by Harley E. Tillitt

Assembly Room, A. K. Smiley Public Library

In our day-to-day lives we encounter many situations in which we make comparisons such as the prices of competitive items in the grocery store, the time of day relative to the time remaining before an appointment, the length and weight of a new-born child, the street address of a person we are trying to locate, or the rate of interest on a bank account.

These and many others are "measured" by means of a specific numerical system which we started to learn when we were children.

But are there other numerical systems than the one we regularly use?

This paper considers that question.

Harley Tillitt is a native of the village of Del Rosa, California located in what is now the North East part of San Bernardino. He attended Del Rosa Grammar school, which was on the same site as the school in which his parents, in about 1900, met in the first grade. He attended Highland Junior High School, and was graduated from San Bernardino High School, the University of Redlands, and the Claremont Graduate School.In 1938 he was married to Sylvia Jewel Payne. There were two children, Jay Lanning and Kay Lynn.After five years in the California Public School System he became involved with the Manhattan Project [A-Bomb] at the University of California, Berkeley, and at Oak Ridge, Tennessee. After War II he was employed by the U.S. Navy, in the Mathematics Division of its largest laboratory, then known as the Naval Ordnance Test Station, at China Lake, California, with the responsibility to establish a center for scientific and technical computation. In 1971, "on loan from China Lake", he became a part of the Headquarters Staff of the Navy Material Command, located in Arlington, Virginia. He was involved with the acquisition and dissemination of information related to the Navy's Research and Development programs.He has given papers on computer-related topics both in the United States and abroad.

He retired in 1975. He was widowed in 1994. In 1996 he married his high school sweetheart, Frances Lucille Hunting Bryan. Frances passed away in 1996.

If: 111 + 110 = 1101

Do: 7 + 6 = 15?

by Harley E. Tillitt

Of course the answer is yes. But it may take a bit of explanation to be convincing. Instead of using slides to display certain topics, there are several insertions of TABLES and references to APPENDICES.

It was in the late fifties or early sixties that an organization known as the School Mathematics Study Group was formed at Yale University. It became known as SMSG and later moved to Stanford University.

Its purpose was to suggest and encourage ways to modify the teaching of arithmetic and mathematics in the schools of America. One outgrowth of these efforts became known as the New Math. In many ways it did not have the results which were hoped for because it was difficult to motivate, and hence educate, teachers to get involved with the new concepts.

There were important objectives associated with these efforts among them being what might be called the development of fundamental concepts for students which were beyond the memorization of arithmetic facts such as the multiplication tables

One of these so-called fundamentals was the understanding of what is meant by the concept of "number" as distinct from the concept of "numeral."

Related to this is the idea of counting. That is, just what is the background for saying one, two, three, four, and so on, which parents are so often proud to hear their children reciting, and exclaim "Johnny can count to eight now."

Of course Johnny can learn to say the words one, two, three, and so on, up to eight, but what, if anything is going on in his mind as he says it? Is there any perception of quantity or magnitude associated with the words? Does the word "eight" convey to him that there is a difference, perhaps in size, between the words "seven" and "eight" beyond the parental admonition that one of them follows the other as he "counts" for them and receives their praise? So, just what is "number" and "counting?"

Chapter I of the book entitled Elementary Number Theory, by Uspensky and Heaslet, published by McGraw Hill in 1939 included the following paragraphs, without citation:

The theory of numbers, also called higher arithmetic, is concerned, at least in its elementary parts, with the properties of whole numbers or integers. As an established discipline it is one of the youngest, yet its roots go deep into history.

The notion of the whole number as an aggregate of units is so primitive that it is hardly possible to imagine human beings who do not possess it in the form of counting, at least within a limited range. As far back as any records can be found, mankind possessed adequate methods for keeping a tally of things, while our knowledge of ancient civilizations reveals an already highly developed art of denoting and operating on numbers as far back as 3500 BC and earlier.

