User:Omegatron/Prettytable examples
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Contents |
[edit] Old caption style
| × | 1 | 2 | 3 |
|---|---|---|---|
| 1 | 1 | 2 | 3 |
| 2 | 2 | 4 | 6 |
| 3 | 3 | 6 | 9 |
| 4 | 4 | 8 | 12 |
| 5 | 5 | 10 | 15 |
| Foo | Bar | Baz | Quux |
|---|---|---|---|
| 100 | Cake | Monster in the closet | NO! |
| Wikipedia | ^______^ | Darth Vader | 42 |
| Moo | 1.618033989 | Pay your bills | Bach |
[edit] Algebraic properties
Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer.
| 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 | |
| distributivity: | a × (b + c) = (a × b) + (a × c) | |
In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + ... 1 or (−1) + (−1) + ... + (−1). In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z.
The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, note that not every integer has a multiplicative inverse; e.g. there is no integer x such that 2x = 1, because the LHS is even, while the RHS is odd. This means that Z under multiplication is not a group.
All the properties from the above table taken together say that Z together with addition and multiplication is a commutative ring with unity. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field. The smallest field containing the integers is the field of rational numbers. This process can be mimicked to form the quotient field of any integral domain, where an integral domain is a commutative ring with unity such that whenever ab = 0, either a = 0 or b = 0.
[edit] Proposed caption style
| × | 1 | 2 | 3 |
|---|---|---|---|
| 1 | 1 | 2 | 3 |
| 2 | 2 | 4 | 6 |
| 3 | 3 | 6 | 9 |
| 4 | 4 | 8 | 12 |
| 5 | 5 | 10 | 15 |
| Foo | Bar | Baz | Quux |
|---|---|---|---|
| 100 | Cake | Monster in the closet | NO! |
| Wikipedia | ^______^ | Darth Vader | 42 |
| Moo | 1.618033989 | Pay your bills | Bach |
[edit] Algebraic properties
Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer.
| 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 element: | a + (−a) = 0 | |
| Distributivity: |
a × (b + c) = (a × b) + (a × c) |
|
In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + ... 1 or (−1) + (−1) + ... + (−1). In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z.
The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, note that not every integer has a multiplicative inverse; e.g. there is no integer x such that 2x = 1, because the LHS is even, while the RHS is odd. This means that Z under multiplication is not a group.
All the properties from the above table taken together say that Z together with addition and multiplication is a commutative ring with unity. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field. The smallest field containing the integers is the field of rational numbers. This process can be mimicked to form the quotient field of any integral domain, where an integral domain is a commutative ring with unity such that whenever ab = 0, either a = 0 or b = 0.
