Cuesta Computer with Software based on Binary Arithmetic and a
Hardware based on Bites I
                 Vladimir Cuesta ...
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              k→       1 0 1...
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4.4. Example n...
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and I ca...
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Cuesta Computer With Software Based On Binary Arithmetic And A Hardware Based On Bites I

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Cuesta Computer With Software Based On Binary Arithmetic And A Hardware Based On Bites I

  1. 1. Cuesta Computer with Software based on Binary Arithmetic and a Hardware based on Bites I Vladimir Cuesta † Instituto de Investigaciones en Matem´ ticas Aplicadas y en Sistemas, Universidad Nacional Aut´ noma de a o M´ xico, 20-726, Ciudad de M´ xico, M´ xico; e e e Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de M´ xico, o e e M´ xico e Abstract. I present a computer with name Cuesta Computer with a Software based on Binary Arithmetic and a Hardware based on Bites I, the set of programs or software are based on the important result of number theory that every positive integer can be represented in an unique way on an arithmetic of base two, I represent numbers based on the previous result and I can define a set of symbols using in a correct way the same result of uniqueness, the hardware or the physical part of the computer consists of a band where I print or I write zeros, ones or nothing (empty spaces), the basic units of my bands are bites. 1. Introduction In our days, the desing and the construction of high technology computers, printers, video camera, photographic camera and so on need mathematical tools for creating better models that the consumer can use in their daily life, in complex applications people use clusters or supercomputers to study physical phenomena in quantum mechanics, classical mechanics, chaos theory, the climate, storms, wind, aeronautic engineering, construction, economics, genetics, astronomical phenomena, artificial intelligence, cryptography, genetic algorithms, robots, industrial robots and so on (see [1], [2] and [3] for a review of the subject). The advantage of my desing lies in the use of the available space, I think that the use of bites like basic units is better than bytes. 2. Binary arithmetic and the representation of numbers I present binary arithmetic with the help of a general theory and examples. † vladimir.cuesta@nucleares.unam.mx
  2. 2. Cuesta Computer with Software based on Binary Arithmetic and a Hardware based on Bites I 2 2.1. Binary system, basic operations Let a be a positive integer, this number can be expressed like the unique following sum (see [4] for instance), a = λn 2n + λn−1 2n−1 + λ1 21 + λ0 20 , (1) where λi can take the values 0 or 1, i = 0, . . . , n, I will represent this number in the way, λn λn−1 . . . λ1 λ0 I will call this a binary representation for the number a. It is easy to describe how to obtain this representation. I show the following example: take the number 17, this is an odd number, take an unit (I mean, make the operation 17 − 1 = 16 ) and write this on a square, 1 later, take 16/2 = 8, in this case I obtain an even number, then I must write a zero to the left of the previous square. I mean, 0 1 if I continue, I must take 8/2 = 4 and is an even number and I write a zero to the left of the previous set of squares, 0 0 1 the following step is to take 4/2 = 2 and is and even number, then I write, 0 0 0 1 the final step is to take 2/2 = 1 and I must write 1 to the left of all numbers and in fact I must finish my reasoning when I obtain the number one in the algorithm then the number 17 can be represented like, 1 0 0 0 1 or 17 = 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20 , (2) I can sum two binary numbers following the rules: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1 and 1 + 1 = 0, but in this case I must transfer one number 1 to the left, for example, 101 + 111 = 1100, (3) another example is, 100101 + 11001 = 111110, (4) and again I understand the number like a number with base two.
