Mathematics in Ancient Egypt: signs, numbers, examples.
The non-positional Egyptian number system, which was used in Ancient Egypt, is clearly introduced to the few surviving papyri. The examples of problems and their solutions in them are so interesting that one can only regret that there are so few of them.
They show that mathematics and the Egyptian number system were closely related to economic needs and practical application... Every year after the flood of the Nile, buildings had to be restored, land plots had to be re-surveyed, calculating the area and boundaries, keeping records of the harvest, and a calendar.
What are positional and non-positional number systems?
The answer lies in the name itself. If the position of a digit affects the result of calculations, we have a positional system of numbers, if not, a non-positional one.
If we write 12, that's twelve, and with the same numbers, 21, that's twenty-one. According to the Egyptian number system: to write 12, you need to use the one symbol twice and the tens symbol once, and 21 will look like one one and two tens, that is, you need to write three digits in total.
Non-positional ones include: the familiar Roman system, in which the numbers were denoted by Roman letters, the Slavic system, where each letter also denoted some number or number. The Roman system coped with its functions in Western Europe until the 16th century.
The number system we use is modern life- positional decimal system.
Non-positional systems were well suited for performing simple arithmetic operations, since complex calculations assumed cumbersome records, which did not interfere with the successful development of algebra and geometry in Ancient Egypt.
How did the Egyptians think?
What is it - the Egyptian number system? To write a number, they used the hieroglyphs denoting certain numbers, the sum of which was equal to the desired value.
There were special designations for the numbers 1, 10, 100, 1000, 10000, 100000, 1,000,000. When writing the desired number, each designation was used up to 9 times. The Egyptian number system was recorded in ascending order: first, one, then tens, hundreds, and so on.
Moreover, they wrote, as a rule, from right to left, but it could have been from left to right, the amount did not change from this. Vertical writing was also used, but then the counting went from top to bottom.
There were two ways of writing:
- Hieroglyphic, in which accepted hieroglyphs were used.
- Hieratic, which was more schematic and convenient in practice.
An excursion into history
The history of the Egyptian number system originated in ancient times, the first manuscripts with numbers date back to the second millennium BC. There was no money then, so the system was used both for incredible complexity and greatness of mathematical problems, and for solving everyday everyday problems.
After all, knowledge of mathematics was used both in land surveying, and in the construction of calendars, maps in astronomy, navigation, in the construction of palaces, canals and military fortifications.
The Egyptian non-positional number system was used until the 10th century AD.
It also had a mystical meaning, the secret of which the priests took with them, but Pythagoras partially revealed to the world. He has works in which he describes the symbolic meanings that are attached to digital hieroglyphs, written by him after his stay in Egypt. Therefore, their description is attributed to the Egyptian number system.
Only a few papyri of those times have survived, by which one can understand that the level of mathematics was high. It is known for certain that the Greeks studied ancient Egyptian mathematics. One of the secret knowledge is the Egyptian non-positional number system.
Ahmes Papyrus
The Ahmes papyrus dates from 1650 BC, contains 84 math problems... It was found in Thebes and is now in the British Museum.
All tasks in papyrus are considered on specific examples Egyptian number system. They show examples of calculations with fractions, with integers, division and multiplication.
Calculations are given to find areas geometric shapes: quadrilateral, circle, triangle.
Information from the papyrus proves that Egyptian mathematicians knew how to extract the root, compose arithmetic and geometric progression, equations with unknowns.
Aliquot fractions
Interestingly, only aliquots were used in the calculations, in which the numerator was equal to one and was denoted by such a sign, and the denominator values were written below it, and all other fractions for calculations had to be first expanded to aliquots. But the fractions 2/3 and 3/4 were used and had a special designation.
To bring ordinary fractions into the state of aliquots according to the Egyptian number system, it was necessary to work hard:
4/5 = 16/20 = 10/20 + 5/20 + 1/20 = 1/2+1/4 + 1/20
2/5 = 1/5 + 1/5, 2/7 = 1/4 + 1/28
3/7 = 12/28 = 24/56 = 14/56+7/56+3/56 = 1/4+1/8+1/18+1/56.