The needs of everyday life were primarily responsible for the rise of practical arithmetic. On the higher level of civilization when urgent needs were satisfied and opportunity and leisure were available to ponder about things, numbers in themselves began to attract attention. Peculiarities of individual numbers or classes of numbers began to be observed. But such speculations on numbers, far from being a real study of their properties, developed at first into a peculiar number mysticism among ancient civilized peoples. Such numbers at 3 and 7 were accepted as omens of good luck.

So ends the partial quotation from the book. The point to emphasize, from my point of view, is that of dealing with a unit, that is, a single thing. Are we able to think first of one thing, then another, then another and finally conceptualize one more thing which somehow conveys to us the collection of all of the single ones? Is this called counting?

Because of our lifetime of experience we might be inclined to say 1 + 1 + 1 = 3. But these are merely numerals used to name the several units and their combination.

Let’s reconsider Uspensky’s and Haslet’s suggestion and go back for a few thousand years to investigate a hypothetical situation having to do with all of this.

The story starts long ago when people, in some regions, lived in caves. In this case there were two families. Since there are no records of these hypothetical families they will be referred to as the Easts and the Wests. That is because their caves were on the east and west sides of a large river.

They had never talked with each other because the river was too wide and fast for them to cross. But they had many habits and customs in common.

In both cases the men of the two families spent a great deal of time hunting animals for food. After each successful hunting trip each man made a mark on the wall of his respective cave. There was no special reason for doing this. It was simply a tabulation of their successes. In both cases a single mark was made for each success. In time each wall had a long string of single marks extending from what we would call left to right.

Then one day there was a massive earthquake. The whole region was shaken. There was no harm done to the caves of the Easts and the Wests, but the river separating them was diverted, some distance upstream, with the result that it became not much more than a rivulet, with the water only about knee-deep.

In time the two families began to make contact with each other. Among other things, the men began to discuss their hunting experiences, and were making comparisons between their respective lines of markings on the walls of the caves.

Each was interested to find out which one had been the most successful. But how could that be done? The marks were not the same size from day to day, and the distance between successive marks varied considerably.

What would many centuries later become known as counting was not known to them. But one day they did conceive an idea! It involved what would some day be known as establishing one-to--one correspondences. In modern terms this meant that they would, in their case, go into the cave and place a finger on one of the marks, then the next finger on the next mark and so on until all of the fingers were touching marks. They were making one-to-one correspondences between the fingers of the hands and the marks on the cave wall. It was not counting.

To keep track of what they were doing, and looking forward to a later comparison, they decided to put a pebble in a small pile every time the correspondences were established for both hands. Then to make a comparison between the two caves all they would have to do would be to establish the one-to-one correspondences between the two piles of pebbles.

They agreed to go back to their respective caves and begin the job of building their piles of pebbles, and to meet again when each of them had finished his work.

In time they met again at East’s cave. West brought his basket of pebbles to make the comparison. The two piles looked something like the display in Table 1 below.

The two men set to work making the one-to-one correspondences between their respective piles.

One East pebble was matched against one West pebble. Then another East pebble was matched against West pebble. This was continued until there were no East pebbles remaining, as shown in Table 2 below.

This seemed to show that West had been more successful in his hunting trips than East had been in his, since West had some pebbles left over that East could not match.

So the two men continued to discuss other things relative to their hunting experiences and how they had kept track of their successes. During this discussion West commented that when he had done his matching in his cave he did not use his thumbs because he did not consider them to be fingers. The two of them had not mentioned the possibility of this situation earlier. In response to this East expressed his opinion that the two piles could not be compared to give a good idea of their respective hunting successes.

West agreed. What to do now they wondered. They decided to return to West’s cave and both start over on the same row of marks with each one using the same method as he had used before. This was done for quite a few marks when it was clear that West had put down more pebbles than East, although they were establishing one-to-one correspondences with the same group of marks. Table 3 below shows the results of their efforts.