  3. 3. Cuesta Computer with Software based on Binary Arithmetic and a Hardware based on Bites I 3 Now, I show the product algorithm for numbers with base two, I can obtain 1011 × 110 in the way, 1 0 1 1 × 1 1 0 − − −− − − − 0 0 0 0 1 0 1 1 1 0 1 1 − − −− − − − 1 0 0 0 0 1 0 I finish my discussion here, but this can help to make further progress in computing with complex technology. I mean, with electronic devices. 3. The computer In the present section I use the mathematical theory of binary arithmetic to construct a new machine that I called Cuesta Computer with a Software based on Binary Arithmetic and a Hardware based on Bites I, I show my convention for programming on it. 3.1. Conventions: numbers According to the previous section, I can think computing like a science based on binary arithmetic (arithmetic of base two). In fact, every positive integer can be represented in an arithmetic of base two (see [5] and [4] for instance) in an unique way, if I use this result, I can represent these kind of numbers in a serie of powers of the number two, I show more examples, 0→ 0 1→ 1 2→ 1 0 3→ 1 1 4→ 1 0 0 5→ 1 0 1 6→ 1 1 0 7→ 1 1 1 8→ 1 0 0 0 9→ 1 0 0 1
  4. 4. Cuesta Computer with Software based on Binary Arithmetic and a Hardware based on Bites I 4 10 → 1 0 1 0 11 → 1 0 1 1 12 → 1 1 0 0 13 → 1 1 0 1 14 → 1 1 1 0 15 → 1 1 1 1 16 → 1 0 0 0 0 17 → 1 0 0 0 1 18 → 1 0 0 1 0 19 → 1 0 0 1 1 20 → 1 0 1 0 0 35 → 1 0 0 0 1 1 2001 → 1 1 1 1 1 0 1 0 0 0 1 10035833 → 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 1 and so on. 3.2. Conventions: symbols If I apply the result that I used in the previous section, I can represent a set of symbols in an unique way. In this case I put two empty spaces before a number and I put one empty space to the final, I choose one convention, although I can use many of them, a→ 0 b→ 1 c→ 1 0 d→ 1 1 e→ 1 0 0 f→ 1 0 1 g→ 1 1 0 h→ 1 1 1 i→ 1 0 0 0 j→ 1 0 0 1
  5. 5. Cuesta Computer with Software based on Binary Arithmetic and a Hardware based on Bites I 5 k→ 1 0 1 0 l→ 1 0 1 1 m→ 1 1 0 0 n→ 1 1 0 1 n→ ˜ 1 1 1 0 o→ 1 1 1 1 p→ 1 0 0 0 0 q→ 1 0 0 0 1 r→ 1 0 0 1 0 s→ 1 0 0 1 1 t→ 1 0 1 0 0 u→ 1 0 1 0 1 v→ 1 0 1 1 0 w→ 1 0 1 1 1 x→ 1 1 0 0 0 y→ 1 1 0 0 1 z→ 1 1 0 1 0 A→ 1 1 0 1 1 B→ 1 1 1 0 0 C→ 1 1 1 0 1 D→ 1 1 1 1 0 E→ 1 1 1 1 1 F → 1 0 0 0 0 0 G→ 1 0 0 0 0 1 H→ 1 0 0 0 1 0 I→ 1 0 0 0 1 1 J→ 1 0 0 1 0 0 K→ 1 0 0 1 0 1 L→ 1 0 0 1 1 0
  6. 6. Cuesta Computer with Software based on Binary Arithmetic and a Hardware based on Bites I 6 M→ 1 0 0 1 1 1 N→ 1 0 1 0 0 0 ˜ N→ 1 0 1 0 0 1 O→ 1 0 1 0 1 0 P → 1 0 1 0 1 1 Q→ 1 0 1 1 0 0 R→ 1 0 1 1 0 1 S→ 1 0 1 1 1 0 T → 1 0 1 1 1 1 U→ 1 1 0 0 0 0 V → 1 1 0 0 0 1 W → 1 1 0 0 1 0 X→ 1 1 0 0 1 1 Y → 1 1 0 1 0 0 Z→ 1 1 0 1 0 1 +→ 1 1 0 1 1 0 −→ 1 1 0 1 1 1 /→ 1 1 1 0 0 0 ∗→ 1 1 1 0 0 1 =→ 1 1 1 0 1 0 (→ 1 1 1 0 1 1 )→ 1 1 1 1 0 0 {→ 1 1 1 1 0 1 }→ 1 1 1 1 1 0 ?→ 1 1 1 1 1 1 .→ 1 0 0 0 0 0 0 ,→ 1 0 0 0 0 0 1 ;→ 1 0 0 0 0 1 0 :→ 1 0 0 0 0 1 1
  7. 7. Cuesta Computer with Software based on Binary Arithmetic and a Hardware based on Bites I 7 !→ 1 0 0 0 1 0 0 Empty space ” ”→ 1 0 0 0 1 0 1 and so on. 4. Entry and exit in the computer (in and out): examples 4.1. First case: Two consecutive symbols If I put together (entry), 1 0 1 0 0 0 1 1 1 1 I find, 1 0 1 0 0 0 1 1 1 1 → No and If I use my convention, I find the word No, in this first case it is important to note that when I put two symbols together it appears three empty spaces in the middle of the set of numbers, the previous sequence occupies 16 bites or equivalently 2 bytes. 4.2. Second case: One symbol and one number If I put together the following sequence, 1 1 0 1 1 1 1 0 1 I find, 1 1 0 1 1 1 1 0 1 → −5 and if I use my convention, I obtain -5, in this case it is important to note that when I put one symbol and one number it appears one empty space in the middle of the set of numbers, the previous sequence occupies 12 bites or 1 byte and 4 bites. 4.3. Third: One number and one symbol If I write one number and one symbol, 1 1 0 0 1 1 I obtain, 1 1 0 0 1 1 → 1s and I can read 1s, in this case it is important to note that when I put one number and one symbol it appears two empty spaces in the middle of the set of numbers, the previous sequence occupies 9 bites or equivalently 1 byte and 1 bite.
