Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on)
"Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.
Integers
The set of integers consists of zero (0), the natural numbers (1, 2, 3, …), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e. −1, −2, −3, …).
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| The Zahlen symbol, often used to denote the set of all integers |
Order
An integer is positive if it is greater than zero and negative if it is less than zero
| Addition | Multiplication | |
|---|---|---|
| Closure: | a + b is an integer | a × b is an integer |
| Associativity: | a + (b + c) = (a + b) + c | a × (b × c) = (a × b) × c |
| Commutativity: | a + b = b + a | a × b = b × a |
| Existence of an identity element: | a + 0 = a | a × 1 = a |
| Existence of inverse elements: | a + (−a) = 0 | An inverse element usually does not exist at all. |
| Distributivity: | a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c) | |
| No zero divisors: (*) | If a × b = 0, then a = 0 or b = 0 (or both) | |
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