They discussed this situation and arrived at two conclusions: (1) That while after each pebble was placed and they "started over" again with another "double hands" worth of matching fingers (and/or thumbs) with marks, a comparison could not be made unless both either used thumbs or did not use thumbs, and (2) It did not matter, for the sake of comparison, whether or not thumbs were used, as long as they both did the same thing.

They began to think of other ways to place pebbles. Suppose, they said, they used only the fingers and thumb of one hand. Or perhaps use the fingers and thumbs of both hands as well as the toes of both feet. They even went one step further to consider using just the thumbs of both hands.

Table 4 below shows how they might match the marks with these new ways of establishing one-to-one correspondences. As has been seen, there can be many ways of expressing the idea of number. Is one of these better than another?

In time they thought it would be a good idea to make a special sign, which we might call a numeral, for each of the correspondences to indicate which finger was used for each mark. This is shown in Table 5 below along the left side of which are shown the numerals they invented when using the "one hand" method.

Table 6 below repeats the left-hand columns of the table above but also shows, under the heading of BASE 5, some squiggly marks. By squiggly marks is meant the numerals we see on these pages, and which we use every day. That is, what we call 1, 2, 3, 4, etc. It also shows, under BASE 10, our familiar counting system.

From most points of view the several ways are the same since each way, although they may not look the same, can represent the same collection of units, which is the same number.

The system we use from day to day has as many squiggly marks as we have thumbs and fingers on both hands. This dictates how often the cycle of numerals repeats.

If the system used only as many marks as the fingers and thumb on one hand the cycle would repeat more often. For example. as shown in table 6, counting to 20 in BASE 5 would be: 1,2,3,4,10,11,12,13,14, 20.

In sharp contrast, if the system used as many marks as all of the appendages on both feet and both hands, the cycle would repeat much less often, but there would be many more, and different, numerals required with which to become acquainted.

Unlike our day-to-day system, which has what we now refer to as having ten different marks before recycling, the both-hands-and-both-feet system would need what we would now refer to as needing twenty different marks before recycling.

[APPENDICES I, II, and III, which follow, give some examples pertinent to systems based on other than what we call ten, that is, the number of fingers and thumbs on both hands.]

Accordingly, the greater the number of distinguishing numerals within a system before recycling begins, the greater the complexity.

Likewise, the fewer the number of distinguishing marks, the less the complexity becomes. This is very well illustrated in the technology of the modern digital computer. In this field there are only two distinguishing numerals: 1 and 0 This is the ultimate in simplicity although the recycling is done every other time.

The column with the heading BASE 2, in table 7, illustrates such a system which is based upon the number of items corresponding to the thumbs on both hands. Compare this with the right-hand column of table 4 where the recycling is done for every other item.

Table 8 below is a rearrangement of the BASE 2 Column in table 7

The table shows how the numerical value of these entries, in our familiar system, can be determined, using the following rules:

If there is a 1 in the right-hand position add 1

If there is a 0 in the right-hand column add 0

If there is a 1 in the second column to the left add 2

If there is a 0 in the second column to the left add 0

If there is a 1 in the third column to the left add 4

If there is a 0 in the third column to the left add 0

If there is a 1 in the fourth column to the left add 8

If there is a 0 in the fourth column to the left add 0

If there is a 1 in the fifth column to the left add 16

If there is a 0 in the fifth column to the left add 0

EXAMPLE:

This system is used in computer technology since electronic circuitry based upon the two-state ON-OFF [or 1- 0] concepts can be made to operate at great speeds. That is, it is not necessary to establish five or ten or twenty different levels of impulses or conditions which must be distinguished. Only two are needed.

This allows, for example, for circuits to be somewhat similar in complexity to a light switch. That is, if the switch is in the UP position the light goes on. If the switch is in the DOWN position the light goes off. In other words, each action merely changes the status of the switch to the opposite of what it was.

