An integer is a number that does not have a fractional part. The set of integers is
The notation for the set of integers comes from the German word Zahlen, which means "numbers". Integers strictly larger than zero are positive integers and integers strictly less than zero are negative integers.
For example, , , , and are all integers (2 and 67 are positive integers and -13 is a negative integer). The values , , , , and are not integers.
The following are the properties of integers:
- The set of integers is closed under the operation of addition: if , then .
- The set of integers is closed under the operation of multiplication: if , then .
- For any integer , the additive inverse is an integer.
- If and are integers such that , then or .
- The set of integers is infinite and has no smallest element and no largest element.
means "belongs to", as means is an element of the set or belongs to the set
Note that the set of integers is not closed under the operation of division. As an example, and are integers, but is not an integer.
Which of the following are integers?
Since , and , the integers in the list are , and
What is the smallest integer that is larger than
Since the smallest integer that is larger than is .
What is the largest integer that is smaller than
Since the largest integer that is smaller than is .
Using the properties of integers above, show that set of integers is closed under the operation of subtraction.
Consider any two integers and . We would like to show is also an integer. By property the additive inverse of is , which is an integer. Then
is an integer by Property Therefore, the set of integers is closed under the operation of subtraction.