June 21, 2015 at 22:01So the ancients believed. Babylon
This is a continuation of the series I conceived about the history of computing and counting. The first article about Egypt is here .
Now I will try to talk a little about another great civilization and culture of the past. The Babylonian kingdom arose at the beginning of the 2nd millennium BC, it replaced Sumer and Akkad and existed before the Persians conquered in 539 BC. They wrote in Babylon, as everyone remembers, on clay tablets with cuneiform writing, which are very well preserved unlike paper, papyrus, and similar things, so we know quite a lot about Babylon and its mathematics. But of course we do not know everything. Unlike the Greeks, the Babylonians did not leave exact algorithms and clear explanations of their tricks. Now we can only guess exactly how the Babylonians acted in a particular case in solving the problem. In this work, I will focus mainly on Babylonian arithmetic, leaving aside geometry, algebra, and astronomy.
The Babylonians in mathematics moved much further than the Egyptians, as far as we know, although they did not equal the Greeks, apparently. They already knew how to solve quadratic equations, in addition, they had some rudiments of numerical algebra. One of their achievements was the introduction of the positional six-decimal number system without zero. This means that the handling of numbers has become much more flexible and simple than in Egypt. It is not known exactly where such a system came from. One version says that a mixture of the 6-decimal and 10-decimal systems of the peoples of Sumer and Akkad led to it. But there are other thoughts on this subject.
Unfortunately, this system (maybe fortunately, I would not want to learn their multiplication table) was not mastered by other peoples of the Ancient World, and I had to wait for the arrival of the Indian positional system. However, some reflection of Babylonian mathematics in our culture remains: dividing the minute by sixty seconds and the hour by 60 minutes is an echo of the ancient Babylonian number system.
The picture shows how the Babylonians denoted 1 and 10. With their help, all the numbers from 1 to 59 were displayed. The number 33 is shown in the picture below. This is similar to Roman and other non-positional number writing systems.
The number 60 was denoted exactly as the unit. In the beginning, it was drawn larger, but later this difference was erased. Numbers greater than 60, but less than 120, were designated as follows: first the number 60 was written, then the rest of the number, less than 60, separated by a space.
Below is an example of the number 63.
Numbers of the form K * 60 + n (1 <= K <60; n = 1 , 2, 3, ... 59) were designated by analogy, as in the example below.
The Babylonians did not have 0, but over time they came up with the use of a sign that denoted missed bits. This sign was used only for digits inside the number and was not placed at the end. Here is an example in the picture.
The problem is that this number could be read both as 2 * 60 ^ 2 +2, and as 2 * 60 ^ 5 + 2 * 60 ^ 3. Very uncomfortable! Such a recording system should have led to numerous errors, do not you think? The Babylonians tried to separate the discharges very carefully to avoid confusion (much more accurate than me). Nevertheless, in some cases, errors are very likely. Examples of large numbers are known when part of the number was transferred to another line. Try here to figure out what was meant! But the number of errors in the Babylonian texts is small, although they are over.
Fractions were also designated in the same way. Only for the very popular 1/2, 1/3 and 2/3 there were special badges.
Everywhere further I will write down the Babylonian numbers, separating the digits with a comma and the integer part from the fractional one using a semicolon. For example: 177 will be 2.57, etc. Missed digits, I will replace 0.
Since the system of the Babylonians was positional, their calculations were very similar to ours. When subtracting and adding, they simply added and subtracted the numbers bit by bit. An additional plus was that six-decimal digits were designated in a non-positional way using units and tens, and in such a system it is much easier to subtract and add than in our abstract notations, which require learning a special addition table.
Multiplication, as you might guess, was also similar to ours. But how did they use their huge multiplication table? Taught her by heart? They had prepared special tables where they could watch works.
Many Babylonian tables have come down to us from the Babylonians, but they did not include all the products of “single-valued” numbers, like our decimal tables. Their tables began from 1 to 20 inclusive, then the works followed by 30, 40, 50. If the Babylonians wanted to multiply 35 by 47, then he needed to find 35 * 40 in the table, and then 35 * 7 and add it. This required unnecessary action, but in this way it was possible to significantly save space.
