# Integral vs. Integer — What's the Difference?

By Tayyaba Rehman & Fiza Rafique — Updated on March 8, 2024

**Integral refers to essential or fundamental components, while integer is a whole number in mathematics.**

## Difference Between Integral and Integer

### Table of Contents

ADVERTISEMENT

## Key Differences

Integral relates to things that are essential or necessary parts of a whole, contributing to completeness or functionality in various contexts, such as mathematics, engineering, or philosophy. On the other hand, an integer represents a whole number in the realm of mathematics, devoid of fractions or decimals, and can be positive, negative, or zero.

In mathematics, the term integral also denotes a specific operation that calculates the area under the curve of a function, reflecting accumulation or continuous addition. Whereas integers form the basic building block of arithmetic, serving as the foundation for more complex numerical concepts and operations.

While the concept of integral extends beyond numbers to imply indispensability or completeness in systems, processes, or theories, integers are strictly numerical entities used in counting, ordering, and basic mathematical operations. This distinction underscores the broader applicational scope of integral versus the specific numerical nature of integers.

The integral concept is applied in various fields to describe components or factors that cannot be omitted without affecting the system's integrity or functionality. In contrast, integers are utilized in mathematical contexts, including algebra and number theory, highlighting their fundamental role in quantitative analysis and reasoning.

Understanding the difference between integral and integer is crucial for grasping their respective applications and implications in mathematics and other disciplines. Integral emphasizes necessity and completeness, while integer focuses on whole numbers and their properties in mathematical operations.

ADVERTISEMENT

## Comparison Chart

### Definition

Essential or necessary part of a whole

A whole number in mathematics

### Context

Mathematics, engineering, philosophy

Mathematics, specifically arithmetic and number theory

### Application

Describes indispensability, completeness, operations in calculus

Used in counting, ordering, basic operations

### Mathematical Significance

In calculus, refers to operation calculating area under a curve

Basic building block of numbers, includes positive, negative numbers, and zero

### Scope

Broad, beyond numbers to imply indispensability

Specific to numerical entities

## Compare with Definitions

#### Integral

Essential component.

Water is integral to life on Earth.

#### Integer

Whole number.

The numbers -2, 0, and 5 are all integers.

#### Integral

Mathematical operation.

The integral of the function gives the area under its curve.

#### Integer

Number line.

Integers are evenly spaced on the number line.

#### Integral

Completeness.

Teamwork is integral for the project's success.

#### Integer

Counting example.

There are 7 integers between -3 and 3, inclusive.

#### Integral

System functionality.

Regular maintenance is integral to keeping the machine operational.

#### Integer

Algebraic foundation.

Integers are fundamental in forming algebraic expressions.

#### Integral

Inherent part.

Trust is integral to a healthy relationship.

#### Integer

Basic arithmetic.

The sum of two integers is always an integer.

#### Integral

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration.

#### Integer

An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.

#### Integral

Of or denoted by an integer.

#### Integer

A member of the set of positive whole numbers {1, 2, 3, ... }, negative whole numbers {-1, -2, -3, ... }, and zero {0}.

#### Integral

Necessary to make a whole complete; essential or fundamental

Games are an integral part of the school's curriculum

Systematic training should be integral to library management

#### Integer

A number which is not a fraction; a whole number

Integer values

#### Integral

A function of which a given function is the derivative, i.e. which yields that function when differentiated, and which may express the area under the curve of a graph of the function.

#### Integer

A thing complete in itself.

#### Integral

Essential or necessary for completeness; constituent

The kitchen is an integral part of a house.

#### Integer

A complete unit or entity.

#### Integral

Possessing everything essential; entire.

#### Integer

(arithmetic) A number that is not a fraction; an element of the infinite and numerable set {..., -3, -2, -1, 0, 1, 2, 3, ...}.

#### Integral

Expressed or expressible as or in terms of integers.

#### Integer

A complete entity; a whole number, in contradistinction to a fraction or a mixed number.

#### Integral

Expressed as or involving integrals.

#### Integer

Any of the natural numbers (positive or negative) or zero

#### Integral

A complete unit; a whole.

#### Integral

A number computed by a limiting process in which the domain of a function, often an interval or planar region, is divided into arbitrarily small units, the value of the function at a point in each unit is multiplied by the linear or areal measurement of that unit, and all such products are summed.

#### Integral

A definite integral.

#### Integral

An indefinite integral.

#### Integral

Constituting a whole together with other parts or factors; not omittable or removable

#### Integral

(mathematics) Of, pertaining to, or being an integer.

#### Integral

(mathematics) Relating to integration.

#### Integral

(obsolete) Whole; undamaged.

#### Integral

(mathematics) One of the two fundamental operations of calculus (the other being differentiation), whereby a function's displacement, area, volume, or other qualities arising from the study of infinitesimal change are quantified, usually defined as a limiting process on a sequence of partial sums. Denoted using a long s: ∫, or a variant thereof.

