LOS MAYAS
Ésta
civilización tenía muchos conocimientos de matemáticas y astronomía, también
tenía un sistema de escritura jeroglífico, el único de la América precolombina.
A pesar de esto, la rueda era usada únicamente en juguetes.
ü En
algunas ciudades como Palenque o Chichén Itzá hay observatorios.
Sus
conocimientos de astronomía eran realmente buenos, incluso podían predecir
eclipses.
Además,
los mayas tenían un calendario muy exacto. Medían el tiempo de tres formas
distintas:
ü Calendario
Tzolkin (calendario sagrado), de 260 días.
ü Calendario
Haab (calendario civil), de 365 días, 5 horas, 48 minutos y 46 segundos.
ü Calendario
de cuenta larga, que es una mezcla del Tzolkin y el Haab.
1
KIN=1 DÍA
1
UINAL=20 KINES
1
TUN=18 UINALES
1
KATÚN=20 TUNES
1
BAKTÚN=20 KATUNES
Las primeras
ideas respecto el calendario posiblemente fueron desarrolladas por los olmecas.
ü Su
sistema matemático se basaba en el número 20:
ü Conocían
la cifra cero, esto es muy importante, porque no todas las culturas la
conocían.
ü Sabían
sumar, restar, multiplicar y dividir.
ü Su
sistema de escritura se está empezando a descifrar pero es una tarea que
llevará mucho tiempo acabar porque es muy complicado.
Ø Se
trataba de un sistema de escritura que combinaba elementos fonéticos (un
símbolo para cada sílaba) y elementos ideográficos (relacionar un glifo con una
cosa).
TABLA NUMÉRICA DE LOS MAYAS
Operaciones
Aritméticas En El Sistema De Numeración Maya
Para
entender la sencillez y precisión de la ciencia matemática de los mayas, la
utilización del tablero es un factor indispensable; sobre esta cuadrícula se
realizaban las operaciones y los cálculos con los que se contabilizaron desde
las pertenencias, los impuestos y la repartición de las cosechas, hasta los
eventos astronómicos y los ciclos del tiempo.
Como
todas las muestras de la cultura maya, el tablero, que es una cuadrícula
semejante a la del ajedrez, es un objeto lleno de significaciones relacionadas
con su cosmovisión; este elemento representaba, en un sentido místico, la
urdimbre del universo; el campo donde suceden los hechos que transforman el
tiempo y el espacio y el lugar donde se asienta el conocimiento humano. Por
eso, al comprender su función y hacer uso de ella, se manifiesta como una
figura que, de forma simbólica, ejemplifica el orden y equilibrio de todo
cuanto existe.
El
posicionamiento dentro del tablero, los cálculos y las operaciones aritméticas
se realizan por medio de mecanismos fáciles de comprender. Los niveles del
tablero incrementan su valor de abajo hacia arriba, de acuerdo a la posición que
tiene el numeral dentro de dicho tablero, como se muestra a continuación,
ordenando los numerales por unidades, veintenas, veintenas de veintenas,
veintenas de veintenas de veintenas, etcétera, por lo que un punto (o unidad)
en cada nivel, tendría la siguiente equivalencia:
Un
punto en la 6ª posición 3,200,000
Un
punto en la 5ª posición 160,000
Un
punto en la 4ª posición 8,000
Un
punto en la 3ª posición 400
Un
punto en la 2ª posición 20
Un
punto en la 1ª posición 1
Este
mecanismo permitió a los mayas hacer cálculos con números estratosféricos; por
ejemplo, el número 25 673 295, se representa en maya de la siguiente manera,
utilizando seis niveles o posiciones del tablero:
El Cero
Las
matemáticas mayas han dejado una huella en el tiempo; antes que cualquier otra
civilización, los mayas originaron un concepto revolucionario: el cero, el cual
es un símbolo comúnmente utilizado para representar la nada; sin embargo, el
concepto maya del cero no implica una ausencia ni una negación; para los mayas,
el cero posee un sentido de plenitud. Por ejemplo, al escribir la cifra 20, el
cero, puesto en el primer nivel, únicamente indica que la veintena está
completa.
OPERACIONES ARITMÉTICAS CON NÚMEROS
MAYAS
SUMA Y RESTA
Para
sumar dos o más números hay que reunir, en una sola columna, las barras y los
puntos de un mismo nivel del tablero y, posteriormente, convertir los grupos de
cinco puntos en barras y las veintenas completas (conjuntos de cuatro barras)
en unidades del nivel superior inmediato. Para mayor claridad enunciaremos las
siguientes reglas:
Se
colocan las cantidades en sus respectivas posiciones, en columnas de izquierda
a derecha sobre una superficie plana, (se puede emplear granos de maíz para
representar los puntos, palillos para las barras y si es posible una concha
para el cero, si no se cuenta con estos elementos se puede emplear entonces
lápiz y papel).
