History of mathematics and some problems

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1 The legend of the chess game (3000 BC)
2 The lunulae of Hippocrates of Chio (5th century BC)
3 A problem of Euclid (3rd century BC)
4 Who was Euclid?
5 The epitaph of Diophantus (3rd century)
6 Who was Diophantus?
7 The Fibonacci Fountain (1175-1240)
8 Who was Fibonacci?
9 A poem by Nicolas Boileau-Despréaux (1636-1711)
10 Varignon’s theorem (1654-1722)
11 Who was Varignon?
12 The Gauss formula (1777-1855)
13 Who was Gauss?
14 The Sierpinski triangle (1882-1969)
15 Who was Sierpinski?

The history of mathematics goes back millennia, with evidence of the use of numerical systems and measurements in ancient civilizations such as the Egyptians and Mesopotamians. Mathematics has continued to evolve over the centuries, with advances in many areas, such as geometry, algebra, and the natural sciences.

One of the first great advances in mathematics was the appearance of Euclidean geometry in ancient Greece. The Greek mathematician Euclid defined the concepts of point, line, plane and space, as well as rules for drawing geometric figures and calculating their properties. These concepts have provided a solid foundation for many other developments in mathematics over the centuries.

In the Middle Ages, mathematicians such as Al-Khwarizmi and Fibonacci contributed to the evolution of algebra by introducing concepts such as Arabic numerals and position notation. These advances made it possible to solve more complex problems using mathematical operations, such as addition, subtraction, multiplication and division.

Over the centuries, mathematics has continued to evolve and expand into new areas. In modern times , mathematicians such as Isaac Newton and René Descartes have contributed to the evolution of physics by using mathematical concepts to explain the laws of nature. Other mathematicians, such as Leonhard Euler and Pierre-Simon Laplace, have contributed to the evolution of natural sciences by using mathematical tools to model complex phenomena.

Today, mathematics continues to play a crucial role in many fields, such as science, engineering, finance and technology. Mathematics is used to solve complex problems, model natural phenomena and develop new technologies. Mathematics is also a subject of study in itself, with many branches and sub-disciplines, each exploring different aspects of the science of numbers and computation.

The legend of the chess game (3000 BC)

According to legend, the origin of the game of chess dates back to around 3000 BC.

The King of India, Belkib, was deeply bored and was looking for a distraction. To motivate his subjects to find him a satisfying hobby, he promised an exceptional reward to the one who would succeed in entertaining him. It was then that the Brahmin Sissa, known as the sage, presented him with the chess set. King Belkib was amazed and as a reward he gave Sissa anything he wanted.

The latter then replied:

“Place one grain of wheat on the first square, two on the second, four on the third, eight on the fourth, and so on, doubling the number of grains until you reach the last of the 64 squares on the board.
.
1) The monarch immediately agreed to this request, which he considered fanciful. Without calculations, give an arbitrary estimate of the total number of wheat grains to be placed on the board.
2) Using powers of two, express the number of grains of wheat to be placed on the 5th and 10th boxes. Deduce the number of grains of wheat in the 64th box. Evaluate this number using a computer.
3) Check that the following equations are correct:
i.
ii.
iii.
4) Deduce the total number of grains of wheat to be placed on the chess set. Evaluate this number using a computer.
5) Jonathan wants to represent this number concretely. To do this, he estimates that 30 grains of wheat occupy a volume of $1\,cm^3$ . Moreover, the area of the chessboard surface is equal to 900 Since it is impossible to arrange precisely a large number of wheat grains in columns on a single square, we will suppose that the total number of wheat grains will be poured into a right block whose base is the chessboard, and whose height is unknown.

Calculate the height of this right block.
6) Compare this height to the distance from the Earth to the Moon.
7) According to the FAO (Food and Agriculture Organization of the United Nations), France produces on average 40 million tons of wheat per year. How many years would it take for French producers to honor King Belkib’s request?
8) What do you think of the answer Sissa requested?

The lunulae of Hippocrates of Chio (5th century BC)

The figure below is obtained from the following construction program:
1 Construct a segment [AB].
2 Construct the semicircle of diameter [AB].
3 Place any point C on this semicircle.
4 Construct the segments [AC] and [BC].
5 Construct the semicircles of diameter [AC] and [BC].
We note d,e,f,g,h the area of each colored part.