The integral of a univariate real-valued function is the area under its curve; but be warned! Not all functions are integrable!

#### Integral

(specifically) Any of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.

#### Integral

(mathematics) A definite integral: the result of the application of such an operation onto a function and a suitable subset of the function's domain: either a number or positive or negative infinity. In the former case, the integral is said to be finite or to converge; in the latter, the integral is said to diverge. In notation, the domain of integration is indicated either below the sign, or, if it is an interval, with its endpoints as sub- and super-scripts, and the function being integrated forming part of the integrand (or, generally, differential form) appearing in front of the integral sign.

The integral of $\backslash frac\{1\}\{x\}$ on $[\backslash frac\{1\}\{2\},\; 1]$ is $\backslash ln(2)$, but the integral of the same function on $(0,\; 1]$ diverges. In notation, $\backslash int\_\backslash frac\{1\}\{2\}^1\backslash frac\{1\}\{x\}\; dx\; =\; \backslash ln(2)$, but $\backslash int\_0^1\backslash frac\{1\}\{x\}\; dx\; =\; \backslash infty$.

#### Integral

(mathematics) An indefinite integral: the result of the application of such an operation onto a function together with an indefinite domain, yielding a function; a function's antiderivative;

The integral of $x^2$ is $\backslash frac\{x^3\}\{3\}$ plus a constant.

#### Integral

The fluent of a given fluxion in Newtonian calculus.

#### Integral

Lacking nothing of completeness; complete; perfect; uninjured; whole; entire.

A local motion keepeth bodies integral.

#### Integral

Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.

Ceasing to do evil, and doing good, are the two great integral parts that complete this duty.

#### Integral

Of, pertaining to, or being, a whole number or undivided quantity; not fractional.

#### Integral

A whole; an entire thing; a whole number; an individual.

#### Integral

An expression which, being differentiated, will produce a given differential. See differential Differential, and Integration. Cf. Fluent.

#### Integral

The result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)

#### Integral

Existing as an essential constituent or characteristic;

The Ptolemaic system with its built-in concept of periodicity

A constitutional inability to tell the truth

#### Integral

Constituting the undiminished entirety; lacking nothing essential especially not damaged;

A local motion keepeth bodies integral

Was able to keep the collection entire during his lifetime

Fought to keep the union intact

## Common Curiosities

#### What does integral mean?

Integral refers to something that is essential or fundamentally necessary to make a whole complete.

#### What is an integer in mathematics?

An integer is a whole number that can be positive, negative, or zero, not including fractions or decimals.

#### How are integers used in everyday life?

Integers are used in counting objects, measuring temperatures below zero, and calculating debts, among other applications.

#### Can you give an example of something integral?

"Honest communication is integral to maintaining strong personal relationships."

#### What makes an integer positive or negative?

An integer's positivity or negativity is determined by its value relative to zero: positive if above zero, negative if below.

#### What role do integers play in mathematics?

Integers serve as the foundation for arithmetic operations, number theory, and are integral in defining more complex number systems.

#### Why is something considered integral?

Something is considered integral if its absence would compromise the whole's completeness or functionality.

#### How are negative numbers represented as integers?

Negative numbers are represented as integers with a minus sign, indicating their position on the negative side of the number line.

#### Is the concept of integral limited to mathematics?

No, while integral has a specific meaning in calculus, it also broadly applies to essential components or aspects in various fields.

#### What distinguishes an integer from other numbers?

Integers include whole numbers without fractional or decimal parts, distinguishing them from real or rational numbers that can include fractions.

#### Can the term integral be used in technology?

Yes, certain components or features can be described as integral to a technology's functionality or design.

#### Is the integration process in calculus related to the integral concept?

Yes, the integration process in calculus embodies the integral concept, calculating quantities that contribute to a whole.

#### How is the integral concept applied in calculus?

In calculus, integrals calculate the area under a curve, representing accumulation or continuous addition of quantities.

#### Can integers be used in statistics?

Yes, integers can be used in statistical data, representing counts, losses, gains, or other discrete variables.

#### Are all numbers integers?

No, only whole numbers are integers; numbers with fractional or decimal components are not.

## Share Your Discovery

Previous Comparison

Concession vs. ConsignmentNext Comparison

Wooly vs. Woolly## Author Spotlight

Written by

Tayyaba RehmanTayyaba Rehman is a distinguished writer, currently serving as a primary contributor to askdifference.com. As a researcher in semantics and etymology, Tayyaba's passion for the complexity of languages and their distinctions has found a perfect home on the platform. Tayyaba delves into the intricacies of language, distinguishing between commonly confused words and phrases, thereby providing clarity for readers worldwide.

Co-written by

Fiza RafiqueFiza Rafique is a skilled content writer at AskDifference.com, where she meticulously refines and enhances written pieces. Drawing from her vast editorial expertise, Fiza ensures clarity, accuracy, and precision in every article. Passionate about language, she continually seeks to elevate the quality of content for readers worldwide.