Representation of Numbers (Part 3)

Representation of Numbers (Part 3)

THE "NATURAL" INTERPRETATION

Technical Point: In the previous section, we explored a possible interpretation of the language, WHOLE BINARY. Although there are an endless number of possible interpretations for this language, we've been using WHOLE BINARY which in itself has no meaning to also include the natural interpretation presented in the last section. We will continue with this slight abuse of notation to avoid introducing further definitions. This notational abuse is actually very common as we often associate a language with its natural meanings (e.g. associating the English language with the "natural" meanings of words such as 'water' being a (clear) liquid).

The important point here is that a language in itself does not come equipped with an interpretation. Why is this interpretation of WHOLE BINARY so "natural" then? Well, natural here means commonplace rather than what would be your first intuitive idea for a representation of a number. In fact the idea of different columns representating different numbers (weighted columns) was far from the first representation. However, as you proceed through this tutorial series, you will see why this type of interpretation of is very useful.

INTERPRETING NUMBERS IN WHOLE BINARY

In the previous section, we built up numbers by increasing by one each time. However, we want to use the observations from before to interpret numbers in WHOLE BINARY so that we don't have to increase by one starting from zero until we reach the number that we're trying to interpret. Let's consider an example.

Example: To interpret 101001 in WHOLE BINARY, we put it in Section 2's concluding table that was built up from the observations:

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

1 0 1 0 0 1

From the table we can see that we have one package of ones, one package of eights, and one package of thirty-twos, so by JOINING THEM TOGETHER, we have thirty-two joined with eight joined with one for a total of forty-one (thirty-two joined with eight is forty or the eighth consecutive number after thirty-two by counting by ones: (1) thirty-three, (2) thirty-four, (3) thirty-five, (4) thirty-six, (5) thirty-seven, (6) thirty-eight, (7) thirty-nine, (8) forty, and one jointed together with forty is the next consecutive number after forty, fourty-one). Therefore 101001 is interpreted as forty-one in WHOLE BINARY.

Now let's make a few more key observations about WHOLE BINARY:

A word consisting of all 1's is the largest number that can be represented with that many letters. (e.g. 111 is the largest three-letter word in WHOLE BINARY.) Also a non-zero word consisting of EXACTLY one 1 (must be in the leftmost column since all non-zero words must have a 1 in its leftmost column) is the smallest number that can be represented with that many letters. (e.g. 1000 is the smallest non-zero four-letter word in WHOLE BINARY.)
The more letters a WORD in WHOLE BINARY contains, the LARGER the number it represents.
A word consisting of all 1's is ONE LESS than the word containing ONE MORE LETTER than it and containing EXACTLY ONE 1 (in the leftmost column). (e.g. 111 is ONE LESS than 1000 in WHOLE BINARY.)

The reasons these properties hold true are as follows:

All strings/words in WHOLE BINARY consist solely of the letters 0 and 1. A 0 represents the absence of a package and 1 represents the presence of a package, and packages only increase the number. Therefore, a word consisting of all 1's is the largest number that can be represented with that many letters. Similarly, other than 0, every word in WHOLE BINARY must contain at least one 1 (the LEFTMOST or LEADING letter) so that a word consisting of EXACTLY one 1 is the smallest non-zero number that can be represented with that many letters.
Recall the assembly line diagram in the previous section. Packages roll on the conveyor belt from right to left so once a column receives one, it will only give it up if it receives another one. However, then it tapes the two packages together and gives it to its closest neighbor on its LEFT. Its closest neighbor will then either receive it and update its column number to 1 to represent that it contains a package, or if it already contains a package, then it will tape the two packages and give it to its closest neighbor on the LEFT. Eventually there will be a column that is in possession of no packages (empty or contain a 0) so that it'll receive a package and update its column to 1 to represent that it is now in possession of a package. Since each column gives packages to columns to the LEFT of it, the column that eventually ends up having a 1 will be to the LEFT of the first column giving up its package. Therefore, by increasing a number, we will always wind up with a number with AT LEAST AS MANY LETTERS as the original one. Consequently, the more letters a WORD in WHOLE BINARY contains, the LARGER the number it represents.
A couple of ways to understand the reasoning behind this third observation are as follows:
- We know that all words with EXACTLY ONE 1 must have the 1 appear in the leftmost column since all non-zero words in WHOLE BINARY must have a 1 there. Then think about the assembly line and the diagram in the previous tutorial to see what happens when we increase a number represented with all 1's in WHOLE BINARY by ONE.
- Since a word consisting of all 1's is the LARGEST number that can be represented with that many letters, a word consisting of EXACTLY ONE 1 is the SMALLEST number that can be represented with that many letters, LONGER words represent LARGER numbers, and numbers are increased by ONE each time in the assembly line, the LARGEST NUMBER we can represent with ONE LESS letter (a string/word of all 1's in WHOLE BINARY) MUST BE ONE LESS than the SMALLEST number WITH ONE MORE letter (a string/word in WHOLE BINARY with 1 in its leftmost column and 0 everywhere else) in WHOLE BINARY.

Example: What number does 1111111 represent in WHOLE BINARY? Again we can use our table, but this time let's use the observation above to figure out the answer quickly.

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
				1	1	1	1	1	1	1
			1	0	0	0	0	0	0	0

From the table, we have that the (smallest) eight-letter word in WHOLE BINARY containing EXACTLY ONE 1 corresponds to the number one hundred twenty-eight given the natural interpretation. Now since 1111111 is the (largest) seven-letter word consisting of all 1's, we know by the above observation that it must be ONE LESS than one-hundred twenty-eight or one hundred twenty-seven so that 1111111 is one hundred twenty-seven in WHOLE BINARY.

Exercise 1.3.1: What are the numbers 100001, 11 and 110 in WHOLE BINARY? Do you see the purpose of 0's as place holders?

Exercise 1.3.2: Refer to the table above. Instead of using the observation, we could have interpreted 1111111 in WHOLE BINARY by joining one sixty-four with one thirty-two with one sixteen with one eight with one four with one two with one one, and we would have still received the same answer of one hundred twenty-seven. Why is this true?

Representation of Numbers (Part 4)

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