Fractions were added in a modern way: by converting to a common denominator, for many values there were numerous ready-made tables.
Multiplication
The Egyptians learned the desired result without knowing the multiplication table, but using the knowledge that if one factor is doubled and the other is reduced, the result will not change:
32*13=16*26=8*52=4*104=2*208=1*416
It is interesting that this method of multiplication was known in Russia, and it was believed that it came from Ancient Egypt, and in Europe it was called Russian.
Golenishchev papyrus
Thanks to the efforts of the Egyptologist V.S. Golenishchev, the papyrus is kept in Moscow for another 200 years. older than papyrus the scribe Ahmes. The scientist bought it during his work at Thebes.
It was written in a hieratic way, in italics, 25 problems are considered, their description according to the Egyptian number system and the solution are given. Its length is more than 5 m and a width of 7 cm. There are no comments to these problems, as in the previous papyrus, there are only mathematical calculations.
He shows that the Egyptians knew how to calculate the areas of a triangle, trapezoid, rectangle, circle, as well as the volumes of a pyramid, prism, parallelepiped, cylinder and truncated pyramid with great accuracy, and many formulas completely coincide with modern ones.
Under the Egyptian number system, the number "pi" 3.16 was calculated, which almost corresponded to modern meaning 3.14, although at that time the value of 3 was used throughout the East.
All things are numbers
It is believed that Pythagoras lived in Egypt for 22 years, deeply studying geometry, philosophy, mysticism of numbers. The discoveries that the Pythagorean school later made could well have been made in Ancient Egypt.
Therefore, it is believed that the works of Pythagoras on the mysticism of numbers, which he wrote later, are based on secret knowledge received by him from the Egyptian priests. They did not accept foreigners for training, he came to them under high patronage, after an interview with the chief priest, who considered him worthy to be initiated into secrets.
Numbers were living entities, reflecting the properties of space, music, energy. Everything can be expressed through mathematics, describing visible phenomena by formulas, predicting invisible ones, relying on logic and mathematical laws.
The height, width of the base, the angle of inclination of the Cheops pyramid in Egypt correspond to mathematical rule the construction of the pyramid of Pythagoras, which also confirms the relationship between the discoveries made by him and the knowledge received from the ancient Egyptian priests who used the Egyptian number system.
Working with numbers, ancient thinkers not only understood the essence of things, but could also influence them.
Studying the mathematics of Ancient Egypt, using the Egyptian number system, one can only admire how much was revealed to people thousands of years before our era.
The official language of modern Egypt is the so-called "high" Arabic.Arabic script, including dialectal, is written and read from right to left. There are no capital letters anywhere - not even in proper names and geographical names... But be careful: numbers are written and read from left to right. If you want to understand coins and prices, it is better to learn Arabic numerals, rather than what we used to call Arabic numerals.
On closer examination of the issue, it turns out that our "Arabic" numbers are partially, but far from completely derived from the real Arabic numerals... According to some sources, the numbers 2, 3, 7 originated from the Arabic ones by rotating them 90 degrees for more convenience in writing. If you do not find fault with much, then it looks like the truth. The numbers 1 and 9 are also of Arabic origin, and no spelling twists have been affected. Indeed, the similarities are obvious here, which cannot be said about 4, 5, 6, and 8.
Sometimes it seems that mathematical symbols are a non-national scientific tool, common and uniform for all countries and peoples.
However, our "Arabic" numbers differ, as you already understood, from the "Arabic" numbers in Egypt. The European positional system for writing numbers from high-order to low-order, from left to right, is also not the only one. In the East, a system of recording numbers from right to left is also used. In Egypt, numbers are written and read from left to right, just like ours.
Car numbers in Egypt with real Arabic numerals.
Both Arabic and Latin scripts are often used on road signs and street names.
