Future Beacon



        
Chapter D

Names for Numbers





Introduction


I feel that it was best to define integers before defining any arithmetic.   In this development, operations like addition and multiplication are not defined at the same time that integers are defined (as operations together with a set) as they are elsewhere.   We have the independent existence of integers defined prior to defining arithmetic operations.   Arithmetic operates on things that have been separately and previously defined.

Like operations, the defining of naming systems for numbers has needed to wait until after the establishment of the meaning of the numbers. As stated earlier, a name is a word, phrase, character, symbol, or mark that is used to refer to something other than itself, and a numeral is a single-character symbol used singly or in combinations only to form names. N is a number if and only if N is a non-set element in an ordered set of non-set elements, and N is an integer if and only if N is an element in an integer set.

For integers to be useful for counting, they must be capable of being applied to any assemblage of discrete things. Not every such assemblage is a set, as will soon be evident; therefore let's take another a look at what it means for T to contain N elements: T contains N elements if and only if T is a set and the size of T equals N. We need a meaning that is more general:

Definition - T contains N things if and only if the following statements are true:
W is the set of whole numbers in integer set Q.
S is the string from 0 to N in W.
T is an assemblage of discrete things.
Element r is in C if and only if r is a unique name for each instance of a thing in T.
C has a one-to-one correspondence with S.
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If T contains a ball, a sandwich, a nail and two wrenches, it contains two things that, initially, have the same name, but they must be called wrench A and wrench B, or in some similar way given unique names for the purpose of inclusion in set C. Sets may not contain duplicate names. Each named element in a set has a name that is unique among the elements named in that set. If the five statements made in the definition are true, then there is the required one-to-one correspondence between C and S, and there are indeed N things in T.

By defining numbers as elements in an ordered set, they are required to have names unique in the set (by the definition of the term set).   Names can be assigned to numbers in various ways.   Integers being elements in an integer set is consistent with the notion that integers are not mere names. They are not simply the names that they may be known by.   The integer that is referred to by the name 5 in the naming system that uses 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 as numerals is the same integer that is referred to by the name 101 in the naming system that uses only 0 and 1 as numerals.   An integer is not its name, but it is an element belonging to a set.   Naming schemes can be defined and redefined without altering the meaning of the term integer and without altering the meaning of any particular integer.

The integers of integer set Q can be given names such that if x is associated with y in Q, y has a name that can be surmised from the name assigned to x. This is done by using a fixed set of numerals (a numeral set) in a systematic way.

To develop names for numbers that are comprised of numerals, and to do so with a minimum of confusion, this chapter displays numerals in italics.

Definition - T is a numeral set if and only if the following statements are true:
Q is an integer set.
T is an ordered set of numerals.
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Notice that the numerals 0 and 1 have each already been separately used to stand for an integer in previously presented definitions. These are widely used as numerals and also widely used as integers.

There are many naming systems. I will not define them all. The development of some naming systems for the integers in an integer set begins here with these common and popular numeral sets:

Definition - Numeral set b is the binary numeral set if and only if b = (0, 1)
Definition - B is a binary numeral if and only if B is an element in the binary numeral set.
Definition - Numeral set d is the decimal numeral set if and only if d = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
Definition - D is a decimal numeral if and only if D is an element in the decimal numeral set.
Definition - Numeral set h is the hexadecimal numeral set if and only if h = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F)
Definition - H is a hexadecimal numeral if and only if H is an element in the hexadecimal numeral set.

Notice that these numeral sets are ordered sets.

Definition - S is an n-tuple if and only if the following statements are true:
S is a list of names of items such that if e such an item then e is an element in some set.
Integer n is a positive integer.
The n in the term n-tuple specifies the number of items listed in S.
A single item j listed in S, and only j listed in S, is such that no other item in S is associated with it.
A single item k listed in S, and only k listed in S is not associated with any item in S.
Each item p listed in S other than item k is associated with a single other item listed in S, and only that single other item.
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Notice that an n-tuple may be contained in another item, and may be an element in a set, and may be an element in an ordered set.

An n-tuple is not a set. It may list a name more than once. Each instance of a name in S has a unique place in the order of S. Each name listed in an n-tuple is required to be the name of an element in some entity other than n-tuple S, and that entity must be a set.