Se
disponen las cantidades una a la par de la otra.
Se
agrupan los granos o cifras de la misma posición conservando sus valores
relativos en la primera columna (es decir la de la izquierda).
Por
cada cinco puntos que se juntan forman una barra, cada cuatro barras forman un
punto en la posición inmediata superior.
La
adición y posiblemente las otras operaciones de la aritmética, las trabajaron
sobre una tabla o en el suelo, en ella se colocan puntos y barras (frijoles y
palitos). León-Portilla propone que en el Codigo De Dresde, se encuentra la
representación de una multiplicación. También Calderón (1966) describe en forma
muy didáctica, las cuatro operaciones de la aritmética, además de la raíz
cuadrada y la raíz cúbica, el único inconveniente es que no indica las fuentes
que utilizó.
MULTIPLICACION
León-Portilla
(1988), señala que en una hoja del código de Dresdre, aparecen diferentes
cantidades que son múltiplos de otra. Algunos autores indican que el proceso de
multiplicación, probablemente se hacía con sumas repetidas, por ejemplo,
Seidenberg (pag. 380). “...a Maya Priest could have multiplied 23457 by 432,
say, by repeated additions of 23457”, estas conclusiones las hacen,
probablemente, por la forma en que se construye la multiplicación en los
números enteros. En los inicios de su desarrollo matemático, probablemente,
esta fue la forma de efectuar multiplicaciones, pero, considerando las grandes
cantidades que ellos manejaban en sus cálculos astronómicos y la exactitud de
los mismos, es muy lógico pensar, que debieron de haber desarrollado un
algoritmo para efectuar la multiplicación. Hasta el momento, no ha sido posible
deducir históricamente dicho algoritmo.
En
lo que respecta a este trabajo presentaremos una simulación de este proceso
para llegar a una propuesta, de lo que pudo haber sido el algoritmo de la
multiplicación en el sistema Maya.
Iniciaremos
con la multiplicación de un número por 2.
Por
ejemplo: 46 por 2. Colocamos en el reticulado el 46 en dos columnas y luego
sumamos. Obtenemos:
El
resultado final se escribe de la forma siguiente, destacando los factores de la
multiplicación:
Ahora
se multiplicará el 46 por 3, como se hizo la multiplicación por dos, ahora se
sumará otra vez 46 a este producto y el resultado será 46 por 3.
De nuevo se coloca el resultado final de la siguiente forma:
EGIPTO
Los sacerdotes egipcios empezaron a ser cultivadas por los sacerdotes egipcios, en sus tiempos libres. Aristóteles.
Debido a las crecidas del Nilo, se dio la necesidad de establecer límites para las tierras, así nació la geometría. Heródoto.
“Rope stretchers” Demócrito.
Según los egipcios las matemáticas era divina, para Aristóteles son creación de la mente humana y según Platón son independientes del ser humano.
El papiro de Moscú: la parte más importante de este escrito es el problema de la pirámide truncada o tronco de pirámide de base cuadrangular, debido a la forma de las pirámides de Egipto.
Describe el método de cálculo para su volumen.
usando la media de Herón.
El papiro de Rhind: describía 84 problemas destinados a los principiantes de escribas.
Usaban 7 símbolos para los números.
Multiplicaban doblando y sacando mitad.
Dividían duplicando hasta pasar el dividendo.
Para el área del círculo usaban una aproximación, (4/3)4 r2 , donde se tomaba una cercanía al número pi.
Fracciones egipcias: escribían las fracciones como la suma de fracciones unitarias. Fibonacci.
BABILONIA
Los sumerios inventaron la escritura cuneiforme (combinación de signos).
- Dieron el primer sistema de numeración posicional (sexagesimal: base 60).
- Multiplicaban usando la fórmula [(a+b)2-a2-b2]/ 2.
- No tenían algoritmos para la división.
- Trabajaron con ternas pitagóricas. Diofanto.
- Conocían el cuadrado con sus diagonales, círculos con triángulos inscritos y cálculos de lados de rectángulos a partir de su área y perímetro.
MAYA
In some cities such as Palenque and Chichén Itzá there observatories.
His knowledge of astronomy were really good , they could even predict eclipses.
In addition , the Maya had a very precise timetable . They measured time in three different ways :
Calendar Tzolkin ( the sacred calendar ) of 260 days
His mathematical system was based on the number 20:
knew the number zero, this is very important because not all cultures knew her.
They knew add, subtract, multiply and divide.
Your writing system is beginning to decipher but it is a task that will take a long time to finish because it is very complicated.
