The "General Games Chat" forum, which includes Retro Game Reviews, has been archived and is now read-only. You cannot post here or create a new thread or review on this forum.
ANYTHING
Page:
If you could answer every question you should know why. And know I'm not going to tell you as I have done enough typing as it is.
c wrote:
> I cant believe your still posting messags in my topic AJ.
---
Well, you'd better believe it because I'm still doing it.
> I cant believe your still posting messags in my topic AJ.
---
Well, you'd better believe it because I'm still doing it.
I cant believe your still posting messags in my topic AJ.
Number are a manmade creation to aid mathamatical problem solving, and what are you doing talking about letters for?
c wrote:
> I cant believe nobody at special reserve has any proper questions to
> ask.
---
I can't believe you're still rambling on about questions.
> I cant believe nobody at special reserve has any proper questions to
> ask.
---
I can't believe you're still rambling on about questions.
I cant believe nobody at special reserve has any proper questions to ask.
WHAT ARE NUMBERS?
ANSWER
Introduction
Informally:
N = { 0,1,... } or N = { 1,2,... }
Wether 0 is in N depends on where you live and what is your field of interest. At the informal level it is a religious topic.
Z = { ..., - 1,0,1,... }
Q = { p/q | p, q in Z and q != 0 }
R = { d_0.d_1d_2... | d_0 in Z and 0 <= d_i <= 9 for i > 0 }
C = { a + b o i | a, b in R and i^2 = -1 }
Construction of the Number System
Formally (following the mainstream in math) the numbers are constructed from scratch out of the axioms of Zermelo Fraenkel set theory (a.k.a. ZF set theory) [Enderton77, Henle86, Hrbacek84]. The only things that can be derived from the axioms are sets with the empty set at the bottom of the hierarchy. This will mean that any number is a set (it is the only thing you can derive from the axioms). It doesn't mean that you always have to use set notation when you use numbers: just introduce the numerals as an abbreviation of the formal counterparts.
The construction starts with N and algebraically speaking, N with its operations and order is quite a weak structure. In the following constructions the structures will be strengthen one step at the time: Z will be an integral domain, Q will be a field, for the field R the order will be made complete, and field C will be made algebraically complete.
Before we start, first some notational stuff:
a pair (a,b) = { { a } , { a,b } } ,
an equivalence class [a] = { b | a == b } ,
the successor of a is s(a) = a U { a } .
Although the previous notations and the constructions that follow are the de facto standard ones, there are different definitions possible.
Construction of N
{ } in N
if a in N then s(a) in N
N is the smallest possible set such that the preceding rules hold.
Informally n = { 0,...,n - 1 } (thus 0 = { } , 1 = { 0 } , 2 = { 0,1 } , 3 = { 0,1,2 } ). We will refer to the elements of N by giving them a subscript _n. The relation <_n on N is defined as: a_n <_n b_n iff a_n in b_n. We can define +_n as follows:
a_n +_n 0_n = a_n
a_n +_n s(b_n) = s(a_n +_n b_n)
Define *_n as:
a_n *_n 0_n = 0_n
a_n *_n s(b_n) = (a_n *_n b_n) +_n a_n
Construction of Z
We define an equivalence relation on N x N: (a_n,b_n) ==_z(c_n,d_n) iff a_n +_n d_n = c_n +_n b_n. Note that ==_z ``simulates'' a subtraction in N . Z = { [(a_n,b_n)]_z | a_n, b_n in N } . We will refer to the elements of Z by giving them a subscript _z. The elements of N can be embedded as follows: embed_n : N --> Z such that embed_n(a_n) = [(a_n,0_n)]_z. Furthermore we can define:
[(a_n,b_n)]_z <_z [(c_n,d_n)]_z iff a_n +_n d_n <_n c_n +_n b_n
[(a_n,b_n)]_z +_z [(c_n,d_n)]_z = [(a_n +_n c_n, b_n +_n d_n)]_z
[(a_n,b_n)]_z *_z [(c_n,d_n)]_z =
[((a_n *_n c_n) +_n (b_n *_n d_n), (a_n *_n d_n) +_n (c_n *_n b_n))]_z
Construction of Q
We define an equivalence relation on Z x (Z { 0_z }): (a_z,b_z) ==_q (c_z,d_z) iff a_z *_z d_z = c_z *_z b_z. Note that ==_q ``simulates'' a division in Z . Q = { [(a_z,b_z)]_q | a_z in Z and b_z in Z { 0_z } } . We will refer to the elements of Q by giving them a subscript _q. The elements of Z can be embedded as follows: embed_z : Z --> Q such that embed_z(a_z) = [(a_z,1_z)]_q. Furthermore we can define:
[(a_z,b_z)]_q <_q [(c_z,d_z)]_q iff a_z *_z d_z <_z c_z *_z b_z
when 0_z <_z b_z and 0_z <_z d_z
[(a_z,b_z)]_q +_q [(c_z,d_z)]_q = [((a_z *_z d_z) +_z (c_z *_z b_z), b_z *_z d_z)]_q
[(a_z,b_z)]_q *_q [(c_z,d_z)]_q = [(a_z *_z c_z, b_z *_z d_z)]_q
Construction of R
The construction of R is different (and more awkward to understand) because we must ensure that the cardinality of R is greater than that of Q .
