**Mathematics in Antiquity. **Thanks to the papyri that remain, the structure and level reached by the Mathematics of Antiquity are well known. Almost without exception, it is an empirical mathematics developed as “recipes”, and which tried to solve obvious practical problems, such as surveying, tax calculation, deposit volume determination, etc .; administrative problems treated mathematically and that belonged to the scope of the scribes.

## Summary

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- 1 Story
- 2 Three classic problems of Greece
- 3 Rhind Papyrus
- 1 Some Rhind Papyrus Problems

- 4 The Moscow papyrus
- 5 Sources
- 6 External link

## History

From the beginning of history, the main mathematical disciplines arose from the need of man to make calculations in order to control taxes and commerce, understand the relationships between numbers, the measurement of land and the prediction of astronomical events.

These needs are closely related to the main properties that mathematics studies – quantity, structure, space, and change. Since then, mathematics has had a profuse development and there has been a fruitful interaction between mathematics and science, for the benefit of both.

Various mathematical discoveries have happened throughout history and continue to occur today. In addition to knowing how to count physical objects, prehistoric men also knew how to count abstract quantities such as time (days, seasons, years, etc.). They also began to master elementary arithmetic ( addition , subtraction , multiplication, and division ).

It is easy to see how the vast majority of parts of mathematics appear in different games:

– Arithmetic is immersed in magic squares, currency exchanges, …

– The elementary number theory is the basis of many divination games based on divisibility criteria, it appears in games that involve different numbering systems, …

– The combinatorics is the key piece of all the games in which you are asked to list the different ways of running a business. Many of them still unresolved, such as the traveling salesman problem.

– Algebra is the basis of many riddles at about ages, measures.

– Group theory is an instrument of vital importance to analyze certain games with tiles on a board in which, like the checkers, tiles are removed when making movements.

Let us now begin a brief tour of the different mathematical games that have been appearing throughout the history of humanity.

The ‘ Papyrus of Rhind ‘ or ‘Ahmes’ (work of the Egyptian civilization, which can be seen in the following figure), found in an ancient building in Thebes, dates from the year 1850 BC . It is a writing that shows us the mathematics of the time. It contains a compilation of various problems whose resolution is carried out mainly through trial and error-based methods. With him it is shown how in the mathematics of that civilization games already appeared as riddles.

## Three classic problems of Greece

- Squaring the circle, doubling the cube, and trisecting the angle. The astute Cretans considered the construction of these three figures only using the ruler and the compass, which has been found to be impossible today.
- The squaring of the circle, which Anaxagoras first considered, consists of making a square with the same area as that of a given circle.

It took more than two thousand years for Ferdinand Lindeman ( 1852 – 1939 ) to demonstrate that such a construction with a ruler without marks and compass was impossible, since pi is a transcendent number.

- Duplication of the cube resides in building the side of a cube whose volume is twice the volume of a given initial cube. That is, given a cube of edge a and volume V, find the edge of a cube of volume 2V.

Many centuries had to pass before they could prove that this problem had no solution in the way the Greeks posed it. And the reason is that if we use Cartesian coordinates, this problem consists in calculating x³ = 2.

The French geometer L. Wantzel was commissioned in 1837 to demonstrate in one of his works that this feat was impossible with the simple use of these two elements.

- Trisection of the angle, this was the third Greek problem. The work consisted of trisecting an angle only with a ruler (not graduated) and a compass.

The Greeks themselves knew that for certain angles, with specific characteristics, this was possible. But in general, this problem, like the previous two, has no solution under those conditions. It was the French mathematician Pierre Wantzel (1814-1848) who formally proved that an angle w is trisecable with ruler and compass if the polynomial 4x³ – 3x – cos (w) is reducible.

In the same way, it was also PL Wantzel who in 1837 published for the first time, in a French mathematics magazine, the first rigorous proof on the impossibility of trisecting the angle with ruler and compass. Even so, there are still mathematicians who reject this test and continue investigating, believing that they have reached the solution to the problem many times.

