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