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What is an Integer?

By Brendan McGuigan
Updated May 21, 2024
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An integer is what is more commonly known as a whole number. It may be positive, negative, or the number zero, but it must be whole. In some cases, the definition of whole number will exclude the number zero, or even the set of negative numbers, but this is not as common as the more inclusive use of the term. Integers are the numbers people are most familiar with, and they serve a crucial role in virtually all mathematics.

To understand what an integer is — that is to say, why it is different than simply a ‘number’ — we must look at the other sets of numbers that can exist. Many of these sets overlap with the integer set in some areas, and some are virtually identical. Others have very little in common with any integer — these sorts of numbers tend to be much less familiar to most people.

The subset of positive integers is probably the oldest set of numbers. This group is often referred to as the set of counting numbers, since these are the numbers used to count things and ideas. The numbers in the positive set are all of the whole numbers above zero. So the set would be listed as {1, 2, 3, 4 ...} and so on, forever. Like the set of integers itself, the positive integers are infinite. Since people have been counting as far back as we know of, this set has also existed for a very long time. Although it may not have been known to be infinite, the set was still essentially the same.

A very closely related set is the set of all non-negative integers. This set is identical to the set of positive integers, except that it also includes zero. Historically, the number zero was an innovation that came about quite a bit after the counting numbers had been in wide-spread use.

Both of these sets may be referred to as the set of natural numbers. Some mathematicians prefer to exclude zero from the natural numbers, while others find it useful to include it. If we consider the more inclusive definition, we can then define an integer as any member of the set of natural numbers, as well as their negative counterparts.

Beyond the integer, we find other sets that are more complicated. The next logical progression is the set of all rational numbers. A rational number is any number that can be discussed as a ratio between two integers. This means that an integer itself would be rational — 2/2 is a ratio, but is also simply equal to 1, while 8/2 is also a ratio, and also equal to 4. It also means that fractions are rational numbers — 3/4 is not an integer, but it is a rational number.

The next step out would be the set of real numbers. These could most easily be described as any number which could be placed on a number line. This would include any integer, as well as any rational number, as fractions can be placed on a number line. It further includes numbers which cannot be expressed simply as the ratio between two numbers — for example, the square root of two produces a string of digits after the decimal place which go on infinitely, so it can never be adequately described as a rational number, but it is a real number.

The final set of numbers commonly dealt with is the set of complex numbers. These numbers have no actual place on a number line, but have a use in many mathematical processes nonetheless. Complex numbers include an imaginary component, usually given as i, where i2 is equal to -1.

There are many different types of numbers, and each have their place in the world of mathematics and the many disciplines in which it is used. An integer can best be described both by what it is, and by what it is not. It is any whole, positive number, from one to an infinitely-large number. An integer is the number zero. It is any whole, negative number, from negative one to an infinitely-large negative number. It is not any number which has a remainder beyond the decimal place. An integer is not a special real number, such as pi or e. And it is not a complex or irrational number.

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Discussion Comments

By anon333652 — On May 07, 2013

How do you work out a ratio with a non-integer? What is a non-integer?

By anon288925 — On Sep 01, 2012

Are integers the set of whole numbers and the negatives?

By anon124149 — On Nov 04, 2010

is 1/2 an integer or not?

By anon112794 — On Sep 22, 2010

Do real numbers, integers, whole numbers, rational numbers, and irrational numbers overlap? How?

By anon107761 — On Aug 31, 2010

If a number is a whole number, is it always an integer. Also, if a number is an integer, is it always a whole number?

By anon106704 — On Aug 26, 2010

what is the relationship between rational integers, whole and counting numbers?

By anon97240 — On Jul 19, 2010

i want to learn integers because that is our lesson.

By anon89656 — On Jun 11, 2010

the integers are denoted with a boldface Z because the German word for "integers" is "Zahlen."

By anon89410 — On Jun 10, 2010

how do i find this number on an invoice?

By anon79322 — On Apr 22, 2010

what is a positive integer? what is a negative integer?

By anon72411 — On Mar 23, 2010

what is 'zero'? even or odd?

By justlearning — On Sep 01, 2009

what is the difference between the set of natural numbers and the set of integers?

By anon43677 — On Aug 31, 2009

why is the set of integers symbolized with a capital Z?

By anon40002 — On Aug 05, 2009

what is the integers of 18

By anon32652 — On May 25, 2009

well if the whole numbers vary from teacher to teacher based on what they consider an integer, how am i ever supposed to know if i get the right answer?

By karin33 — On Apr 29, 2009

yes

By cmood — On Mar 30, 2009

Answer for karin33

Is the new product 0?

By karin33 — On Jan 22, 2009

Can you please answer this question? There are two integers. If you multiply them together, you get 48. If you take one of the integers, multiply it by two, subtract two thirds of the other integer, what is the new product?

By mendocino — On May 20, 2008

Jeffrey1997 - Your math teacher is right, and, this article is right. To get a good grade in your class, I recommend that you take on the definition that your teacher provides. :) But, to understand the issue completely, read on.... People disagree about the definition of a whole number. Not everything is as clear, simple, and straightforward as it should be! Some people define the term "whole number" in the loosest way -- that it's a number that doesn't include decimals or fractions, meaning {...-2, -1, 0, 1, 2...}. Others, like your math teacher, I assume, define it to mean non-negative integers, meaning {0, 1, 2, 3...}. Still others define it to mean counting numbers {1, 2, 3...}. So whether a whole number and an integer are completely the same or there is just some overlap between the terms but not totally, is a matter of dispute.

By jeffrey1997 — On May 19, 2008

My teacher in 6th grade mathematics right now always says that a whole number and integer are totally diiferent. I'm not sure if you're right, or that my teacher is right.

By anon11634 — On Apr 20, 2008

Think in this The number -0.36 is between the consecutive integers -1 and 0. You choose the left integer -1.That's is why is -1.

By anon1582 — On Jun 07, 2007

What is the integer component of -0.36? My Excel spreadsheet says it is -1, but I would have thought it was 0

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