Definition - Item e is an element in n-tuple S if and only if e is an instance of a name being listed in S.
Definition - Item e is in n-tuple S or e is a member of n-tuple S if and only if e is an element in S.
Definition - Item s is the starting element of n-tuple R if and only if the following statements are true:
R is an n-tuple.
No item listed in R is associated with s.
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Definition - Item z is the ending element of n-tuple S if and only if the following statements are true:
S is an n-tuple.
Element z in S is not associated with any item listed in S.
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Definition - S is an n-tuple string in T if and only if the following statements are true:
T is an n-tuple.
S is an n-tuple.
Each element in S is also an element in T.
If x and y are in S, and x is associated with y in T, then x is associated with y in S.
If p is associated with q in T, and p is not is S, and q is in S, then there is no element y in S such that y is associated with q.
If q is the starting element of T, and q is in S, then there is no element y in S such that y is associated with q.
If p is associated with q in T, and q is not in S, and p is in S, then p is not associated with any element in S.
If p is the ending element of T, and p is in S, then p is not associated with any element in S.
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Definition - Element s in S is the starting element of n-tuple string R if and only if the following statements are true:
T is an n-tuple.
R is an n-tuple string in T.
There is no element in R that is associated with s.
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Definition - Element z in S is the ending element of n-tuple string S if and only if the following statements are true:
T is an n-tuple.
S is an n-tuple string in T.
Element z in S is not associated with any element in S.
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Definition - W is a written name if and only if the following statements are true:
W is an n-tuple.
Q is a numeral set.
W is a written entity comprised exclusively of numerals belonging to numeral set Q.
The numerals of W are arranged horizontally without intervening spaces.
If s and t are in W, and s is associated with t in W, then t is written immediately to the right of s.
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Notice that a written name is not a set. For example, there is no prohibition against a numeral appearing more than once in a written name.



Binary Names


Definition - W is a binary name if and only if the following statements are true:
W is a written name.
W is comprise exclusively of numerals belonging to the binary numeral set.
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Binary Name Examples - 100101, 1, 101, 0, 11000110101.

Definition - Y is a binary string if and only if the following statements are true:
Q is an integer set.
P is the set of positive integers in Q.
Y is an ordered set or a set containing a single element.
If element s is in Y, then s is in P.
The ending element of Y is equal to 1.
If s and t are in Y, and s is associated with t in Y, then s = 2 ⋅ t.
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Binary String Examples - Y = {1}; Y = (2, 1); Y = (2 ⋅ 2, 2, 1); Y = (2 ⋅ 2 ⋅ 2, 2 ⋅ 2, 2, 1)

Definition - Sn is the non-negative binary name for whole number z if and only if the following statements are true:
Q is an integer set containing integers j and k.
Sn is a binary name containing j numerals where j is an integer.
Y is a binary string containing j elements where j is an integer.
The starting element of Sn is s.
The starting element of Y is t.
K is a set.
Integer N is in K if and only if X is in Sn; and the string from 0 to y in Q has a one-to-one correspondence with the string from 0 to X in the binary numeral set; and the n-tuple string from s to X in Sn contains k elements; and the string from t to Yc in Y contains k elements; and N = Yc ⋅ y.
Whole number z = the sum of all of the whole numbers N in K.
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Definition - The negative binary name of negative integer z is Minus Sn or -Sn if and only if the following statements are true:
Q is an integer set.
Negative integer z and positive integer y are in Q.
Sn is the non-negative binary name of positive integer y.
Integer y and integer z are symmetric.
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Definition - Sn is the binary name for integer z if and only if either z is a whole number and Sn is the non-negative binary name of whole number z; or z is a negative integer and Sn is the negative binary name of negative integer z.
Definition - B is a bit if and only if B is a binary numeral of a binary name.
Definition - Sn is a base two name or a base 2 name if and only if Sn is a binary name.

Notice that the notation introduced in chapter B for the number two was used in the examples of binary strings, but that notation is not used in base two (2 becomes 10).



Decimal Names


Definition - W is a decimal name if and only if the following statements are true:
W is a written name.
W is comprise exclusively of numerals belonging to the decimal numeral set.
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Decimal Name Examples - 70432, 5, 16, 1, 986, 0, 12749310253.

Binary names (the only systematic names we have so far) are used in the following definitions concerning decimal names for numbers.

Definition - Y is a decimal string if and only if the following statements are true:
Q is an integer set.
P is the set of positive integers in Q.
Y is an ordered set or a set containing a single element.
If element s is in Y, then s is in P.
The ending element of Y is equal to 1.
If s and t are in Y, and s is associated with t in Y, then s = 110 ⋅ t where 110 is an integer named with a binary name.
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Decimal String Examples - Y = {1}; Y = (110, 1); Y = (110 ⋅110, 110, 1); Y = (110 ⋅ 110 ⋅ 110, 110 ⋅ 110, 110, 1)

Definition - Sn is the non-negative decimal name for whole number z if and only if the following statements are true:
Q is an integer set containing integers j and k.
Sn is a decimal name containing j numerals where j is an integer.
Y is a decimal string containing j elements where j is an integer.
The starting element of Sn is s.
The starting element of Y is t.
K is a set.
Integer N is in K if and only if X is in Sn; and the string from 0 to y in Q has a one-to-one correspondence with the string from 0 to X in the decimal numeral set; and the n-tuple string from s to X in Sn contains k elements; and the string from t to Yc in Y contains k elements; and N = Yc ⋅ y.
Whole number z = the sum of all of the whole numbers N in K.
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Definition - The negative decimal name of negative integer z is Minus Sn or -Sn if and only if the following statements are true:
Q is an integer set.
Negative integer z and positive integer y are in Q.
Sn is the non-negative binary name of positive integer y.
Integer y and integer z are symmetric.
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Definition - Sn is the decimal name for integer z if and only if either z is a whole number and Sn is the non-negative decimal name of whole number z; or z is a negative integer and Sn is the negative decimal name of negative integer z.
Definition - D is a digit if and only if D is a decimal numeral of a decimal name.
Definition - Sn is a base ten name or a base 10 name if and only if Sn is a decimal name.



