The common language can generate misimpressions about numbers. Each integer is an element in a succession. Element x may be separated from zero, for instance, by many or few associations relative to that same kind of separation in the case of another element, y. This is the essential difference between any pair of integers. We often speak of "big numbers" and "bigger numbers" as though big numbers were somehow larger than smaller numbers. Element y is said to be greater than element x whenever element y is next in order from element x, but the common association between

The term

The concept of the one-to-one correspondence has facilitated comparison of sets. We may recite the names of positive integers, in their defined order, as we place a sting of these numbers in a one-to-one correspondence with objects. Integers can be used for counting the size of sets:

We use the term

Since a number is a non-set element in an ordered set containing only non-set elements, the term

Although it exists in the common language, we have not made a formal connection between "1 thing" and "a single thing" or "0 things" and "nothing" or "2 things" and "a pair of things." So here it is:

Likewise (because of the meaning of

Accountants can describe both dollar profits, and dollar losses with integers. They name negative amounts with negative integers. There are points of possible confusion. Any string of negative integers (and any string is a set) has a size. This size is always represented by a whole number. There are no sets defined as containing a negative number of elements. There are no anti-elements to balance actual elements. Each association between any given element and the element that is next in order from it is part of a succession of such associations that does not proceed consistently toward zero or away from zero. The succession proceeds away from zero within the positive integers and toward zero within the negative integers. Graphically, if an element p is associated with an element q, element q is just to the right of p whether p is negative, zero or positive. In the graphical sense, the positive direction is toward the right, and the negative direction is toward the left. The accountant assigns negative numbers to debts because then the sum will reveal the balance. The physicist assigns negative numbers to opposing forces so that the sum of the forces will reveal the balance. The description of kinds or directions as being positive or negative is up to those applying the math; but given these choices, the arithmetic of integers is determined by the mathematical definitions involved. Everybody applying math anticipates that 7 minus 12 is equal to a negative number, and that -2 plus 8 is a positive number; but whether a negative result means a debt, or a move downward, or something else is not something that the meanings defined in the arithmetic itself can determine.

While the size of sets cannot be defined as negative, the negative integers are non-set elements that have been defined. Integers, unlike set sizes, have both absolute value and sign. Elements in a set can be counted; but some amounts are not described by counting the elements in a set. We may, however, quantify an account balance or the sum of various forces by assigning a number, whether positive, zero or negative. However, this set-size restriction should not distract us from the realization that sets have provided a path to the creation of integers. In turn, integers provide us with a means of counting and quantifying all kinds of things in the world.

Now we need a way to deal with parts of things. A tomato may weight only part of a pound or cost only part of a dollar. The business of counting and quantifying often requires distinctions other than "a single thing" or "a pair of things" or "many things." Quantifying goes beyond whole numbers to negative numbers, and soon to numbers that can describe parts of things. The remainder of the present chapter is devoted to defining such numbers.

M is an ordered set.

T is an ordered set of strings.

Every element in T is a string in M.

Every element in M belongs to one and only one string in T.

If S1 and S2 are strings in T, then S1 and S2 have a one-to-one correspondence.

If S1 and S2 are strings in T, then the intersection of S1 and S2 is empty.

The concatenation of all of the strings in T is equal to M.

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C is an ordered set and V is an ordered set.

Each element in V is a string.

If A is associated with B in V, then the ending element of A in C is associated with the starting element of B in C.

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Q is an integer set.

M is an ordered set.

M is not an integer set.

The intersection of Q and M is empty.

T is a partition of M.

V is an ordered set of strings such that if string A is in V, then A is the concatenation of x in Q and S in T.

If A is the concatenation of x in Q and S in T, and B is the concatenation of y in Q and R is T, and A is associated with B in V, then x is associated with y in Q and S is associated with R in T.

C is the concatenation of all of the elements in V.

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..........-3..........-2..........-1..........0..........1..........2..........3..........

The integers along this line are the integers of Q, the integer set interleaved in C.

Unless stated otherwise, the integers referred to below are named in base 10.

Three terms have been defined in the above definition. The same set of statements could have been repeated to form the definition of each term separately. This as was done earlier in the case of a shorter definition which gave us the integers and the major subsets of integers in Chapter II.

Integer set Q is a subset of counting set C. Q is interleaved in C in such a way as to include one integer from Q at regular intervals along the succession of elements in C. These integers from Q appear in C in the same order that they appear in Q from the starting element of Q to the ending element of Q. The elements between integers from Q in C are non-set elements that come from ordered set M. The elements between integers from Q in C appear in the same order that they do in M.

S is an ordered set of non-set elements.

S contains elements x, y, and z.

Element y is in the string from x to z

Element y is not equal to x, and element y is not equal to z.

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Q is an ordered set.

Elements x and y are in Q.

S is a string in Q.

Each element in S is between x and y in Q.

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C is a counting set.

Q is an integer set interleaved in C.

Element n is in Q.