Set c is a Dedekind cut iff
{ } subset c subset Q (strict inclusions!)
c is closed downward:
if a_q in c and b_q <_q a_q then b_q in c
c has no largest element:
there isn't an element a_q in c such that b_q <_q a_q for all b_q != a_q in c
You can think of a cut as taking a pair of scissors and cutting Q in two parts such that one part contains all the small numbers and the other part contains all large numbers. If the part with the small numbers was cut in such a way that it doesn't have a largest element, it is called a Dedekind cut. R = { c | c is a Dedekind cut } . We will refer to the elements of R by giving them a subscript _r. The elements of Q can be embedded as follows: embed_q : Q --> R such that embed_q(a_q) = { b_q | b_q <_q a_q } . Furthermore we can define:
a_r <_r b_r iff a_r subset b_r (strict inclusion!)
a_r +_r b_r = { c_q +_q d_q | c_q in a_r and d_q in b_r }
-_r a_r = ; { b_q | there exists an c_q in Q such that b_q <_q c_q and (-1)_q *_q c_q in a_r }
|a_r|_r = a_r U -_r a_r
*_r is defined as:
if not a_r <_r 0_r and not b_r <_r 0_r
then a_r *_r b_r = 0_r U { c_q *_q d_q | c_q in a_r and d_q in b_r }
if a_r <_r 0_r and b_r <_r 0_r then a_r *_r b_r = |a_r|_r *_r |b_r|_r
otherwise a_r *_r b_r = -_r (|a_r|_r *_r |b_r|_r)
There exists an alternative definition of R using Cauchy sequences: a Cauchy sequence is a s : N --> Q such that s(i_n) +_q((-1)_q *_q s(j_n)) can be made arbitrary near to 0_q for all sufficiently large i_n and j_n. We will define an equivalence relation ==_r on the set of Cauchy sequences as: r ==_r s iff r(m_n) +_q((-1)_q *_q s(m_n)) can be made arbitrary close to 0_q for all sufficiently large m_n. R = { [s]_r | s is a Cauchy sequence } . Note that this definition is close to ``decimal'' expansions.
Construction of C
C = R x R. We will refer to the elements of C by giving them a subscript _c. The elements of R can be embedded as follows: embed_r : R --> C such that embed_r(a_r) = (a_r,0_r). Furthermore we can define:
(a_r,b_r) +_c (c_r,d_r) = (a_r +_r c_r, b_r +_r d_r)
(a_r,b_r) *_c (c_r,d_r) = ((a_r *_r c_r) +_r -_r (b_r * d_r), (a_r *_r d_r) +_r (b_r *_r c_r))
There exists an elegant alternative definition using ideals. To be a bit sloppy: C = R [x]/< (x *_r x) +_r 1_r > , i.e. C is the resulting quotient ring of factoring ideal < (x *_r x) +_r 1_r > out of the ring R [x] of polynomials over R . The sloppy part is that we need to define concepts like quotient ring, ideal, and ring of polynomials. Note that this definition is close to working with i^2 = -1: (x *_r x) +_r 1_r = 0_r can be rewritten as (x *_r x) = (-1)_r.