Hexadecimal Names


Definition - W is a hexadecimal name if and only if the following statements are true:
W is a written name.
W is comprise exclusively of numerals belonging to the hexadecimal numeral set.
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Hexadecimal Name Examples - A50B9, 5, 16, 1, 986, 0, 12F493D0258.

Decimal names are used in the following definitions concerning hexadecimal names for numbers.

Definition - Y is a hexadecimal string if and only if the following statements are true:
Q is an integer set.
P is the set of positive integers in Q.
Y is an ordered set or a set containing a single element.
If element s is in Y, then s is in P.
The ending element of Y is equal to 1.
If s and t are in Y, and s is associated with t in Y, then s = 16 ⋅ t where 16 is an integer named with a decimal name.
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Hexadecimal String Examples - Y = {1}; Y = (16, 1); Y = (16 ⋅ 16, 16, 1); Y = (16 ⋅ 16 ⋅ 16, 16 ⋅ 16, 16, 1)

Notice that the integer named with a decimal name can only be named with a hexadecimal name after the definition of hexadecimal name is presented. Until then, the decimal name refers to the same integer.

Definition - Sn is the non-negative hexadecimal name for whole number z if and only if the following statements are true:
Q is an integer set containing integers j and k.
Sn is a hexadecimal name containing j numerals where j is an integer.
Y is a hexadecimal string containing j elements where j is an integer.
The starting element of Sn is s.
The starting element of Y is t.
K is a set.
Integer N is in K if and only if X is in Sn; and the string from 0 to y in Q has a one-to-one correspondence with the string from 0 to X in the decimal numeral set; and the n-tuple string from s to X in Sn contains k elements; and the string from t to Yc in Y contains k elements; and N = Yc ⋅ y.
Whole number z = the sum of all of the whole numbers N in K.
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Definition - The negative hexadecimal name of negative integer z is Minus Sn or -Sn if and only if the following statements are true:
Q is an integer set.
Negative integer z and positive integer y are in Q.
Sn is the non-negative hexadecimal name of positive integer y.
Integer y and integer z are symmetric.
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Definition - Sn is the hexadecimal name for integer z if and only if either z is a whole number and Sn is the non-negative hexadecimal name of whole number z; or z is a negative integer and Sn is the negative hexadecimal name of negative integer z.
Definition - H is a hex digit if and only if H is a hexadecimal numeral of a hexadecimal name.
Definition - Sn is a base sixteen name or a base 16 name if and only if Sn is a hexadecimal name.



Base b Names for Integers


In a particular style, three kinds of names for integers have been defined (base two, base ten and base sixteen). This section will generalize this style of naming system to base b, where b is an integer equal to or greater than two. Signed-digit bases and other unusual naming schemes will not be defined in this section.

Definition - A base b numeral set is an ordered set of symbols containing b elements.
Definition - A base b numeral is an element in a base b numeral set.
Definition - W is a base b name if and only if the following statements are true:
W is a written.
W is comprise exclusively of numerals belonging to a base b numeral set.
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Definition - Y is a base b string if and only if the following statements are true:
Q is an integer set.
P is the set of positive integers in Q.
Element b is in P.
Y is an ordered set or a set containing a single element.
If element s is in Y, then s is in P.
The ending element of Y is equal to 1.
If s and t are in Y, and s is associated with t in Y, then s = b ⋅ t.
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Definition - Sn is the non-negative base b name for whole number z if and only if the following statements are true:
Q is an integer set containing integers j and k.
Sn is a base b name containing j numerals where j is an integer.
Y is a base b string containing j elements where j is an integer.
The starting element of Sn is s.
The starting element of Y is t.
K is a set.
Integer N is in K if and only if X is in Sn; and the string from 0 to y in Q has a one-to-one correspondence with the string from 0 to X in the binary numeral set; and the n-tuple string from s to X in Sn contains k elements; and the string from t to Yc in Y contains k elements; and N = Yc ⋅ y.
Whole number z = the sum of all of the whole numbers N in K.
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Definition - The negative base b name of negative integer z is Minus Sn or -Sn if and only if the following statements are true:
Q is an integer set.
Negative integer z and positive integer y are in Q.
Sn is the non-negative base b name of positive integer y.
Integer y and integer z are symmetric.
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Definition - Sn is the base b name for integer z if and only if either z is a whole number and Sn is the non-negative base b name of whole number z; or z is a negative integer and Sn is the negative base b name of negative integer z.

Notation - To designated the base of a name, square brackets are used this way:

2A[base 16] = 42[base 10] = 101010[base 2]

Notice that the bracketed subscripts use base ten names. In the absence of this type of subscript or an explicit statement specifying the base of a given section or document, base ten is to be assumed.

Verbal Names - The verbal names for the numerals are the same ones used in English: zero, one, two, three, four, five, six, seven, eight, nine, A, B, C, D, E, and F. Five may now be freely used in place of 5 and similarly for the rest of the numerals.