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.