Arabic alphabet - the alphabet used for writing Arabic and (most often in modified form) some other languages, in particular Persian and some Turkic languages. It consists of 28 letters and is used for right-to-left writing. The Arabic alphabet originated from the Phoenician alphabet by including all its letters and adding letters to them reflecting specifically Arabic sounds. These are the letters - sa, ha, zal, dad, za, gain.
The letters have four graphic positions (style, spelling):
- independent(isolated, isolated from other letters), when the letter has no connection either to the right of itself or to the left;
- initial, that is, having a connection only on the left (except for alif, zal, dal, zane, pa, vav);
- median, that is, having a connection on both the right and left;
- the final(with connection on the right side only).
Consonant notation
Each of the 28 letters, except for the letter Alif, denotes one consonant. The shape of the letters changes depending on the location within the word. All letters of one word are written together, with the exception of six letters (alif, dal, zal, ra, zai, vav), which do not connect with the next letter.
"Alif" is the only letter of the Arabic alphabet that does not denote any consonant sound. Depending on the context, it can be used to indicate a long vowel a, or as an auxiliary spelling sign that does not have its own sound.
Vowel notation
The three long vowel sounds of the Arabic language are designated by the letters "alif", "vav", "ya". Short vowels are usually not conveyed in writing. In cases where it is necessary to convey the exact sound of a word (for example, in the Qur'an and in dictionaries), superscript and subscript vowels (harakat) are used to indicate vowels.
The 28 letters above are called Huruf. In addition to them, three additional characters are used in the Arabic script, which are not independent letters of the alphabet.
1. Hamza (guttural stop) can be written as a separate letter, or on a "stand" letter ("alif", "vav" or "ya"). The way of writing hamza is determined by its context in accordance with the series spelling rules... Regardless of the way of writing, hamza always means the same sound.
2. Ta-marbuta ("tied ta") is the form of the letter ta. It is written only at the end of a word and only after the vowel is made. When the letter ta-marbuta has no vowel (for example, at the end of a phrase), it is read like the letter ha. The common form of the letter ta is called "open ta".
3. Alif-maksura ("shortened alif") is the shape of the letter Alif. It is written only at the end of a word, and is shortened to a short sound a before alif-wasla next word(in particular, before the prefix al-). The usual form of the letter Alif is called "elongated alif".
The Egyptians invented this system about 5,000 years ago. This is one of the most ancient systems of notation for numbers, known to man.
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1. Like most people, the Egyptians used sticks to count a small number of objects. |
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If several sticks need to be depicted, then they were depicted in two rows, and in the lower one there should be the same number of sticks as in the upper one, or one more. |
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10.The Egyptians used these bonds to tie the cows |
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If you need to depict several dozen, then the hieroglyph was repeated the right amount once. The same applies to the rest of the hieroglyphs. |
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100. This is a measuring rope that was used to measure land after the Nile flood. |
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1,000. Have you ever seen a blooming lotus? If not, then you will never understand why the Egyptians assigned such a meaning to the image of this flower. |
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10,000. "B large numbers be careful! "says the raised index finger. |
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100,000. This is a tadpole. Common frog tadpole. |
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1 000 000. Seeing such a number a common person will be very surprised and raise his hands to the sky. This is what this hieroglyph depicts. |
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10,000,000. The Egyptians worshiped Amun Ra, the god of the Sun, and this is probably why they depicted their largest number in the form of the rising sun |
The digits of the number were written starting with large values and ending with smaller ones. If there were no tens, units, or some other category, then they went on to the next category.
- 1207, - 1 023 029
Try to add these two numbers, knowing that you cannot use more than 9 of the same kanji.
Ancient greek numbering
In ancient times, the so-called Attic numbering was widespread in Greece. In this numbering, the numbers 1, 2, 3, 4 were depicted by the corresponding number of vertical stripes : , , , . The number 5 was written with a sign (the ancient outline of the letter "Pi", with which the word "five" began - "pente". Numbers 6, 7, 8, 9 were designated by combinations of these signs: .
