Integer
Integers are the natural numbers, their negative values, and zero. Essentially, integers are numbers that can be written without a fractional component, such as 0, 1, -1, 2, -2, 3, -3 and so on.
Example
1. Which of the following are positive integers?
1, 2, -1, 4.5, 7, -3.2, 4¼
The positive integers in this list are: 1, 2, and 7.
The rest of the numbers in the list are not positive integers; -1 is a negative integer, 4.5 and -3.2 are not integers because they have a decimal component, and 4¼ has a fractional component.
2. Which of the following are integers?
-1.5, -76, 1,000,000, 15¾, 0, 48.27, -5,700, 12
-76, 1,000,000, 0, -5700, and 12 are all integers. The remaining numbers contain either a fractional or decimal component, and therefore are not integers.
Arithmetic properties of integers
Below is a table of some of the properties of integers undergoing arithmetic operations. The properties in the table are dependent on a and b being integers.
| Addition | Multiplication | |
|---|---|---|
| Closure | a + b is an integer | a × b is an integer |
| Commutativity | a + b = b + a | a × b = b × a |
| Associativity | a + (b + c) = (a + b) + c | a × (b × c) = (a × b) × c |
| Existence of an identity | a + 0 = a | a × 1 = a |
| Existence of an inverse | a + (-a) = 0 | only -1 and 1 are invertible |
| Distributivity | a × (b + c) = a × b + a × c | |