Negative Number - Formal Construction of Negative Integers

Formal Construction of Negative Integers

See also: Integer#Construction

In a similar manner to rational numbers, we can extend the natural numbers N to the integers Z 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:

(a, b) + (c, d) = (a + c, b + d)
(a, b) × (c, d) = (a × c + b × d, a × d + b × c)

We define an equivalence relation ~ upon these pairs with the following rule:

(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 Z to be the quotient set N²/~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. Note that Z, equipped with these operations of addition and multiplication, is a ring, and is in fact, the prototypical example of a ring.

We can also define a total order on Z by writing

(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

(a, b) − (c, d) = (a + d, b + c).

This construction is a special case of the Grothendieck construction.

Read more about this topic:  Negative Number

Famous quotes containing the words formal, construction and/or negative:

    “This is no argument against teaching manners to the young. On the contrary, it is a fine old tradition that ought to be resurrected from its current mothballs and put to work...In fact, children are much more comfortable when they know the guide rules for handling the social amenities. It’s no more fun for a child to be introduced to a strange adult and have no idea what to say or do than it is for a grownup to go to a formal dinner and have no idea what fork to use.”
    —Leontine Young (20th century)

    “Striving toward a goal puts a more pleasing construction on our advance toward death.”
    —Mason Cooley (b. 1927)

    “Most literature on the culture of adolescence focuses on peer pressure as a negative force. Warnings about the “wrong crowd” read like tornado alerts in parent manuals. . . . It is a relative term that means different things in different places. In Fort Wayne, for example, the wrong crowd meant hanging out with liberal Democrats. In Connecticut, it meant kids who weren’t planning to get a Ph.D. from Yale.”
    —Mary Kay Blakely (20th century)