The number 10 was designated - capital "Delta" from the word "deck" - "ten". Numbers 100, 1,000 and 10,000 were designated H, X, M. Numbers 50, 500, 5,000 were designated by combinations of numbers 5 and 10, 5 and 100, 5 and 1,000.
Around the third century BC, Attic numbering in Greece was supplanted by another, the so-called "Ionian" system. In it, the numbers 1 - 9 are denoted by the first letters of the Greek alphabet:
numbers 10, 20, ... 90 were represented by the following nine letters: ѓ
numbers 100, 200, ... 900 with the last nine letters:
To designate thousands and tens of thousands, they used the same numbers, but only with the addition of a special sign. "Any letter with this sign immediately became a thousand times larger.
To distinguish numbers and letters, dashes were written above the numbers.
In antiquity, Jews, Arabs and many other peoples of the Middle East had an organized number system based on approximately the same principle.
Few people think that the techniques that we use for writing and invoicing have been formed over many thousands of years. They seem obvious to us, well, just think, multiply in a column, transfer all the terms with the unknown to one side. It's so easy! In fact, these are huge intellectual conquests of mankind, which were often inaccessible. the smartest people of the past. I'm going (if I have the patience and time) to write a few notes about how they believed in the past. In this I will tell you about how the Egyptians believed.
I've always been a little interested in ancient Egypt. Well, firstly, Egypt is one of the first states that we know a lot about, and besides, it is a very great state that left a huge legacy. I do not mean the sheer size of the pyramids. Even our writing, both Latin and Cyrillic, dates back to ancient egypt... I've also always liked Egyptian sculpture and the fashion for women and men to shave their heads. It sounds very modern. But this article is not about artistic culture. So let's get started.
Numbers and numbers
Egyptians used non-positional decimal system reckoning. The numbers looked something like this:
These figures refer to the so-called. hieroglyphic writing, which was later replaced by hieratic. I love hieratic writing very much. It looks very stylish. But here I will use hieroglyphic style.
All integers were formed by repeating the signs above (and some others for even higher digits). For example 3215 would be:
A very clear system, although not too laconic. It is easy to learn, but the numbers are not very convenient. It's hard to grasp at first glance exact value numbers. The Egyptians wrote in different directions, but I write here as we are accustomed to from left to right.
Now about fractions. There were special icons for three fractions:
All other fractions that had one in the numerator were indicated by the denominator and an eye-like icon at the top. For example, below I wrote 1/14 
All correct fractions were written as the sum of such fractions. For instance: 
On one site I read that "in some cases" Egyptian fractions are "better than ours." And even in the English wiki, there is such a wonderful example: “Egyptian fractions sometimes make it easier to compare the size of fractions. For example, if someone wants to know if 4/5 is greater than ¾, he can turn them into Egyptian fractions:
4/5= 1/2 + 1/4+ 1/20
3/4 = 1/2 +1/4 "
To me this " easy way"Reminds a joke about Feynman, who for the sake of some task school course summed up the ranks in my mind. I am a humanitarian and I am not particularly good at counting, but comparing in my mind regular fractions in their normal way it seems to me much easier than translating them into Egyptian form. Perhaps, for the Egyptians, comparisons of this kind were more convenient, since they did not know our fractions.
Addition and multiplication
Well, here we come to the main point. How did the Egyptians think? The addition and subtraction of integers for them was the same as for us, and maybe even easier, because they just needed to combine the hieroglyphs and take into account the change in digits. What about multiplication and division? In the ancient Egyptian world, this was not at all a trivial task.
The Egyptians used such an algorithm for multiplication. Numbers were written in two columns. The first column started at one, and the second at the multiplicable. Then each number in the column was doubled until a factor could be added from some of the numbers in the first column. Did you understand? An example is better understood. For example, 7 by 22
1+4+8=13
and 57+228+456=741
Sometimes, to speed up the process, they resorted to multiplying by 10.