A good way to demonstrate this is for five volunteers to come forward, face the audience, and form a single row from left to right. The person farthest to the right, from the perspective of the audience, will be referred to as A. To person the left of him, from the perspective of the audience will be referred to as B. Following the same procedure, the others will be referred to as C, D, and E.

Imagine a series of five switches which are designated, from left to right, E D C B A. Consider that these switches are interconnected in such a way that if A is OFF and receives an impulse, it will turn ON. If A receives another impulse while it is ON it will pass the impulse to B and then turn itself OFF. B now is ON but A is OFF. The next impulse to A will just turn A ON again. Now both A and B are ON. The next impulse to A will pass to B and A will go OFF. But when B receives this impulse from A it will pass it to C as it turns itself OFF. Now A and B are OFF and C is ON. Or represented in another way: ON OFF OFF, or 1 0 0. This is what, in table 7, is referred to as 4.

This analog of the electrical impulses, mentioned above, will be a tap on a person’s left shoulder, as will be shown.

At the beginning all hands will be held DOWN, pointing to the floor. A hand that is DOWN will represent OFF. A hand that is UP will represent ON. So, all hands being DOWN at the start means: OFF OFF OFF OFF OFF. Only right hands will be used, and the following rules are to be applied.

The RULES:

RULE 1: If you get a tap on your left shoulder change the position of your hand. That is, if it is UP put it DOWN. If it is DOWN put it UP.

RULE 2: Whenever you are putting your hand DOWN, tap the left shoulder of the person to your right and leave your hand DOWN until the next tap on your left shoulder.

As these people analogs to the electrical circuit go into action, the audience should look at table 8 and compare the successive hand positions as the impulses are applied.

This demonstration is intended to show the level of simplicity associated with the BASE 2 system. table 9 below shows a comparison between the BASE 2 and the BASE 10 systems for the adding only two digits which can be visualized as being placed one above the other, on paper, with the single digit of the answer going directly underneath with the possibility of the carry of a 1 to the next column to the left.

In the United States there are two systems in use which are based upon 12 [called duodecimal]. They are used in the measurement of quantity when reference is made to the dozen, and to length when reference is made to the foot. That is, there are 12 eggs in a dozen and 12 inches to the foot. Such a system would need to have two additional "squiggly marks" as shown in table 10 BELOW.

Here they are printed as L and Z, and pronounced LAM and ZUG. In table 10, ZUG + ZUG = L*. This would be pronounced LAMTEEN. Accordingly, LAMTEEN + 1 = ZUGTEEN, and ZUGTEEN + 1 = 20.

To count to 20 in this system one would say: 1,2,3,4,5,6,7,8,9,LAM, ZUG,10,11,12,13,14,15,16,17,18,19, LAMTEEN, ZUGTEEN, 20.

[This is demonstrated at the bottom of the first page of table 11 below. Table 11 also shows, in addition to the duodecimal system, a comparison in "Counting to 100" for some other systems. One of these to be referred to later, is based upon 16.]

COUNTING TO ONE HUNDRED

The tables below illustrate the similarities among five number systems, in counting to "100." In each case the "100" represents a number of items corresponding to the square of the number of distinguishing numerals used in the system. BASE2: 4=[2x2] entries. BASE8: 64= [8x8] entries. BASE10: 100= [10x10] entries.BASE12:144= [12x12] entries. BASE16:256 = 116x16] entries.

Counting to 100 in BASE 2 [Binary] [Thumbs Only]

1 10

11 100

Counting to 100 in BASE 8 [Octal] [Fingers Only]

1 2 3 4 5 6   7    10

11 12 13 14 15 16 17 20

21 22 23 24 25 26 27 30

31 32 33 34 35 36 37 40

41 42 43 44 45 46 47 50

51 52 53 54 55 56 57 60

61 62 63 64 65 66 67 70

71 72 73 74 75 76 77 100

---

Counting to 100 in BASE10 [Decimal] [Thumbs and Fingers]