Rounding things up
At this moment we don't have that N is a subset of Z , Z of Q , etc. But we can get the inclusions if we look at the embedded copies of N , Z , etc. Let
N' = ran embed_r o embed_q o embed_z o embed_n
Z' = ran embed_r o embed_q o embed_z
Q' = ran embed_r o embed_q
R' = ran embed_r
For these sets we have N' subseteq Z' subseteq Q' subseteq R' subseteq C. Furthermore these sets have all the properties that the ``informal'' numbers have.
ANSWER
Introduction
Informally:
N = { 0,1,... } or N = { 1,2,... }
Wether 0 is in N depends on where you live and what is your field of interest. At the informal level it is a religious topic.
Z = { ..., - 1,0,1,... }
Q = { p/q | p, q in Z and q != 0 }
R = { d_0.d_1d_2... | d_0 in Z and 0 <= d_i <= 9 for i > 0 }
C = { a + b o i | a, b in R and i^2 = -1 }
Construction of the Number System
Formally (following the mainstream in math) the numbers are constructed from scratch out of the axioms of Zermelo Fraenkel set theory (a.k.a. ZF set theory) [Enderton77, Henle86, Hrbacek84]. The only things that can be derived from the axioms are sets with the empty set at the bottom of the hierarchy. This will mean that any number is a set (it is the only thing you can derive from the axioms). It doesn't mean that you always have to use set notation when you use numbers: just introduce the numerals as an abbreviation of the formal counterparts.
The construction starts with N and algebraically speaking, N with its operations and order is quite a weak structure. In the following constructions the structures will be strengthen one step at the time: Z will be an integral domain, Q will be a field, for the field R the order will be made complete, and field C will be made algebraically complete.
Before we start, first some notational stuff:
a pair (a,b) = { { a } , { a,b } } ,
an equivalence class [a] = { b | a == b } ,
the successor of a is s(a) = a U { a } .
Although the previous notations and the constructions that follow are the de facto standard ones, there are different definitions possible.
Construction of N
{ } in N
if a in N then s(a) in N
N is the smallest possible set such that the preceding rules hold.
Informally n = { 0,...,n - 1 } (thus 0 = { } , 1 = { 0 } , 2 = { 0,1 } , 3 = { 0,1,2 } ). We will refer to the elements of N by giving them a subscript _n. The relation <_n on N is defined as: a_n <_n b_n iff a_n in b_n. We can define +_n as follows:
a_n +_n 0_n = a_n
a_n +_n s(b_n) = s(a_n +_n b_n)
Define *_n as:
a_n *_n 0_n = 0_n
a_n *_n s(b_n) = (a_n *_n b_n) +_n a_n
Construction of Z
We define an equivalence relation on N x N: (a_n,b_n) ==_z(c_n,d_n) iff a_n +_n d_n = c_n +_n b_n. Note that ==_z ``simulates'' a subtraction in N . Z = { [(a_n,b_n)]_z | a_n, b_n in N } . We will refer to the elements of Z by giving them a subscript _z. The elements of N can be embedded as follows: embed_n : N --> Z such that embed_n(a_n) = [(a_n,0_n)]_z. Furthermore we can define:
[(a_n,b_n)]_z <_z [(c_n,d_n)]_z iff a_n +_n d_n <_n c_n +_n b_n
[(a_n,b_n)]_z +_z [(c_n,d_n)]_z = [(a_n +_n c_n, b_n +_n d_n)]_z
[(a_n,b_n)]_z *_z [(c_n,d_n)]_z =
[((a_n *_n c_n) +_n (b_n *_n d_n), (a_n *_n d_n) +_n (c_n *_n b_n))]_z
Construction of Q
We define an equivalence relation on Z x (Z { 0_z }): (a_z,b_z) ==_q (c_z,d_z) iff a_z *_z d_z = c_z *_z b_z. Note that ==_q ``simulates'' a division in Z . Q = { [(a_z,b_z)]_q | a_z in Z and b_z in Z { 0_z } } . We will refer to the elements of Q by giving them a subscript _q. The elements of Z can be embedded as follows: embed_z : Z --> Q such that embed_z(a_z) = [(a_z,1_z)]_q. Furthermore we can define:
[(a_z,b_z)]_q <_q [(c_z,d_z)]_q iff a_z *_z d_z <_z c_z *_z b_z
when 0_z <_z b_z and 0_z <_z d_z
[(a_z,b_z)]_q +_q [(c_z,d_z)]_q = [((a_z *_z d_z) +_z (c_z *_z b_z), b_z *_z d_z)]_q
[(a_z,b_z)]_q *_q [(c_z,d_z)]_q = [(a_z *_z c_z, b_z *_z d_z)]_q
Construction of R
The construction of R is different (and more awkward to understand) because we must ensure that the cardinality of R is greater than that of Q .
