Representation of Numbers (Part 2)

Representation of Numbers (Part 2)

INTERPRETATION OF WORDS IN WHOLE BINARY

In the previous section, we addressed the syntax (alphabet, what constitutes a word) or structure of the language, WHOLE BINARY. Now we'll define a possible interpretation or meaning of the words of the language (semantics).

Now let's try representing some numbers in WHOLE BINARY by starting with zero and representing each successive number with as few letters as possible. Remember however that we can only assign numbers to words because the other strings in the language are supposed to be gibberish. To do this, we'll adopt an assembly line approach of sorts as described below:

Every empty column or column containing 0 has zero packages. Every column containing a 1 contains one package.
When we increase the number by one, a package rolls up on the rightmost column.
When a column receives a package, it performs one of two actions:
- If it is empty or has a 0 in its column, it is not in possession of any packages so it takes that one received package and increases its number to 1 to represent that it is now holding one package.
- If it has a 1 in its column, it already holds 1 package and since there is no letter 2 in WHOLE BINARY and 0 represents the absence of any packages, it has no more space. Therefore, it tapes the two packages together and gives it to its closest neighbor on its left and then resets its column to 0 since it gave away all of its packages.

Just as in a real assembly line, each column does not concern itself with what's going on in other columns and simply performs its assigned task repeatedly. Now let's see this assembly line in (static) action when we add one to the number 111. A blurred image represents the absence of a package from that column and is represented by 0. Similarly, the filled-in package represents the column's possession of a package and is represented by 1.

Example: Adding one to 111 --> 1000

(1) A package rolls on to the conveyor belt.

(2) The rightmost column receives a package, but already has one. Therefore, it can't hold anymore so it tapes the two packages together to give to its closest neighbor on its left.

(3) The second column from the right is also already in possession of a package so it too can hold no more. It therefore tapes the packages together so that it can give it to its closest neighbor on its left.

(4) The third column from the right is also already in possession of a package so it too can hold no more. It therefore tapes the packages together so that it can give it to its closest neighbor on its left.

(5) The leftmost column has no packages so it accepts the package from its closest neighbor on the right. The process is now complete.

Now let's see this assembly line in action for the sixteen numbers zero through fifteen. For the below table, imagine a 1 being fed into the rightmost column of each row, the above procedure being performed, and the successive row representing the final result of the procedure.

Numbers in WHOLE BINARY
zero:					0
one:					1
two:				1	0
three:				1	1
four:			1	0	0
five:			1	0	1
six:			1	1	0
seven:			1	1	1
eight:		1	0	0	0
nine:		1	0	0	1
ten:		1	0	1	0
eleven:		1	0	1	1
twelve:		1	1	0	0
thirteen:		1	1	0	1
fourteen:		1	1	1	0
fifteen:		1	1	1	1
sixteen:	1	0	0	0	0

OBSERVATIONS

We now make some observations about the columns:

The rightmost column holds the number of ones in its own package.
The package of a column (that has a neighbor on its right) holds two packages of its closest neighbor on its right.

From the above procedure, do you see why these observations are true? Before you think about it, let's examine a concrete example to see what is being claimed.

Example: 1101 contains one individually wrapped one, and the package in the leftmost column of 1101 contains twice as much as the its closest neighbor on the right.

The intuitive reasoning for these observations is explained below:

The above procedure has 0 representing no packages and 1 representing one package. Since the rightmost column only receives a delivery when we increase the number by one, we can conclude that the rightmost column represents the number of ones in their own package. We'll therefore call the rightmost column the ones place in WHOLE BINARY.
Since a column tapes two of its packages together and gives it to its closest left neighbor, a 1 in every column EXCEPT for the rightmost one represents the possession of two taped packages from its closest right neighbor. We'll therefore call every column with a neighbor on its right the (double the closest neighbor on its right)s place in WHOLE BINARY.

Example: Let's determine the place name of the third column from the right in WHOLE BINARY. The rightmost column is the ones place. Therefore, the second column from the right is the (double one)s place or twos place. Then, the third column from the right is the (double two)s place or fours place.

We can use the observations that the rightmost column represents the number of ones and that every other column is double its closest right neighbor recursively (getting the place name of a column by getting the names of all the names of the column places to the right of it as in the above example) to determine each column's place name. Below is a table that represents the results of this procedure for the first 11 columns.

WHOLE BINARY COLUMN PLACE NAMES
one thousand twenty-fours	five hundred twelves	two hundred fifty-sixes	one hundred twenty-eights	sixty-fours	thirty-twos	sixteens	eights	fours	twos	ones

Exercise 1.2.1: Extend the top table that represents the numbers from zero (0) through sixteen (10000) so that it also represents the numbers seventeen through thirty-two in WHOLE BINARY.

Exercise 1.2.2: What are the relationships between the second column and the number of two-letter strings, the third column and the number of three-letter strings, and the fourth column and the number of four-letter strings that can be expressed with WHOLE BINARY? Do you see a pattern?

Exercise 1.2.3: How many one-letter words and strings, two-letter words and strings with two or less letters, three-letter words and strings with three or less letters, and four-letter words and strings with four or less letters in WHOLE BINARY? Do you see a pattern?

Representation of Numbers (Part 3)

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