The question may arise, is it always possible to represent the factor in this form? Yes, in fact we are actually dealing with a binary number system: 1*2 0 +0*2 1 +1*2 2 +1*2 3
those. 1+100+ 1000=1101
The division was performed using a similar algorithm. Divide 238 by 17:
Again we draw up a plate on one side, which is 17 on the other, a unit. The doubling process stops at a number that, when doubled, will be greater than the dividend.
We will stop here, because 128 by 2 = 256, which is more than 213.128 + 64<213. 128+64+32 уже опять больше. Не подходит. 128+64+16<213 Пока все ОК. 128+64+16+8 уже больше. Значит мы смогли набрать только 208=128+64+16 из 213. И нам осталось разделить 213-208=5
We divide the divisor by sex using the already familiar table. Luckily, 5 is 1 + 4.
| 1/2* | 4 |
| 1/4 | 2 |
| 1/8* | 1 |
So the final result will be
213/8 = 2+8+16+1/2+1/8 =26+1/2+1/8
Now we have a good case, but this is not always the case.
The origin of mathematical knowledge among the ancient Egyptians is associated with the development of economic needs. Without mathematical skills, ancient Egyptian scribes could not provide for the conduct of land surveying, calculate the number of workers and their maintenance, or produce the layout of tax deductions. So the emergence of mathematics can be dated to the era of the earliest state formations in Egypt.
Egyptian numeric designations
The decimal counting system in Ancient Egypt was based on the use of the number of fingers on both hands for counting objects. Numbers from one to nine were indicated by the corresponding number of dashes, for tens, hundreds, thousands, and so on, there were special hieroglyphic signs.
Most likely, digital Egyptian symbols arose as a result of the consonance of one or another numeral and the name of an object, because in the era of the formation of writing, pictogram signs had a strictly objective meaning. So, for example, hundreds were designated by a hieroglyph depicting a rope, tens of thousands - by a finger.
In the era (the beginning of the 2nd millennium BC), a more simplified, convenient for writing on papyrus, hieratic form of writing appears, and the writing of digital signs changes accordingly. The famous mathematical papyri are written in hieratic script. Hieroglyphics were used mainly for wall inscriptions.
Hasn't changed for thousands of years. The ancient Egyptians did not know the positional way of writing numbers, since they had not yet approached the concept of zero, not only as an independent quantity, but simply as the absence of quantity in a certain category (mathematics reached this initial stage in Babylon).
Fractions in Ancient Egyptian Mathematics
The Egyptians knew about fractions and knew how to perform some operations with fractional numbers. Egyptian fractions are numbers of the form 1 / n (so-called aliquots) because the Egyptians thought of a fraction as one part of something. The exceptions are the fractions 2/3 and 3/4. An integral part of the recording of a fractional number was a hieroglyph, usually translated as "one of (a certain amount)". For the most common fractions, there were special signs.
The fraction, the numerator of which is different from one, the Egyptian scribe understood literally, as several parts of a number, and literally wrote it down. For example, twice in a row 1/5, if you wanted to represent the number 2/5. So the Egyptian system of fractions was quite cumbersome.
Interestingly, one of the sacred symbols of the Egyptians - the so-called "eye of Horus" - also has a mathematical meaning. One version of the myth of the battle between the deity of rage and destruction, Seth and his nephew, the sun god Horus, says that Seth gouged Horus out of his left eye and tore or trample it. The gods restored the eye, but not completely. The Eye of Horus personified various aspects of the divine order in the world order, such as the idea of fertility or the power of the pharaoh.
The image of the eye, revered as an amulet, contains elements that denote a special series of numbers. These are fractions, each of which is half the size of the previous one: 1/2, 1/4, 1/8, 1/16, 1/32 and 1/64. The symbol of the divine eye thus represents their sum - 63/64. Some mathematical historians believe that this symbol reflects the Egyptians' concept of a geometric progression. The constituent parts of the image of the Eye of Hoare have been used in practical calculations, for example, when measuring the volume of bulk solids such as grain.