  8. 8. Cuesta Computer with Software based on Binary Arithmetic and a Hardware based on Bites I 8 4.4. Example number four I will consider a larger example, If I put together, 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 I obtain, 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 Hi, I am a computer. (5) I obtain Hi, I am a computer. If I read from left to the right, I begin with two empty spaces. I mean, I begin with a word, If I continue I will find three spaces and according to the previous discussion the following set of zeros and ones represent another word, If I continue I will find three empty spaces and the next set of zeros and ones represents one word and so on. 4.5. Example number five If I put together the following set of bands, 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 I obtain, 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 1 + 18 = 19,
  9. 9. Cuesta Computer with Software based on Binary Arithmetic and a Hardware based on Bites I 9 and I can read 1 + 18 = 19, with this I finish my examples, but let me say a little more. If I read the total band from left to the right I begin with a set of numbers and according to my convention I begin with a number, the following set of empty spaces consist of two spaces, then the following object is a symbol, If I continue I will find one empty space, then I will find a number and I can continue reading my band in a predictable way. 5. Conclusions and perspectives In the present paper I have presented a computer, it has the advantage that I can define in an unique way numbers and using a convention where I can represent symbols in an unique way, too. Another important point is that I can make programs saving my files in bites, not in bytes like people usually make and like the reader can see this procedure permits to save files in little space. In fact, this is just the first step of many of them because like the specialist can note, people can construct complex computer. I mean, digital computers using my desing and in this case, all the programs must be based in a binary arithmetic, like I did. The desing of digital technology, compilers, programs and so on it is part of future work, and like I said my desing is better because it permits the use of less physical space, I recommend make some lectures (see [2], [1] and [3] for citing some of them), the computer specialist can use another conventions for symbols or some variations of my design, and this is the future work. Acknowledgments I appreciate all the comments of my brothers, my children and their mothers and all my family and friends I want to dedicate this work to the interested reader in computing I hope that this paper can help to the specialist in complex computer. I mean, for the computer and supercomputer worker, I have contact with people working in some high technology companies and I want to name and to send greetings for some of these high tech companies: Cray Research, Cray Incorporated, Silicon Graphics International, The Open Group and specially to the UNIX group and so on. References [1] Harris, John W., Stocker Horst, Handbook of mathematics and Computacional Science, New Haven, Connecticut and Frankfurt, Germany, Springer-Verlag (1998), [2] Manna, Zohar, Mathematical Theory of computation Stanford University, Dover (1974). [3] Alan Mathison Turing, Computing machinery and intelligence, Mind. 59, 433-460, (1950). [4] William J. LeVeque, Elementary theory of numbers, University of Michigan, Dover Publications, Inc., (1990). [5] Vladimir Cuesta, Monoids, computer science and evolution, (2010).

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