# What is an integer? (with photos)

The pi sign.

An integer is what is more commonly known as an integer. It can be positive, negative or the number zero, but it must be an integer. In some cases, the definition of an integer will exclude the number zero, or even the set of negative numbers, but this is not as common as the broader use of the term. Integers are the numbers that people are most familiar with, and they play a crucial role in virtually all mathematics.

Fractions are not integers because they are not integers.

To understand what an integer is – that is, why it is different from simply a ‘number’ – we must look at the other sets of numbers that may exist. Many of these sets overlap the integer set in some areas and some are virtually identical. Others have very little in common with any whole number – these types 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 counting number set, as these are the numbers used to count things and ideas. The numbers in the positive set are all integers above zero. So the set would be listed as {1, 2, 3, 4…} and so on forever. Like the set of integers itself, positive integers are infinite. Since people have been counting for as long as we know, this set has also been around for a long time. Although it was not known to be infinite, the set was still essentially the same.

A closely related set is the set of all nonnegative 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 emerged shortly after counting numbers became widely used.

Both sets can 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 define an integer as any member of the set of natural numbers, as well as their negative counterparts.

In addition to 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 of two integers. This means that an integer in itself would be rational – 2/2 is a proportion, but it is also simply equal to 1, while 8/2 is also a proportion and also equals 4. This also means that fractions are rational numbers – 3 / 4 is not an integer, but it is a rational number.

The next step would be the set of real numbers. They could more easily be described as any number that could be placed on a number line. This would include any whole number as well as any rational number since fractions can be placed on a number line. It also includes numbers that cannot be expressed simply as the ratio of two numbers – for example, the square root of two produces a sequence of digits after the decimal place that continues infinitely, so it can never be properly described as a rational number, but it’s a real number.