and Tests 4 Solutions and Tables Champion 6th Class Answers.

Frequently asked questions