Math from Words

Chapter Six - Decimal Notation

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The special characters called *numerals* include 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Numerals may also be characters in a text item.

Below, there are three vertical successions of text items comprised only of the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These can be used to represent whole numbers. These text items are *decimal text items* that form a *decimal text set*

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 . . . 95 96 97 98 99 100 101 102 . . . 170199 170200 170201 . . .

**Definition** - S is a *Circular Text Set* if and only if each following statement is true:

S is a set.

S contains at least a pair of elements.

Each element in S is a text item.

Each element y, in S, is asymmetrically associated with some element x, in S, and is next in order from that x, in S.

If element q, in S, is next in order from p, in S, then q is not next in order from any element other than p, in S.

If q, in S, is next in oder from p, in S, then no element other than q is next in order from p.

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**Notation** - These brackets enclose the elements of a Circular Text Set: [A, B, C, D, E, F, G]. The leftmost element listed is arbitrary. The following is the same circular text set: [D, E, F, G, A, B, C].

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**Definition** - Set T is the *Decimal Text Set for Whole Numbers* if and only if each following statement is true:

T is a text set. T contains an unending succession of elements following the first element in T.

The first text item in T is the text item 0.

Circular text set Z = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Text item k is next in order from j if and only if either the statements in Case A, or the statements in Case B are true:

**Case A**

Each character in text item j is the numeral 9.

The leftmost character in k is the numeral 1.

Each other character in k is the numeral 0.

The numerals in k that are each the numeral 0 and have a one-to-one correspondence with the numerals in j that are each the numeral 9.

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**Case A Example**

99 100-----------------------------------------------

**Case B**

Text item j contains a numeral, s in j, that is the rightmost non-9 in j.

(Presuming that text item j is not all 9’s, since Case A and Case B are mutually exclusive, we know there is a right-most non-9 in j. If there are any numerals to the right of s in j, then they are all 9’s.)

Text item k contains a numeral, u in k, such that the numerals to the right of u, if any, are 0's that have a one-to-one correspondence with the numerals, if any, to the right of s in j. Note that any numerals to the right of s in j are 9’s.

(Text item k, to be next in order from j in T, must have a numeral u such that the numerals to the right of u in k, if any, have a one-to-one correspondence with the numerals, if any, to the right of s in j; and numeral u is such that those numerals to the right of u, in k, are all 0’s.)

The numeral u is next in order from s in circular text set Z.

(When changing from text item j to text item k, there is roll-over of 9’s on the right of s, to 0’s on the right of u, unless s is the rightmost numeral of j, in which case u is the rightmost numeral of k. Whether they are rightmost in their respective text-objects or not, it is necessary that u be next in order from s in the circular text set Z, for k to be next in order from j in alpha subset T.)

Text item X, comprised of the characters to the left of s, if any, in j, and the text item Y, comprised of the characters to the left of u, if any, in k, are identical.

**Case B Examples**

Example 1: If j=374, then Case B conditions are satisfied if and only if s=4, k=375 and u=5.

Example 2: If j=25999, then Case B conditions are satisfied if and only if s=5, k=26000 and u=6.

Example 3: If j=7879999, then Case B conditions are satisfied if and only if s=7, k=7880000 and u=8.

Example 4: If j=8, then Case B conditions are satisfied if and only if s=8, k=9 and u=9.

Example 5: If j=199999, then Case B conditions are satisfied if and only if s=1, k=200000 and u=2.

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