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 have carried out land surveying, calculated the number of workers and their content, or laid out tax deductions. So the appearance of mathematics can be dated to the era of the earliest state formations in Egypt.
Egyptian numerals
The decimal system of counting 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, the Egyptian digital symbols arose as a result of the consonance of a particular numeral and the name of an object, because in the era of writing, pictogram signs had a strictly objective meaning. So, for example, hundreds were marked with a hieroglyph depicting a rope, tens of thousands - with a finger.
In the era of the Middle Kingdom (the beginning of the 2nd millennium BC), a more simplified hieratic form of writing, convenient for writing on papyrus, appears, and the spelling of digital signs changes accordingly. The famous mathematical papyrus are written in hieratic writing. Hieroglyphics was used mainly for wall inscriptions.
The system of ancient Egyptian numbering has not changed for thousands of years. The ancient Egyptians did not know the positional way of recording numbers, since they had not yet come to the concept of zero, not only as an independent quantity, but simply as the absence of quantity in a certain category (the mathematician in Babylon reached this initial stage).
Fractions in Mathematics of Ancient Egypt
The Egyptians had a concept about fractions and were able to perform some operations with fractional numbers. Egyptian fractions are numbers of the form 1 / n (the so-called aliquot fractions), since the fraction was represented by the Egyptians as one part of something. The exceptions are fractions 2/3 and 3/4. An integral element of the notation 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 unity, the Egyptian scribe understood literally as several parts of any number, and literally wrote down. For example, twice in a row 1/5, if you wanted to depict the number 2/5. So the Egyptian fraction system was very cumbersome.
Interestingly, one of the sacred symbols of the Egyptians - the so-called "eye of the Choir" - also has a mathematical meaning. One version of the myth of the battle between the deity of rage and destruction of Seth and his nephew, the solar god Chorus, says that Seth knocked out Chora's left eye and tore or trampled 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 denoting a special series of numbers. These are fractions, each of which is half the previous one: 1/2, 1/4, 1/8, 1/16, 1/32 and 1/64. The symbol of the divine eye, therefore, represents their sum - 63/64. Some mathematics historians believe that this symbol reflects the concept of the Egyptians about geometric progression. The components of the image of the Eye of Horus were used in practical calculations, for example, when measuring the volume of bulk solids, such as grain.
Principles of Arithmetic
The method that the Egyptians used when performing simple arithmetic operations was to calculate the total number of characters denoting digits of numbers. Units were added up with units, tens with tens and so on, after which the final record of the result was made. If the summation resulted in more than ten characters in any category, the "extra" ten passed into the highest category and was recorded with 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 computing the product of two numbers, especially many-valued ones, was extremely cumbersome. As a rule, the Egyptians used the method of sequential doubling. One of the factors was decomposed by the sum of numbers, which we would today call powers of two. For the Egyptian, this meant the number of consecutive doubles of the second factor and the total summation of the results. For example, multiplying 53 by 46, the Egyptian scribe would have decomposed 46 by 32 + 8 + 4 + 2 and compiled a tablet, which 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 amount as we are today, but in a different way. Interestingly, such a binary method of multiplication is used in our time in computer technology.
Sometimes, in addition to doubling, the number could be multiplied by ten (since the decimal system was used) or by five, as by half a dozen. Here is another example of multiplication with the record by Egyptian characters (the slash marks marked the summed results).
The division operation was also carried out according to the principle of doubling the divider. The required number, when multiplied by the divisor, should give the dividend indicated in the condition of the problem.
Egyptian mathematical knowledge and skills
It is known that the Egyptians knew exponentiation, and also used the inverse operation - the extraction of the square root. In addition, they had an idea of progression and solved problems that were reduced to equations. True, the equations as such were not compiled, since there was still no understanding that the mathematical relations between the quantities are universal. Tasks were grouped by topic: land demarcation, distribution of products, and so on.
Under the conditions of the tasks, there is an unknown quantity to be found. It is indicated by the hieroglyph “set”, “heap” and is an analogue of the value “X” in modern algebra. The conditions are often stated in a form that, it would seem, simply requires the preparation and solution of a simple algebraic equation, for example: a “heap” adds up to 1/4, which also contains a “heap”, and it turns out 15. But the Egyptian did not solve the equation x + x / 4 = 15, and selected the desired value, which would satisfy the conditions.
Mathematics of Ancient Egypt achieved significant success in solving geometric problems associated with the needs of construction and land surveying. We know about the circle of tasks that scribes faced and about how to solve them due to the fact that several written monuments on papyrus have been preserved, containing examples of calculations.