Principles of arithmetic operations
The method used by the Egyptians when performing the simplest arithmetic operations was to calculate the total denoting digits of numbers. Units were added with ones, tens with tens, and so on, after which the final recording of the result was made. If, when summing up, more than ten characters were obtained in any category, the "extra" ten passed into the highest category and was written in the corresponding hieroglyph. Subtraction was performed in the same way.
Without the use of the multiplication table, which the Egyptians did not know, the process of calculating the product of two numbers, especially multi-valued ones, was extremely cumbersome. As a rule, the Egyptians used the method of successive doubling. One of the factors was expanded into the sum of numbers, which today we would call powers of two. For the Egyptian, this meant the number of consecutive doublings of the second factor and the final summation of the results. For example, multiplying 53 by 46, the Egyptian scribe would factor 46 into 32 + 8 + 4 + 2 and make up the tablet you can see below.
| * 1 | 53 |
| * 2 | 106 |
| * 4 | 212 |
| * 8 | 424 |
| * 16 | 848 |
| * 32 | 1696 |
Summing up the results in the marked lines, he would get 2438 - the same as we do today, but in a different way. It is interesting that such a binary multiplication method is used in our time in computing.
Sometimes, in addition to doubling, the number could be multiplied by ten (since the decimal system was used) or by five, like half a ten. Here is another example of multiplication with Egyptian symbols (the results to be added were marked with a slash).
The division operation was also carried out according to the principle of doubling the divisor. The required number, when multiplied by the divisor, should have given the dividend specified in the problem statement.
Egyptian mathematical knowledge and skills
It is known that the Egyptians knew exponentiation, and also used the inverse operation - extraction of the square root. In addition, they had an idea of the progression and solved problems that reduce to equations. True, the equations as such were not compiled, since the understanding of the fact that the mathematical relations between quantities are universal in nature has not yet developed. The tasks were grouped by subject: demarcation of lands, distribution of products, and so on.
In the conditions of the problems, there is an unknown quantity that needs to be found. It is designated by the hieroglyph "set", "heap" and is analogous to the value "x" in modern algebra. The conditions are often stated in a form that would seem to simply require the compilation and solution of the simplest algebraic equation, for example: "heap" is added to 1/4, which also contains "heap", and it turns out 15. But the Egyptian did not solve the equation x + x / 4 = 15, and selected the desired value that would satisfy the conditions.
The mathematician of Ancient Egypt achieved significant success in solving geometric problems associated with the needs of construction and land surveying. We know about the range of tasks that the scribes faced, and about the ways to solve them, thanks to the fact that several written monuments on papyrus have survived, containing examples of calculations.
Ancient Egyptian problem book
One of the most complete sources on the history of mathematics in Egypt is the so-called Rinda mathematical papyrus (named after the first owner). It is kept in the British Museum in two parts. Small fragments are also in the Museum of the New York Historical Society. It is also called the Ahmes Papyrus, after the scribe who copied this document around 1650 BC. e.
The Papyrus is a collection of problems with solutions. In total, it contains over 80 mathematical examples in arithmetic and geometry. For example, the problem of an equal distribution of 9 loaves among 10 workers was solved as follows: 7 loaves are divided into 3 parts each, and the workers are given 2/3 of the bread, while the remainder is 1/3. Two loaves of bread are divided into 5 parts each, 1/5 per person is given out. The remaining third of the bread is divided into 10 parts.
There is also a problem of unequal distribution of 10 measures of grain among 10 people. The result is an arithmetic progression with a difference of 1/8 of the measure.
The geometric progression problem is humorous: 7 cats live in 7 houses, each of which ate 7 mice. Each mouse ate 7 spikelets, each ear brings 7 measures of bread. You need to calculate the total number of houses, cats, mice, ears of corn and grain measures. It is 19607.
Geometric problems
Of considerable interest are mathematical examples demonstrating the level of knowledge of the Egyptians in the field of geometry. This is finding the volume of a cube, the area of a trapezoid, calculating the slope of the pyramid. The slope was not expressed in degrees, but was calculated as the ratio of half the base of the pyramid to its height. This value, similar to the modern cotangent, was called "seked". The main units of length were the cubit, which was 45 cm ("king's cubit" - 52.5 cm) and the hat - 100 cubits, the main unit of area - seshat, equal to 100 square cubits (about 0.28 hectares).
