Math from Words

Chapter Five - Binary Notation

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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.

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**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.

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**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-----------------------------------------------

**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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