## Rhind Papyrus

Rhind Papyrus.jpg

According to Herodotus, the Egyptians are the parents of Geometry , but thanks to their monuments and papyri we also know today that they had an additional numbering system that allowed them to work with fractions in a very special way since the numerator was always the unit. Egyptian papyrus is less resistant to the passage of time than Babylonian tablets. However, some have reached us.

The most popular are the Rhind papyrus and the Moscow papyrus . In them appears a collection of more than 100 problems that provide us with valuable information on Egyptian mathematics. The scribe Ahmes, who wrote as a child in the famous Rhind Papyrus, a kind of book in which 87 mathematical problems appear, which are supposed to have been elaborated by 2 or 3 mathematicians of the time, the Egyptians, like the Babylonians, also worked with fractions, with parts of the unit.

But the funny thing is that they only used fractions with numerators for the unit, that is, in the form: 1/2, 1/3, 1/4, 1/7, 1/15, 1/47 … Any part of the unit they expressed it as the sum of fractions of this type. The Rhind Papyrus contains a table for converting unit parts to these fractions.

It is the 3,000-year-old equivalent of our multiplication tables, only to work with fractions. This material was purchased by the English Egyptologist with the last name Rhind, in the middle of the 19th century and later acquired by the English museum where it is still preserved, which arithmetic problems appear.

### Some Rhind Papyrus Problems

- Problem 25: A quantity and half of this quantity equals 16. What is that quantity?

The solution is immediate if you see that half the amount is one third of the total, then that third part is 16/3 (16: 3) and the amount is 32/3 (double). At that time it was not solved thus, but applying a more complex method of the False position and the answer Ahmes gave it in the following way 10 + 1/2 + 1/6 because, in Egypt, at that time, only the fractions of numerator 1 were worked (= 1/8). The sum is now reduced to 1/2 1/4 1/8 1/8 and then makes equivalent sums to apply the reduction method 1 / 1/4 1/8 1/8 = 1/2 1/4 1 / 4 = 1/2 1/2 = 1

- Problem 26. A quantity and its quarter become 15, and you are asked to calculate the quantity.

For us this problem translates into solving the equation x + 1 / 4x = 15. We reproduce the steps of the papyrus, and below the explanation of each of them. Ahmes writes: “Take 4 and then you get 1/4 of it in 1, in total 5” Ahmes starts in this case from an estimated value of x = 4, the easiest to cancel the fraction, and calculates 4+ 1 / 4 * 4 = 5.

“Divide by 5 15 and you get 3” Now to find out the real value you have to find a number N such that when multiplied by the result of applying the estimated value, we get 15, that is, 5 * N = 15, N = 15/5 = 3 3. “Multiply 3 by 4 obtaining 12” The value sought is the result of multiplying the previous N by the initial estimated value, this is 3 * 4 which is the quantity sought.

Ahmes follows later: “whose (referred to the previous 12) 1/4 is 3, in total 15”

- Problem 31. It literally says: “A quantity, its 2/3, its 1/2, its 1/7, its total amounts to 33”

For us this means an equation 2x / 3 + x / 2 + x / 7 + x = 33, x = quantity Ahmes solves the problem using complicated division operations.

- Problem 79: There was a property consisting of 7 houses, each house had 7 cats, each cat ate 7 mice, each mouse ate 7 barley grains, each grain had produced 7 measurements. How much did all of this add up to?

## The Moscow papyrus

Moscow papyrus.jpg

Also known as the Golenischev Papyrus, it is almost as long as the Rhid Papyrus but only about seven centimeters wide. It is written by an unknown scribe of the 12th dynasty (about 1890 BC) and was purchased in Egypt in 1893 , retaining in Moscow, hence the name. It is a collection of twenty-five solved problems, on daily issues, that do not differ much from those of Ahmés .

Compounded more sloppily than the previous one, however, there are two geometric problems of particular importance: In Problem 10, the scribe asks for the area of a surface of what appears to be a hemispherical basket of diameter 4 1/2, and proceeds to calculate it, surprising result for the time. Other analyzes of the problem suggest that it could have a simpler interpretation and could be the estimation of the area of a semi-cylindrical surface of length and diameter 4 1/2.

Number 14 presents a figure that seems to represent a trapezoid, but the associated calculations indicate that it is actually a pyramid trunk whose volume it calculates.