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

---

Counting to 100 in BASE 12 [Duo-decimal] [Thumbs + Fingers + Two Morel

1 2 3 4 5 6 7 8 9 L Z 10

11 12 13 14 15 16 17 18 19 1L 1Z 20

21 22 23 24 25 26 27 28 29 2L 2Z 30

31 32 33 34 35 36 37 38 39 3L 3Z 40

41 42 43 44 45 46 47 48 49 4L 4Z 50

51 52 53 54 55 56 57 58 59 5L SZ 60

61 62 63 64 65 66 67 68 69 6L 6Z 70

71 72 73 74 75 76 77 78 79 7L 7Z 80

81 82 83 84 85 86 87 88 89 8L 8Z 90

91 92 93 94 95 96 97 98 99 9L 9Z L0

L1 L2 L3 L4 L5 L6 L7 L8 L9 LL LZ Z0

Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 ZL ZZ 100

---

Counting to 100 in BASE 16 [Hexadecimal]
[Fingers only, both hands, twice.]

1 2 3 4 5 6 7 8 9 A B C D E F 10

11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 20

21 22 23 24 25 26 27 28 29 2A 2B 2C 2D 2E 2F 30

31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F 40

41 42 43 44 45 46 47 48 49 4A 4B 4C 4D 4E 4F 50

51 52 53 54 55 56 57 58 59 5A 5B 5C SD SE SF 60

61 62 63 64 65 66 67 68 69 6A 6B 8C 6D 6E 6F 70

71 72 73 74 75 76 77 78 79 7A 7B 7C 7D 7E 7F 80

81 82 83 84 85 86 87 88 89 8A 8B 8C 8D 8E 8F 90

91 92 93 94 95 96 97 98 99 9A 9B 9C 9D 9E 9F AO

A1 A2 A3 A4 A5 A6 A7 A8 A9 AA AB AC AD AE AF BO

B1 B2 B3 B4 B5 B6 B7 B8 B9 BA BB BC BD BE BF CO

C1 C2 C3 C4 C5 C6 C7 C8 C9 CA CB CC CD CE CF DO

D1 D2 D3 D4 D5 D6 D7 D8 D9 DA DB DC DD DE DF EO

E1 E2 E3 E4 E5 E6 E7 E8 E9 EA EB EC ED EE EF FO

F1 F2 F3 F4 FS F6 F7 F8 F9 FA FB FC FD FE FF 100

The title of this paper raised the question of whether or not 111 + 110 = 1101. This is demonstrated in APPENDIX IV, making use of the addition table at the bottom of table 9.

Under some circumstances, what is shown as the second step, in APPENDIX IV may also produce some carry digits. In this case the steps would be repeated until there are no more carry requirements. This produces what might be considered a "ripple" effect, from right to left, until all of the carries are taken care of.

The title of this paper also raised the question of whether or not 7 + 6 = 15. To show that this is so, the sum above, 1101, is re-written in groups of three, as shown, and the 4-2-1 rule, as shown in table 8, is applied. See the bottom of APPENDIX IV.

0 0 1       1 0 1
_____     _____
   1             5   

There is a relationship between the binary, [BASE 2], the octal. [BASE 8], and the hexadecimal [BASE 16] number systems which is useful to those computer programmers affectionately referred to a "bit twiddlers." These individuals, in the development of software, often need to examine the contents of parts of the computer systems in their most basic form, that is, at the level of just 1’s and 0’s. But these binary representations are long and cumbersome with which to deal. So a kind of "shorthand" is used.

The decimal value of 2000 as expressed in binary is:

11111010000

This follows the pattern shown in Figure 8 when reading from right to left.

0 x 1 +0 x 2+0 x 4+0 x 8+1 x 16+0 x 32+1 x 64+1 x 128+1 x 256+1 x 512 +1 x 1024= 2000.

By separating the binary number into groups of three, and applying the

4-2-1 rule to each triplet, it looks like this:

Accordingly, 3720 is the octal equivalent of 2000 in decimal.