Divisions, as an independent action of the Babylonians did not know. Instead, they used inverse multiplication. To do this, of course, they needed tables of inverse numbers. For example, if it was necessary to divide 1.15 by 5, then the Babylonian found 1/5, which in our record would be 0; 12 and multiplied 1.15 by 0; 12. If such a number was not expressed by a finite hexadecimal fraction, then the Babylonians were looking for a number which, when multiplied by a divisor, gave a dividend.
For example, you need to divide 22.45.0 by 6.30. In this case, the following condition is formulated: “What should I take from 6.30 to get 22.45.0? ”The answer is 3.30. Of course, the Babylonians used approximate values when necessary.
Inverse tables looked something like this:
Etc.
In addition to the table of inverse values, the Babylonians had many other tables: squares, cubes, square and cubic roots, and some others.
What tasks were the Babylonians able to solve?
For example, these are:
“10 brothers and 1 whole and 2/3 mines of silver. Brother is higher than brother. How much higher is it, I do not know. The share of the eighth brother is 6 shekels. Brother over brother how much higher? “
The task is to divide the sum between the brothers so that the share of each is an arithmetic progression and find the difference of this progression.
Of course, the Babylonians also solved the problem of interest. Including tasks for compound interest:
“He gave one gur in growth. In how many years will he grow on himself? ”The
percentage is assumed to be 0; 12 per annum. Some scholars have suggested that the Babylonians owned the rudiments of logarithms. Others disagree with them.
Another example includes quadratic equations:
“I add up the area of two squares, and this is 37.5. The side of one square is 2/3 of the side of the other square. 10 added to the side of the larger, 5 added to the side of the smaller. Are these squares the essence of what? ”
In the tables, these tasks are given with an explanation of their solutions. It can be seen that the Babylonians knew quadratic equations and systems of linear equations.
The Babylonians also knew the square roots, which were calculated by approximate formulas:
“The diagonal of the square is 10. Find the side of the square. 10 s 0; 42.30 multiply 7; 5 is the side. 7; 5 s 1; 25 multiply. 10; 25 it gives. "
Now I will try to talk a little about another great civilization and culture of the past. The Babylonian kingdom arose at the beginning of the 2nd millennium BC, it replaced Sumer and Akkad and existed before the Persians conquered in 539 BC. They wrote in Babylon, as everyone remembers, on clay tablets with cuneiform writing, which are very well preserved unlike paper, papyrus, and similar things, so we know quite a lot about Babylon and its mathematics. But of course we do not know everything. Unlike the Greeks, the Babylonians did not leave exact algorithms and clear explanations of their tricks. Now we can only guess exactly how the Babylonians acted in a particular case in solving the problem. In this work, I will focus mainly on Babylonian arithmetic, leaving aside geometry, algebra, and astronomy.
The Babylonians in mathematics moved much further than the Egyptians, as far as we know, although they did not equal the Greeks, apparently. They already knew how to solve quadratic equations, in addition, they had some rudiments of numerical algebra. One of their achievements was the introduction of the positional six-decimal number system without zero. This means that the handling of numbers has become much more flexible and simple than in Egypt. It is not known exactly where such a system came from. One version says that a mixture of the 6-decimal and 10-decimal systems of the peoples of Sumer and Akkad led to it. But there are other thoughts on this subject.
Unfortunately, this system (maybe fortunately, I would not want to learn their multiplication table) was not mastered by other peoples of the Ancient World, and I had to wait for the arrival of the Indian positional system. However, some reflection of Babylonian mathematics in our culture remains: dividing the minute by sixty seconds and the hour by 60 minutes is an echo of the ancient Babylonian number system.
Numbers and number system
The picture shows how the Babylonians denoted 1 and 10. With their help, all the numbers from 1 to 59 were displayed. The number 33 is shown in the picture below. This is similar to Roman and other non-positional number writing systems.
The number 60 was denoted exactly as the unit. In the beginning, it was drawn larger, but later this difference was erased. Numbers greater than 60, but less than 120, were designated as follows: first the number 60 was written, then the rest of the number, less than 60, separated by a space.
Below is an example of the number 63.
Numbers of the form K * 60 + n (1 <= K <60; n = 1 , 2, 3, ... 59) were designated by analogy, as in the example below.
