Math from Words
Chapter Five - Binary Notation

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Definition - A text set is a set of text items.

The definition of the term text item requires a text item to be either a single character, or characters that are adjacent to each other, written and read from left to right. The characters of such a text item need not be of any particular kind. The special characters naming certain amounts and mentioned earlier called numerals include 0 (which represents the word none), or zero, and 1 (which represents the word one). Numerals may also be characters in a text item. Nothing precludes the use of numerals as characters in text items, or as single-character text items.

Notice that 0 and 1 are text items well precedented in naming whole numbers elsewhere. While any other character could represent none, or zero, and any other character could be used to represent a single thing or one, these text items are very familiar and used in the most common systems.

Below, there are three vertical successions of text items comprised only of the numerals 0 and 1. These can be used to represent whole numbers, although this text set is not the only set of text items that can be used to represent the whole numbers. These text items are binary text items that form a binary text set

```   0
1
10
11
100
101
110
111
1000
.
.
.
100010100110
100010100111
100010101000
100010101001
100010101010
100010101011
100010101100
.
.
.
11101111100001010
11101111100001011
11101111100001100
.
.
.

```

The following explains the order of these binary text items:

Definition - Character r is the rightmost zero (0) in text item E if and only if each following statement is true:

Character r is the numeral 0.

Character r is in text element E.

If q is in E, and q is a 0, then q is not to the right of r in E.

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

Definition - Character v is the rightmost one (1) in text item E if and only if each following statement is true:

Character v is the numeral 1.

Character v is in text item E.

If q is in E, and q is a 1, then q is not to the right of v in E.

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

Definition - Set B is the Binary Text Set for Whole Numbers if and only if each following statement is true:

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

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

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

Case I

Each character in text item j is the numeral 1.

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, have a one-to-one correspondence with the numerals in j that are each the numeral 1.

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Case I Example

``` 111
1000
```
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Case II

Text item j contains a numeral, s in j, that is the rightmost zero in j.

Text item k contains a numeral, u in k, that is the rightmost 1 in k.

The numerals that are the numeral 1 to the right of s, if any, in j, have a one-to-one correspondence with the numerals that are zero to the right of u, if any, in k.

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

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Case II Example

```100010100111
100010101000
```
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