Math from Words

Chapter Seven - Whole Numbers

And Hexadecimal Names

by

The hexadecimal character set includes the numerals together with the letters A, B, C, D, E, and F.

Below, there are three vertical successions of text items comprised only of the characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

0 1 2 3 4 5 6 7 8 9 A B C D E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C . . . 9A 9B 9C 9D 9E 9F 100 101 102 . . . 170199 17019A 17019B . . .

-----------------------------------------------

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

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

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

Circular text set Z = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F]

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 letter F.

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 letter F.

-----------------------------------------------

**Case A Example**

FF 100-----------------------------------------------

**Case B**

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

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

(Text item k, to be next in order from j in alpha subset T, must have a numeral u such that 1- 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 2- 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 F’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=37A, then Case B conditions are satisfied if and only if s=A, k=37B and u=B.

Example 2: If j=25FFF, then Case B conditions are satisfied if and only if s=5, k=26000 and u=6.

Example 3: If j=787FFFF, 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=1FFFFF, then Case B conditions are satisfied if and only if s=1, k=200000 and u=2.

-----------------------------------------------

The decimal text set for whole numbers is unbounded. It provides text items for every whole number in any set of whole numbers, however large. It is a type of unbounded set, but it does not define *unbounded set* for all types of sets that might involve an unending succession of elements.

**Contact**

https://www.futurebeacon.com/jamesadrian.htm