Set c is a Dedekind cut iff
{ } subset c subset Q (strict inclusions!)
c is closed downward:
if a_q in c and b_q <_q a_q then b_q in c
c has no largest element:
there isn't an element a_q in c such that b_q <_q a_q for all b_q != a_q in c
You can think of a cut as taking a pair of scissors and cutting Q in two parts such that one part contains all the small numbers and the other part contains all large numbers. If the part with the small numbers was cut in such a way that it doesn't have a largest element, it is called a Dedekind cut. R = { c | c is a Dedekind cut } . We will refer to the elements of R by giving them a subscript _r. The elements of Q can be embedded as follows: embed_q : Q --> R such that embed_q(a_q) = { b_q | b_q <_q a_q } . Furthermore we can define:
a_r <_r b_r iff a_r subset b_r (strict inclusion!)
a_r +_r b_r = { c_q +_q d_q | c_q in a_r and d_q in b_r }
-_r a_r = ; { b_q | there exists an c_q in Q such that b_q <_q c_q and (-1)_q *_q c_q in a_r }
|a_r|_r = a_r U -_r a_r
*_r is defined as:
if not a_r <_r 0_r and not b_r <_r 0_r
then a_r *_r b_r = 0_r U { c_q *_q d_q | c_q in a_r and d_q in b_r }
if a_r <_r 0_r and b_r <_r 0_r then a_r *_r b_r = |a_r|_r *_r |b_r|_r
otherwise a_r *_r b_r = -_r (|a_r|_r *_r |b_r|_r)
There exists an alternative definition of R using Cauchy sequences: a Cauchy sequence is a s : N --> Q such that s(i_n) +_q((-1)_q *_q s(j_n)) can be made arbitrary near to 0_q for all sufficiently large i_n and j_n. We will define an equivalence relation ==_r on the set of Cauchy sequences as: r ==_r s iff r(m_n) +_q((-1)_q *_q s(m_n)) can be made arbitrary close to 0_q for all sufficiently large m_n. R = { [s]_r | s is a Cauchy sequence } . Note that this definition is close to ``decimal'' expansions.
Construction of C
C = R x R. We will refer to the elements of C by giving them a subscript _c. The elements of R can be embedded as follows: embed_r : R --> C such that embed_r(a_r) = (a_r,0_r). Furthermore we can define:
(a_r,b_r) +_c (c_r,d_r) = (a_r +_r c_r, b_r +_r d_r)
(a_r,b_r) *_c (c_r,d_r) = ((a_r *_r c_r) +_r -_r (b_r * d_r), (a_r *_r d_r) +_r (b_r *_r c_r))
There exists an elegant alternative definition using ideals. To be a bit sloppy: C = R [x]/< (x *_r x) +_r 1_r > , i.e. C is the resulting quotient ring of factoring ideal < (x *_r x) +_r 1_r > out of the ring R [x] of polynomials over R . The sloppy part is that we need to define concepts like quotient ring, ideal, and ring of polynomials. Note that this definition is close to working with i^2 = -1: (x *_r x) +_r 1_r = 0_r can be rewritten as (x *_r x) = (-1)_r.
Rounding things up
At this moment we don't have that N is a subset of Z , Z of Q , etc. But we can get the inclusions if we look at the embedded copies of N , Z , etc. Let
N' = ran embed_r o embed_q o embed_z o embed_n
Z' = ran embed_r o embed_q o embed_z
Q' = ran embed_r o embed_q
R' = ran embed_r
For these sets we have N' subseteq Z' subseteq Q' subseteq R' subseteq C. Furthermore these sets have all the properties that the ``informal'' numbers have.
You just did AJ
c wrote:
> If u all hate this topic so much then why don't u just leave and let
> someone who wants to ask a question have a go.
---
Here's another question:
Do you seriously think someone is going to ask you a question?
> If u all hate this topic so much then why don't u just leave and let
> someone who wants to ask a question have a go.
---
Here's another question:
Do you seriously think someone is going to ask you a question?
If u all hate this topic so much then why don't u just leave and let someone who wants to ask a question have a go.
Page: