Integers are positive, negative, or zero whole numbers. The set of positive integers is also known as counting numbers. The set of non-negative integers includes zero. Integers are a subset of natural numbers. Rational numbers are ratios of integers, while real numbers can be placed on a number line. Complex numbers include an imaginary component.
An integer is what is more commonly known as an integer. It can be positive, negative or the number zero, but 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 more inclusive use of the term. Integers are the numbers people are most familiar with and they play a crucial role in virtually all of mathematics.
To understand what an integer is, that is, why it is different from a simple “number”, we need to look at the other sets of numbers that may exist. Many of these ensembles overlap with the entire ensemble in some areas and some are virtually identical. Others have very little in common with any integer—these kinds 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, as these are the numbers used to count things and ideas. The numbers in the positive set are all integers above zero. Then 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’ve known, this set has also been around for a long time. While it may not have been known to be infinite, the set was still essentially the same.
A 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 arose somewhat after counting numbers had become widely used.
Both of these 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 take 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.
In addition to the whole, we find other more complicated sets. 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 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 would be the set of real numbers. These could most 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 string of digits after the decimal place that goes on forever, so it can never adequately be described as a number rational, but it is a real number.
The last set of numbers commonly covered is the set of complex numbers. These numbers have no actual place on a number line, but still have a use in many mathematical processes. Complex numbers include an imaginary component, usually denoted as i, where i2 equals -1.
There are many different types of numbers, and each has its place in the world of mathematics and the many disciplines in which it is used. A whole can be best described by both what it is and what it is not. It is any positive integer, from one to an infinitely large number. An integer is the number zero. It is any integer, negative number, from a negative one to an infinitely large negative number. It is not a number that has a remainder beyond the decimal point. An integer is not a special real number, such as pi or e. And it’s not a complex or irrational number.
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