The Egyptians were successful in calculating the areas of triangles using a method similar to the modern one. Here is a problem from the Rinda papyrus: What is the area of a triangle that has a height of 10 hetes (1000 cubits) and a base of 4 hetes? As a solution, it is proposed to multiply ten by half of four. We see that the solution method is absolutely correct, it is presented in a concrete numerical form, and not in a formalized one - to multiply the height by half the base.
The problem of calculating the area of a circle is very interesting. According to the solution given, it is equal to 8/9 of the diameter squared. If we now calculate the number "pi" from the resulting area (as the ratio of the quadrupled area to the square of the diameter), then it will be about 3.16, that is, quite close to the true value of "pi". Thus, the Egyptian way of solving the area of a circle was quite accurate.
Moscow papyrus
Another important source of our knowledge about the level of mathematics among the ancient Egyptians is the Moscow Mathematical Papyrus (aka the Golenishchev Papyrus), which is kept in the Museum of Fine Arts. A.S. Pushkin. This is also a problem book with solutions. It is not so extensive, contains 25 tasks, but is older - about 200 years older than the Rinda papyrus. Most of the examples in papyrus are geometric, including the problem of calculating the area of a basket (that is, a curved surface).
In one of the problems, a method for finding the volume of a truncated pyramid is presented, which is completely analogous to the modern formula. But since all the solutions in the Egyptian problem books have a "recipe" character and are given without intermediate logical stages, without any explanation, it remains unknown how the Egyptians found this formula.
Astronomy, mathematics and calendar
Ancient Egyptian mathematics is also associated with calendar calculations based on the recurrence of certain astronomical phenomena. First of all, it is the prediction of the annual rise of the Nile. The Egyptian priests noticed that the beginning of the flooding of the river at the latitude of Memphis usually coincides with the day when Sirius becomes visible in the south before sunrise (this star is not observed at this latitude for most of the year).
Initially, the simplest agricultural calendar was not tied to astronomical events and was based on a simple observation of seasonal changes. Then he received an exact reference to the rise of Sirius, and with it the possibility of refinement and further complication appeared. Without mathematical skills, the priests could not have specified the calendar (however, the Egyptians did not succeed in completely eliminating the shortcomings of the calendar).
No less important was the ability to choose favorable moments for holding certain religious festivals, also timed to coincide with various astronomical phenomena. So the development of mathematics and astronomy in Ancient Egypt, of course, is associated with the maintenance of calendar calculations.
In addition, mathematical knowledge is required for timekeeping when observing the starry sky. It is known that such observations were carried out by a special group of priests - "watch managers".
An integral part of the early history of science
Considering the features and level of development of mathematics in Ancient Egypt, one can see a significant immaturity, which has not yet been overcome in the three thousand years of the existence of the ancient Egyptian civilization. Any informative sources of the era of the formation of mathematics have not reached us, and we do not know how it happened. But it is clear that after some development, the level of knowledge and skills froze in a “recipe”, subject form without signs of progress for many hundreds of years.
Apparently, a stable and monotonous range of issues solved using already established methods did not create a "demand" for new ideas in mathematics, which already coped with solving problems of construction, agriculture, taxation and distribution, primitive trade and maintenance of the calendar, and early astronomy. In addition, archaic thinking does not require the formation of a strict logical, evidence base - it follows the recipe as a ritual, and this also affected the stagnant nature of ancient Egyptian mathematics.
At the same time, it should be noted that scientific knowledge in general and mathematics in particular took the first steps, and they are always the most difficult. In the examples that the papyri with tasks demonstrate to us, the initial stages of generalization of knowledge are already visible - so far without any attempts at formalization. We can say that the mathematics of Ancient Egypt in the form as we know it (due to the lack of a source base for the late period of ancient Egyptian history) is not yet science in the modern sense, but the very beginning of the path to it.