There are two existence and uniqueness conditions of WHOLE BINARY that we want to verify.

• Existence
  • Every number you can reach by counting by ones starting with zero can be represented by AT LEAST ONE word in WHOLE BINARY.
  • Every WORD in WHOLE BINARY represents AT LEAST ONE number.
• Uniqueness
  • Every number you can reach by counting by ones starting with zero is represented by AT MOST ONE word in WHOLE BINARY. (Numbers are uniquely represented in WHOLE BINARY.)
  • Every WORD in WHOLE BINARY represents AT MOST ONE number. (Numbers have representations in WHOLE BINARY that's unique to them.)

• Every number you can reach by counting by ones starting with zero is represented by EXACTLY ONE word in WHOLE BINARY.
• Every WORD in WHOLE BINARY represents EXACTLY ONE number.

• Every number you can reach by counting by ones starting with zero can be represented in WHOLE BINARY by using the iterative process described in Section 2 of rolling individual packages on the conveyor belt from the right and letting each column carry out its job until you reach the number.
• We can use the table of Section 2 with all of the column place names, enter the word into the table, and then the number that is received by joining the place names of all columns with 1's is the number that is represented by the word in question. (Recall Section 3's example with the word 101001 in WHOLE BINARY and the corresponding interpretation of fourty-one.)
• CHALLENGING: The reasoning for this idea is a bit cumbersome so it is presented through an example. The crux of the argument rests in Section 2's table of place names used to interpret words in WHOLE BINARY along with the observation noted in Section 3 that LARGER WORDS REPRESENT LARGER NUMBERS IN WHOLE BINARY.
  • First we note that zero and one are unique represented as zero can be represented with 0 in WHOLE BINARY, one can be represented with 1 in WHOLE BINARY, and 0 and 1 are the only words in WHOLE BINARY with length one (one letter) since the only letters in WHOLE BINARY are 0 and 1. We know that zero and one cannot be represented by any other word in WHOLE BINARY because zero and one are both represented by one-letter words, and LARGER WORDS REPRESENT LARGER NUMBERS IN WHOLE BINARY so that all words containing two or more letters in WHOLE BINARY must represent numbers larger than zero and one.
  • EXAMPLE FOR THE MORE GENERAL CASE: Now let's try to find two words in WHOLE BINARY that represent the same number. Well, in order to have a chance to succeed at this endeavor, they must have the same length (same number of letters) since LARGER WORDS REPRESENT LARGER NUMBERS IN WHOLE BINARY. Now since we already showed that all numbers represented with one letter are uniquely represented in WHOLE BINARY (Zero is only represented by 0, and one is only represented by 1 in WHOLE BINARY, and there are no other one-letter words in WHOLE BINARY other than 0 and 1), let's pick words of a greater length, say seven. Let's try the words 1011001 and 1010111 and let's insert them into our table:

    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	1	0	0	1
    				1	0	1	0	1	1	1

    Now note that the letter in the leftmost place of both words will BE THE SAME as all words in WHOLE BINARY that are not 0 must begin with a 1. Now since they are both of the same length, they will both have their rightmost 1's be in the same column. In the example above, 1011001 and 1010111 both have a 1 in the sixty-fours place. Now if we interpret these words, then we have the following:

    • 1011001 represents the number formed by joining one sixty-four with one sixteen with one eight with one one.
    • 1010111 represents the number formed by joining one sixty-four with one sixteen with one four with one two with one one
    IF these two words were to represent the same number in WHOLE BINARY, then since the formation of BOTH numbers involve a sixty-four being joined to them, by IGNORING the joining of the sixty-four for BOTH numbers, we should get the same answer. After all, if we were to join sixty-four to two different numbers, we'd clearly get different numbers. Therefore by IGNORING the joining of the sixty four in both cases, we can draw the conclusion that if the two WORDS represent the same number, then the number formed by joining one sixteen with one eight with one one should be the same as the number formed by one sixteen with one four with one two with one one. We can now represent these new numbers in WHOLE BINARY by using the table again:

    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	0	0	1
    						1	0	1	1	1

    
    
    Now note that by getting rid of the joining of the two sixty fours we essentially eliminated the first 1 and all 0's preceding the next 1 in both words. We will always be forced to get rid of these 0's up to the leftmost column since all words that are NOT 0 in WHOLE BINARY must begin with a 1. Also note that in order for these resulting words to be the same, they must also be of the same length because again, LARGER WORDS REPRESENT LARGER NUMBERS IN WHOLE BINARY so that all words containing two or more letters in WHOLE BINARY. However, in order for these new words to have the same length, the next occurrences of the 1's when reading from left to right must be in the same columns, which means that all the columns after the leading 1 in both numbers (leftmost column) had to contain 0's and hence match. Therefore, so far, we've had to match the letters of all columns in both words for the numbers to be the same. In this specific example, we've matched the first two columns starting from the right. Now since these two new numbers 11001 and 10111 must represent the same number in order for the original numbers of 1011001 and 1010111 to be the same, we have the following:

    11001, which represents the number formed by joining one sixteen with one eight with one one must be the same as 10111, which represents the number formed by joining one sixteen with one four with one two with one one.

    However, we can then make the same claim that sixteens are being joined in both cases and therefore in order for those two numbers to be the same, we can IGNORE the joing of the sixteen and hence conclude that joining one eight with one one must be the same as joining one four with one two with one one. However, when we represent these two numbers in WHOLE BINARY by using the table, we get the following:

    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	0	1
    								1	1	1

    
    
    However, we know that these two words canNOT represent the same number since the top one is represented with four letters whereas the bottom one is only represented by three, and IN WHOLE BINARY, LARGER WORDS REPRESENT LARGER NUMBERS. Therefore, the original two words 1011001 and 1010111 canNOT represent the same number in WHOLE BINARY.
  As you can see from the preceeding example, we were able to determine that the original two words did NOT represent the same number once we found two columns that didn't match. Note also that the process shrunk the word size after each step. Combining these two observations, it can be seen that the words must match in all columns in order for both of them to represent the same number in WHOLE BINARY so that it is impossible for two DIFFERENT words in WHOLE BINARY to represent the same number.