There were special desk calculators made during the early days of computers, without any 8's or 9's on the keyboard, to carry out calculations in this system. This allowed the programmers to perform operations on "almost" binary numbers as programs were being checked, without worrying about the long binary representations within the computer itself.

More recently the same concept is applied to the separating of the binary numbers into groups of four instead of three. The above binary number for 2000 is shown in this pattern. The same rule as shown in Table 8 is applied but now for four positions instead of three:

By looking at the top row in the "COUNTING TO 100" Section of Figure 11, which has to do with the hexadecimal system, it can be seen that the 13th numeral is D.

Accordingly, 7DO is the hexadecimal equivalent of 2000 in decimal.

So, it can be seen that different sets of numerals may be used to designate a given quantity. The number stays the same, but the numerals which represent the number may be different. Let us add the above numbers, 7 and 6, on our fingers in the BASE 8 system. That is, count to 7 then add 6 more, as follows: 1,2,3,4,5,6,7 then 10,11,12,13,14,15.

Accordingly consider, that under certain circumstances:

1 + 1 = 10.

This is another way of saying that since we have one thumb on each hand, we have a total of ten thumbs. This depends upon what number system is being used, of course. In this case it is what has been referred to as having the BASE 2. Look at the second row at the top of table 8. The 2 at the left, which is in the BASE 10 system is represented by the 1 and the 0 at the right, both of which are in the BASE 2 system.

By way of summarizing what has been presented in this paper, it is hoped that two things have been illustrated: (1) that the idea of number can be expressed in more than one way, and (2) that one of these ways, the binary, or BASE 2 system, is at the heart of modern computer technology.

The latter point being that electronics has been developed to take advantage of the simplicity of manipulation of what might be thought of as the two-level representation of quantities.

An attempt has been made to show that arithmetic in the binary system can be very simple. There is no "rotating of wheels" so to speak, and as shown at the bottom of table 9, there are only four possibilities shown which need to be considered for the addition of two single-position numbers. Considering that there might also be carry bits to include, there are actually eight possibilities since each of the four might, or might not, have to be involved with a "carry" from the operation one cycle earlier. The table at the top of table 13 below shows these eight situations.

THE EIGHT POSSIBILITIES FOR ADDING TWO BITS

AND ALSO INCLUDING A CARRY BIT FROM A PREVIOUS CYCLE

[The * means there is a carry bit to be applied to the next position to the left.]

For any single transaction involving two bits, only one of the above combinations provides any output. This output is the result of the interconnections of a previously "wired" network of electrical components through which impulses travel. There is no electronic equivalent of "rotating wheels."

A very important event in this field was the invention of the transistor. This is credited to Bell Telephone Laboratory scientists: William Shockley, John Bardeen, and Walter Brattain. In 1956 they shared the Nobel Prize in physics for their work.

A transistor can be visualized as a kind of sandwich of three pieces of silicon. The two pieces of the "bread" of the sandwich are made of one kind of silicon, and the "cheese" of the sandwich is made of another kind of silicon. These two types have different conduction characteristics. The "cheese" part of the sandwich has what are called electron holes and the "bread" parts have what are known as free electrons.

When conditions are right the holes will receive free electrons. This flow of electrons is an electrical current. When conditions are otherwise this flow is stopped. The transistor is therefore a kind of electronic "gate". It is either open or closed. The 1's and 0's in the binary representation of numbers are used to manipulate these various combinations through complex networks of transistors to produce operations such as addition, subtraction, multiplication, and division, among all others.

Following the development of the transistor the next big step in this technology was the ability to fabricate many components into a single piece of semiconductor. This led to what came to be known as the integrated circuit. This was invented by an American engineer named Jack Kilby, in 1958, and led to what we now know as the microchip into which many thousands of components are packed. The first microprocessor employing this technology was produced in about 1970.

Modern technology can now "pack" up to one million transistors into a space