It was a writing system combining phonetic elements (a symbol for each syllable) and ideographic elements (relating a glyph with a thing).
TABLE NUMBER OF MAYA
Arithmetic Operations In the numbering system Maya
To understand the simplicity and accuracy of mathematical science of the Maya, the use of the board is an indispensable factor; this grid operations and calculations that were counted from the belongings, taxes and the distribution of crops, to astronomical events and time cycles were performed.
As all samples of the culture, the board, which is similar to chess grid, it is full of meanings related to their worldview object; This item represented, in a mystical sense, the fabric of the universe; the field where the facts happen that transform time and space and place where human knowledge is based. Therefore, to understand their role and make use of it, it appears as a figure, symbolically, exemplifies the order and balance of all that exists.
Positioning within the board, calculations and arithmetic operations are performed through mechanisms easy to understand. Levels board increase in value from the bottom up, according to the position having the numeral within that board, as shown below, ordering paragraphs units, scores, scores of scores, scores of scores of scores, etc., so that a point (or drive) at each level would have the following method:
A point in the 6th position 3,200,000
A point in the 5th position 160,000
A point in the 4th position 8,000
A point in the 3rd position 400
A point in the 2nd position 20
A point in the 1st position 1
This mechanism allowed the Mayan calculations with stratospheric numbers; for example, the number 25673295, is represented in Maya follows, using six levels or board positions:
The zero
Mayan mathematics has left a mark in time; before any other civilization, the Maya originated a revolutionary concept: zero, which is a symbol commonly used to represent anything; however, the Mayan concept of zero does not imply an absence or negation; for the Maya, zero has a sense of fullness. For example, typing the number 20, the zero position at the first level, only indicates that the score is complete
MAYA WITH NUMBERS ARITHMETIC OPERATIONS
ADD AND SUBTRACT
To add two or more numbers must be gathered in a single column, bars and points of the same board level and then convert groups of five points in bars and complete scores (sets of four bars) in units the next higher level. For clarity we enunciate the following rules:
the amounts are placed in their respective positions, in columns from left to right on a flat surface, (can be used corn kernels to represent points, sticks for bars and if a shell for zero is possible, if you do not have these elements can be used then pen and paper).
quantities are arranged alongside one another.
grains or figures are grouped the same position retaining their relative values in the first column (ie the left).
For every five points that together form a bar, four bars form a point at the next higher position.
And possibly adding other operations of arithmetic, worked on a table or on the floor, her dots and bars (beans and sticks) are placed. Leon-Portilla suggests that the Code of Dresden, is the representation of a multiplication. Calderon (1966) also describes in very didactic way, the four arithmetic operations plus square root and cube root, the only drawback is that it does not indicate the sources he used.
MULTIPLICATION
Leon-Portilla (1988) notes that in a sheet Dresdre code, different amounts that are multiples of another appear. Some authors suggest that the multiplication process, probably done with repeated addition, for example, Seidenberg (p. 380). "... Could have Maya Priest 23457 Multiplied by 432, say, by Repeated additions of 23457", these findings do the same, probably, by the way multiplication in integers is constructed. In his early mathematical development, probably, this was the way to make multiplications, but considering the large amounts that they ran in their astronomical calculations and the accuracy thereof, it is very logical to think, they should have developed an algorithm to perform multiplication. So far, it has not been possible historically deduce that algorithm.
Regarding this paper we present a simulation of this process to reach a proposal of what might have been the multiplication algorithm in the Maya system.
We begin with the multiplication of a number by 2.
For example: 46 2. We placed in the lattice 46 in two columns and then add. We obtain:
Due to the flooding of the Nile, it was given the need to set limits for land, geometry was born. Herodotus.
"Rope stretchers" Democritus.
According to Egyptian mathematics was divine, for Aristotle are creation of the human mind and Plato are independent of human beings.
Moscow papyrus: the most important part of this paper is the problem of the truncated pyramid or truncated pyramid of square base, due to the shape of the pyramids of Egypt.
Describes the calculation method for volume.
The Sumerians invented cuneiform (combination of signs).
They took the first positional number system (sexagesimal: base 60).
They multiplied using the formula [(a + b) 2-a2-b2] / 2.
They had no algorithms for division.
They worked with Pythagorean triples. Diophantus.
They knew the square with its diagonals, circles with inscribed triangles and rectangles sides calculations from its area and perimeter.
His mathematical system was based on the number 20:
knew the number zero, this is very important because not all cultures knew her.
They knew add, subtract, multiply and divide.
Your writing system is beginning to decipher but it is a task that will take a long time to finish because it is very complicated.
It was a writing system combining phonetic elements (a symbol for each syllable) and ideographic elements (relating a glyph with a thing).
