# What are the Different Types of Integers?

There are a number of different ways to look at integers, and thus a number of different ways to classify them into types. Integers are sometimes called “whole numbers,” referencing the fact that an integer represents a number without a fraction or decimal. These numbers can be plotted on a number line, and they are not abstract, like the so-called “irrational numbers.” Three, for example, is an integer, while 5.87 is not, because it is represented with a decimal, and neither is ¾. π is an example of an irrational number.

One thing to be aware of when discussing integers is that the terminology surrounding these numbers is not standardized in the math community. People may use the same term to refer to different sets of numbers, for example. For wiseGEEK readers in a math class, it is advisable to go with the definition being used by the instructor.

One common grouping is the non-negative or positive integers, of the set {1, 2, 3...}. This set extends to infinity, for those who have some spare time and enjoy counting. Some people also include 0 in this set, although 0 is technically neither positive nor negative, for the set {0, 1, 2, 3...}. People may also use the term “natural numbers” to refer to the set of all positive numbers, with some people including the number zero in this term, while others do not.

Another type of integer is a negative integer. Negative integers are found in the set {-1, -2, -3...}. The set of negative numbers is also infinite in nature. An example of a negative integer might be a number such as -37 or -9,520.

The set of all integers, including positive numbers, negative numbers, and the number zero, may simply be known as “integers,” although this can sometimes result in confusion as some people may assume that one is referring only to the set of positive numbers. In math, the letter Z is sometimes used to symbolize the complete set of integers. Z stands for *Zahren*, the German word for “number,” reflecting the influence of German mathematicians on mathematics terminology. Z is an all-inclusive term which includes all numbers recognized as integers.

These numbers are the building blocks of mathematics. The set of positive integers, not including zero, has been used by humans for thousands of years. Zero is actually a relatively recent introduction to the mathematics world, and it proved to be a revolutionary one. The ability to represent zero paved the way to developing advanced mathematics such as algebra.

## Discussion Comments

That’s a good point, but it looks really confusing when you type it out like that. When I was a kid, just reading about numbers didn’t help me learn math. Luckily, integer games came to my rescue. We had a computer lab and if it wasn’t for those games, I probably still wouldn’t know how to subtract correctly.

@ qwertyq – Good explanation. Negative integers do some strange things. An important difference between positive and negative integers is the fact that a big positive number represents a bigger amount, while a big negative number represents a smaller amount. To make it easier to understand, I’ll line some integers up in their natural order:

-3,-2,-1,0,1,2,3

As you can see, when the negative numbers move to the left, they get larger, but each negative integer has a value that is less than zero. So, -3 equals three less than zero, but -1 only equals one less than zero, and therefore -1 is bigger than -3.

On the positive integer side, however, the numbers (and amounts they represent) get bigger as you go to the right. So 1 is smaller than 3.

I’ll add to this great article by explaining integer rules. When performing math functions, the outcome will be different depending on the mixture of integer types involved. For the sake of time, I’ll only explain Multiplication rules here.

If you multiply two or more positive integers, you get a positive product. Multiplying a positive with a negative will yield a negative, but two negatives yield a positive. And as always, any integer multiplied by zero will equal zero.

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