Editing Negative number (section)

==Formal construction of negative integers==
{{See also|Integer#Construction}}
In a similar manner to [[rational number]]s, we can extend the [[natural number]]s <math>\mathbb{N}</math> to the integers '''<math>\mathbb{Z}</math>''' by defining integers as an [[ordered pair]] of natural numbers (''a'', ''b''). We can extend addition and multiplication to these pairs with the following rules:
{{block indent | em = 1.5 | text = {{math|1=(''a'', ''b'') + (''c'', ''d'') = (''a'' + ''c'', ''b'' + ''d'')}} }}
{{block indent | em = 1.5 | text = {{math|1=(''a'', ''b'') × (''c'', ''d'') = (''a'' × ''c'' + ''b'' × ''d'', ''a'' × ''d'' + ''b'' × ''c'')}} }}

We define an [[equivalence relation]] ~ upon these pairs with the following rule:
{{block indent | em = 1.5 | text = (''a'', ''b'') ~ (''c'', ''d'') if and only if ''a'' + ''d'' = ''b'' + ''c''. }}
This equivalence relation is compatible with the addition and multiplication defined above, and we may define <math>\mathbb{Z}</math> to be the [[quotient set]] <math>\mathbb{N}^2/\sim</math>, i.e. we identify two pairs (''a'', ''b'') and (''c'', ''d'') if they are equivalent in the above sense. Note that '''<math>\mathbb{Z}</math>''', equipped with these operations of addition and multiplication, is a [[Ring (mathematics)|ring]], and is in fact, the prototypical example of a ring.

We can also define a [[total order]] on '''<math>\mathbb{Z}</math>''' by writing
{{block indent | em = 1.5 | text = {{math|1=(''a'', ''b'') ≤ (''c'', ''d'') if and only if ''a'' + ''d'' ≤ ''b'' + ''c''}}. }}

This will lead to an ''additive zero'' of the form (''a'', ''a''), an ''[[additive inverse]]'' of (''a'', ''b'') of the form (''b'', ''a''), a multiplicative unit of the form (''a'' + 1, ''a''), and a definition of [[subtraction]]
{{block indent | em = 1.5 | text = {{math|1=(''a'', ''b'') − (''c'', ''d'') = (''a'' + ''d'', ''b'' + ''c'')}}. }}
This construction is a special case of the [[Grothendieck group#Explicit constructions|Grothendieck construction]].

===Uniqueness===
The additive inverse of a number is unique, as is shown by the following proof. As mentioned above, an additive inverse of a number is defined as a value which when added to the number yields zero.

Let ''x'' be a number and let ''y'' be its additive inverse. Suppose ''y′'' is another additive inverse of ''x''. By definition,
<math display="block">x + y' = 0, \quad \text{and} \quad x + y = 0.</math>

And so, ''x'' + ''y′'' = ''x'' + ''y''. Using the law of cancellation for addition, it is seen that ''y′'' = ''y''. Thus ''y'' is equal to any other additive inverse of ''x''. That is, ''y'' is the unique additive inverse of ''x''.