Ancient Egyptian problem book
One of the most comprehensive sources on the history of mathematics in Egypt is the so-called mathematical papyrus Rinda (named after its first owner). It is stored 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 papyrus of Ahmes - after the scribe who rewrote this document around 1650 BC. e.
Papyrus is a collection of tasks with solutions. In total, it contains more than 80 mathematical examples in arithmetic and geometry. For example, the task of equal distribution between 10 employees of 9 loaves was solved as follows: 7 loaves are divided into 3 parts each, and 2/3 of the bread is given to the workers, with 1/3 remaining. Two breads are divided into 5 parts each, issued 1/5 per person. The remaining third of the bread is divided into 10 parts.
There is a task for the unequal distribution of 10 measures of grain between 10 people. The result is an arithmetic progression with a difference of 1/8 measure.
The task of geometric progression is comic: 7 cats live in 7 houses, each of which ate 7 mice. Each mouse ate 7 ears of corn, each ear brings 7 measures of bread. It is necessary to calculate the total number of houses, cats, mice, ears of corn and bread measures. It is 19607.
Geometric tasks
Of considerable interest are mathematical examples that demonstrate the level of knowledge of the Egyptians in the field of geometry. This is finding the volume of the cube, the area of the 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 modern cotangent, was called "seked". The main units of length were the elbow, which was 45 cm (“royal elbow” - 52.5 cm) and the hat — 100 cubits, the main unit of area — seshat, equal to 100 square cubits (about 0.28 ha).
The Egyptians successfully coped with calculating the area of triangles, using a method similar to the modern one. Here is the problem from Rind's papyrus: what is the area of a triangle having a height of 10 hatts (1000 cubits) and a base of 4 hatts? As a solution, it is proposed to multiply ten by half from four. We see that the solution method is absolutely correct, it is presented in a specific numerical form, and not in a formalized one - to multiply the height by half the base.
A very interesting problem is the calculation of the area of a circle. According to the above solution, it is equal to the value of 8/9 of the diameter squared. If we now calculate the number "pi" from the resulting area (as the ratio of the quadruple area to the square of the diameter), then it will be about 3.16, that is, pretty close to the true value of "pi". Thus, the Egyptian method of solving the area of the circle was quite accurate.
Moscow papyrus
Another important source of our knowledge of the level of mathematics among the ancient Egyptians is the Moscow Mathematical Papyrus (aka Golenishchev Papyrus), which is stored 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 has a more ancient age - about 200 years older than papyrus Rinda. Most examples in papyrus are geometric, including the task of calculating the area of a basket (that is, a curved surface).
In one of the tasks, a method for finding the volume of a truncated pyramid is given, which is completely similar to the modern formula. But since all the solutions in the Egyptian problem books have a “prescription” character and are given without intermediate logical steps, without any explanation, it remains unknown how the Egyptians found this formula.
Astronomy, math and calendar
Ancient Egyptian mathematics is also associated with calendar calculations based on the recurrence of certain astronomical phenomena. First of all, this is a prediction of the annual rise of the Nile. Egyptian priests noticed that the beginning of a river spill at the latitude of Memphis usually coincides with the day when Sirius becomes visible in the south before sunrise (most of the year this star is not observed at this latitude).
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 clarification and further complication appeared. Without mathematical skills, the priests could not refine the calendar (however, the Egyptians did not succeed in finally eliminating the flaws of the calendar).
No less important was the ability to choose favorable moments for holding various religious festivals, also timed to various astronomical phenomena. So the development of mathematics and astronomy in ancient Egypt, of course, is associated with calendar calculations.
In addition, mathematical knowledge is required for timekeeping when observing the starry sky. It is known that such observations were made by a special group of priests - “watch managers”.
An integral part of the early history of science
When considering the features and level of development of mathematics in Ancient Egypt, significant immaturity is visible, which has not been overcome for three thousand years of the existence of ancient Egyptian civilization. Some informative sources of the era of the emergence 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 “prescription”, subject form without signs of progress for many hundreds of years.
Apparently, a stable and monotonous circle of problems solved using already established methods did not create “demand” for new ideas in mathematics, which already coped with solving the 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.
However, 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 papyruses show us with tasks, the initial stages of generalizing knowledge are already visible - so far without formalization attempts. We can say that the mathematics of Ancient Egypt in the form as we know it (due to the insufficient source base for the late period of ancient Egyptian history) is not a science in the modern sense, but the very beginning of the path to it.