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An integer in a counting set is not associated with the integer next in order within an integer set. Each integer in a counting set is associated with a non-integer element in the counting set and some non-integer element is associated with that integer. If x is associated with y in integer set Q, then a succession of non-integer elements intervene between x and y in counting set C. These elements have a definite order because they are strings obtained from ordered set M. The purpose of this structure is to provide numbers (non-set elements in an ordered set of such elements) that might be used to represent amounts different from those represented by integers.

C is a counting set.

Q is the integer set interleaved in C.

Element x is an integer in counting set C.

Element x = 0 in C.

Element x in C is associated with element s in C.

Element e is the ending element of C.

R is the string from s to e in C.

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Notice that s is the starting element of R.

C is a counting set.

Q is the integer set interleaved in C.

Element x is an integer in counting set C.

Element x = 0 in C.

Element e in C is associated with element x in C.

Element s is the starting element of C.

L is the string from s to e in C.

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Notice that e is the ending element of L.

C is a counting set containing m and n.

S is a string in C such that n is the starting element of string S, and 0 is the ending element of S.

W is a string in C such that 0 is the starting element of string W, and m is the ending element of W.

S has a one-to-one correspondence with W.

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C is a counting set containing x.

If x is in the set of positive counting numbers of C, or of x = 0 in C, then v = x.

If x is in the set of negative counting number of C, then x and v are symmetric.

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The absolute value of x is non-negative whether x is positive, zero or negative.

C is a counting set containing x and y.

Neither x nor y is positive; or, neither x nor y is negative.

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C is a counting set containing x and y.

Integer j is the starting element of C, and k is the ending element of C.

If y is not negative, the string in C from x to z has a one-to-one correspondence with the string in C from 0 to y.

If y is not positive, the string in C from z to x has a one-to-one correspondence with the string in C from y to 0.

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C is a counting set and Q is the integer set interleaved in C.

S is a string in C.

Elements n and m are integers in counting set C, and n is associated with m in Q.

Element n is the starting element of S.

Element z is the ending element of S.

Element z is associated with m in C.

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Integer set Q is interleaved in C and contains s.

The size of each unit string in C is s.

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C is a counting set.

T is a unit string in C.

Element x is the counting set integer in T.

If y is not in T, then y is not in S.

If m is in T and m is not equal to x, then m is in S.

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In this explanation, the term

An integer in a counting set is followed by a collection of non-integer elements. That integer together with the other elements (the string mantissa), comprises the a unit string. This leaves open the question of how many non-integer elements are in the unit string.

The accountant, wishing to quantify amounts in dollars and pennies (dollars and cents), may appreciate a counting set that has 100 elements per unit string. After all, there are 100 pennies (100 cents) in a dollar. However, there are 16 ounces in a pound and there are 12 inches in a foot. A surveyor who is paid in dollars and cents but who must measure distances in feet and inches might prefer a counting set that has a total of 120 elements per unit string. The grosser might argue that standard unit string should be 16 times this number. One could imaging a unit string size that is the product of every integer up to1000, or a very much greater number, just so that nobody is left out. Some do think of parts that way, conceiving of a number line that has every possible relationship as an abstract given. Engineers and scientist tend to select a unit string size that is a high exponent of a small number like 16, or 10 or 2. This makes the quantifying of some things approximate, but approximations can be arbitrarily accurate.

Here is an example counting set:

C is a counting set and Q is the integer set interleaved in C.

S is a unit sting in C.

N is a positive integer in Q.

The size of S is 10

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To generalize this concept, we have the following:

C is a counting set and Q is the integer set interleaved in C.

S is a unit sting in C.

N is a positive integer in Q.

Positive integer b in Q is greater than 1.

The size of S is b

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Consider 31 cents. This would be somewhere between no dollars and 1 dollar. If b = 10 and N = 2, then the size of the unit string is 100. The 31st element in the unit string having a starting element equal to zero would be a good number to represent 31 cents. There are always b

C is a counting set and Q is the integer set interleaved in C.

Element x is in C.

Element j is an integer in counting set C.

If j is associated with n in Q, then either x is between j and n in C or x = j.

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C is a counting set and Q is the integer set interleaved in C.

C is a base b counting set and the elements in Q are given base b names.

The size of the unit string in C is b

Element x is in C.

Elements j and k are in Q.

Element j is an integer in counting set C.

The greatest integer in counting set C that is not greater than x is j.

The size of the string from 1 to k in Q equals the size of the string from j to x in C.

Y is the base b name of j.

If x equals j, then Z is a name comprised of N numerals, all of them

If x is not equal to j, then Z is the base b name of k.

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Consider three dollars and 31 cents. The base 10 name for 31 aptly names the number of parts because any numeral except the numeral on the extreme right represents 10 times the amount as would the same numeral immediately to its right. This holds all the way to the left as j is reached. As j = 3, it represent 10 times the amount that the 3 in 31 represents. The dot only tells us where 1 is - that is, where whole things end and parts begin. In this case, X.Y is 3.31 (the amount of dollars, or the number of dollars).