TABLE NUMBER OF MAYA
Arithmetic Operations In the numbering system Maya
To understand the simplicity and accuracy of mathematical science of the Maya, the use of the board is an indispensable factor; this grid operations and calculations that were counted from the belongings, taxes and the distribution of crops, to astronomical events and time cycles were performed.
As all samples of the culture, the board, which is similar to chess grid, it is full of meanings related to their worldview object; This item represented, in a mystical sense, the fabric of the universe; the field where the facts happen that transform time and space and place where human knowledge is based. Therefore, to understand their role and make use of it, it appears as a figure, symbolically, exemplifies the order and balance of all that exists.
Positioning within the board, calculations and arithmetic operations are performed through mechanisms easy to understand. Levels board increase in value from the bottom up, according to the position having the numeral within that board, as shown below, ordering paragraphs units, scores, scores of scores, scores of scores of scores, etc., so that a point (or drive) at each level would have the following method:
A point in the 6th position 3,200,000
A point in the 5th position 160,000
A point in the 4th position 8,000
A point in the 3rd position 400
A point in the 2nd position 20
A point in the 1st position 1
This mechanism allowed the Mayan calculations with stratospheric numbers; for example, the number 25673295, is represented in Maya follows, using six levels or board positions:
The zero
Mayan mathematics has left a mark in time; before any other civilization, the Maya originated a revolutionary concept: zero, which is a symbol commonly used to represent anything; however, the Mayan concept of zero does not imply an absence or negation; for the Maya, zero has a sense of fullness. For example, typing the number 20, the zero position at the first level, only indicates that the score is complete
MAYA WITH NUMBERS ARITHMETIC OPERATIONS
ADD AND SUBTRACT
To add two or more numbers must be gathered in a single column, bars and points of the same board level and then convert groups of five points in bars and complete scores (sets of four bars) in units the next higher level. For clarity we enunciate the following rules:
the amounts are placed in their respective positions, in columns from left to right on a flat surface, (can be used corn kernels to represent points, sticks for bars and if a shell for zero is possible, if you do not have these elements can be used then pen and paper).
quantities are arranged alongside one another.
grains or figures are grouped the same position retaining their relative values in the first column (ie the left).
For every five points that together form a bar, four bars form a point at the next higher position.
And possibly adding other operations of arithmetic, worked on a table or on the floor, her dots and bars (beans and sticks) are placed. Leon-Portilla suggests that the Code of Dresden, is the representation of a multiplication. Calderon (1966) also describes in very didactic way, the four arithmetic operations plus square root and cube root, the only drawback is that it does not indicate the sources he used.
MULTIPLICATION
Leon-Portilla (1988) notes that in a sheet Dresdre code, different amounts that are multiples of another appear. Some authors suggest that the multiplication process, probably done with repeated addition, for example, Seidenberg (p. 380). "... Could have Maya Priest 23457 Multiplied by 432, say, by Repeated additions of 23457", these findings do the same, probably, by the way multiplication in integers is constructed. In his early mathematical development, probably, this was the way to make multiplications, but considering the large amounts that they ran in their astronomical calculations and the accuracy thereof, it is very logical to think, they should have developed an algorithm to perform multiplication. So far, it has not been possible historically deduce that algorithm.
Regarding this paper we present a simulation of this process to reach a proposal of what might have been the multiplication algorithm in the Maya system.
We begin with the multiplication of a number by 2.
For example: 46 2. We placed in the lattice 46 in two columns and then add. We obtain:
EGYPT
The Egyptian priests began to be cultivated by the Egyptian priests, in their free time. Aristotle.Due to the flooding of the Nile, it was given the need to set limits for land, geometry was born. Herodotus.
"Rope stretchers" Democritus.
According to Egyptian mathematics was divine, for Aristotle are creation of the human mind and Plato are independent of human beings.
Moscow papyrus: the most important part of this paper is the problem of the truncated pyramid or truncated pyramid of square base, due to the shape of the pyramids of Egypt.
Describes the calculation method for volume.
using the average of Herón.
The Rhind papyrus: 84 problems described intended for beginners scribes.
They used seven symbols for numbers.
They multiplied by bending and pulling half.
They divided doubling until you pass the dividend.
For the area of a circle using an approximation, (4/3) 4 r2, where a closeness to the number pi was taken.
Egyptian fractions: writing fractions as the sum of unit fractions. Fibonacci.
BABYLON
They took the first positional number system (sexagesimal: base 60).
They multiplied using the formula [(a + b) 2-a2-b2] / 2.
They had no algorithms for division.
They worked with Pythagorean triples. Diophantus.
They knew the square with its diagonals, circles with inscribed triangles and rectangles sides calculations from its area and perimeter.
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