The Babylonians did not have 0, but over time they came up with the use of a sign that denoted missed bits. This sign was used only for digits inside the number and was not placed at the end. Here is an example in the picture.
The problem is that this number could be read both as 2 * 60 ^ 2 +2, and as 2 * 60 ^ 5 + 2 * 60 ^ 3. Very uncomfortable! Such a recording system should have led to numerous errors, do not you think? The Babylonians tried to separate the discharges very carefully to avoid confusion (much more accurate than me). Nevertheless, in some cases, errors are very likely. Examples of large numbers are known when part of the number was transferred to another line. Try here to figure out what was meant! But the number of errors in the Babylonian texts is small, although they are over.
Fractions were also designated in the same way. Only for the very popular 1/2, 1/3 and 2/3 there were special badges.
Everywhere further I will write down the Babylonian numbers, separating the digits with a comma and the integer part from the fractional one using a semicolon. For example: 177 will be 2.57, etc. Missed digits, I will replace 0.
Calculations
Since the system of the Babylonians was positional, their calculations were very similar to ours. When subtracting and adding, they simply added and subtracted the numbers bit by bit. An additional plus was that six-decimal digits were designated in a non-positional way using units and tens, and in such a system it is much easier to subtract and add than in our abstract notations, which require learning a special addition table.
Multiplication, as you might guess, was also similar to ours. But how did they use their huge multiplication table? Taught her by heart? They had prepared special tables where they could watch works.
Many Babylonian tables have come down to us from the Babylonians, but they did not include all the products of “single-valued” numbers, like our decimal tables. Their tables began from 1 to 20 inclusive, then the works followed by 30, 40, 50. If the Babylonians wanted to multiply 35 by 47, then he needed to find 35 * 40 in the table, and then 35 * 7 and add it. This required unnecessary action, but in this way it was possible to significantly save space.
Divisions, as an independent action of the Babylonians did not know. Instead, they used inverse multiplication. To do this, of course, they needed tables of inverse numbers. For example, if it was necessary to divide 1.15 by 5, then the Babylonian found 1/5, which in our record would be 0; 12 and multiplied 1.15 by 0; 12. If such a number was not expressed by a finite hexadecimal fraction, then the Babylonians were looking for a number which, when multiplied by a divisor, gave a dividend.
For example, you need to divide 22.45.0 by 6.30. In this case, the following condition is formulated: “What should I take from 6.30 to get 22.45.0? ”The answer is 3.30. Of course, the Babylonians used approximate values when necessary.
Inverse tables looked something like this:
| 2 | thirty |
| 3 | 20 |
| 4 | fifteen |
| 5 | 12 |
| 6 | 10 |
| 8 | 7; 30 |
| 9 | 6; 40 |
| 12 | 5 |
| fifteen | 4 |
| 16 | 3; 45 |
| 18 | 3; 20 |
| 20 | 3 |
Etc.
In addition to the table of inverse values, the Babylonians had many other tables: squares, cubes, square and cubic roots, and some others.
Tasks
What tasks were the Babylonians able to solve?
For example, these are:
“10 brothers and 1 whole and 2/3 mines of silver. Brother is higher than brother. How much higher is it, I do not know. The share of the eighth brother is 6 shekels. Brother over brother how much higher? “
The task is to divide the sum between the brothers so that the share of each is an arithmetic progression and find the difference of this progression.
Of course, the Babylonians also solved the problem of interest. Including tasks for compound interest:
“He gave one gur in growth. In how many years will he grow on himself? ”The
percentage is assumed to be 0; 12 per annum. Some scholars have suggested that the Babylonians owned the rudiments of logarithms. Others disagree with them.
Another example includes quadratic equations:
“I add up the area of two squares, and this is 37.5. The side of one square is 2/3 of the side of the other square. 10 added to the side of the larger, 5 added to the side of the smaller. Are these squares the essence of what? ”
In the tables, these tasks are given with an explanation of their solutions. It can be seen that the Babylonians knew quadratic equations and systems of linear equations.
The Babylonians also knew the square roots, which were calculated by approximate formulas:
“The diagonal of the square is 10. Find the side of the square. 10 s 0; 42.30 multiply 7; 5 is the side. 7; 5 s 1; 25 multiply. 10; 25 it gives. "