1) What is the nature of triangle ABC? Justify.
2) The purpose of this question is to show that (d+f)+(e+g)=f+g+h.

The following questions make extensive use of literal calculation.

a) The radius of the semicircle of diameter [AC] is equal to ?
b) Recall the formula for the area of a circle of radius R: $\pi\times ,R^2$ .
Therefore, we have: d+f = ?
c) Similarly, the radius of the semicircle of diameter [BC] is equal to ?
d) Therefore, we have: e+g = ?
e) We deduce that: (d+f)+(e+g)= ?
f) Similarly, the radius of the semicircle of diameter [AB] is equal to ?
g) Therefore, we have: f+g+h = ?
h) Conclude using question 1.

A problem of Euclid (3rd century BC)

Euclid, an ancient Greek mathematician and author of the Elements, demonstrated the following result:
“The straight lines that join two opposite vertices of a parallelogram in the middle of the opposite sides divide the
diagonal that joins two other vertices in three equal parts”.

1. Show that the quadrilateral IBJD is a parallelogram.
2. Use the triangle ALB to show that G is the middle of [AL].
3. Choose the appropriate triangle to show that L is the middle of [GC].
4. Deduce that GA=GL=LC.

Who was Euclid?

Mathematician of Ancient Greece, he lived in Alexandria between -325 and -265.
His mathematical treatise The Elements has been considered as the most important work of
reference until the beginning of the 20th century, and sometimes earned it the
nickname of father of geometry. It is a collection of thirteen books in
in which Euclid tries to rigorously expose the whole of the knowledge
of his time with the help of a hypothetical-deductive system where the properties and
Theorems are proven from definitions and basic axioms.

The epitaph of Diophantus (3rd century)

The epitaph of Diophantus (3rd century)
An epitaph is an inscription engraved on a tomb.
The legend says that on the tomb of the mathematician Diophantus, it was written:

“Passing under this tomb lies Diophantus.
These few verses traced by a learned hand
Will let you know at what age he died.
Quite a few days that counted him the fate,
The sixth marked the time of his childhood.
The twelfth was taken by his adolescence.
Of the seven parts of his life, one more passed,
Then having married, his wife gave him
Five years after a son, who, from the severe fate,
Received of days alas! twice less than his father.
Of four years, in the tears, this one survived.
Say, if you can count, at what age he died.

Answer the question!

Who was Diophantus?

Diophantus lived in Alexandria in the 3rd century. He is notably the author of the Arithmetics, a work that considerably influenced the Arab and Renaissance mathematicians in their developments of algebra, which earned him the title of father of algebra. He is known for his study of the so-called Diophantine equations, which are still on the science curriculum today. His famous epitaph appears for the first time in the Palatine Anthology of Metrodore in the 6th century.

The Fibonacci Fountain (1175-1240)

The following problem is inspired by the “Liber abbaci” (book of the abacus) published by Fibonacci in 1202.
We consider two towers 50 steps apart. The first one is 30 steps high, and the second one is 40 steps high. Between the two towers, there is a fountain towards which two birds fly from each tower. The birds leave at the same time, they fly at the same speed and they arrive at the fountain at the same time.
How far from each tower is the fountain located?

1) Compare the distances AE and CE.
2) Express as a function of CE and CD , then as a function of AE and CE .
3) Express DE as a function of BD and BE. Using this expression for DE, expand .
4) Deduce from the previous questions the relationship :

$100\times ,BE=50^2+40^2-30^2$
5) Solve the problem.

Who was Fibonacci?

Leonardo of Pisa, better known as Fibonacci, was an Italian mathematician. His “Liber abbaci” contributed, in 1202, to the diffusion in the West of the mathematical science of the Arabs and Greeks.

A poem by Nicolas Boileau-Despréaux (1636-1711)

Nicolas Boileau-Despréaux was a French poet, writer and critic. The following poem is both original and pessimistic, because it paints a particularly laborious picture of life.
Fill in the missing durations in this poem and find out how much good time man spends in a day according to the author.

The man, whose whole life
Is ninety-six years old
Sleeps a third of his career,
It’s just _________.
Add for illness, lawsuits, travel, accidents
At least a quarter of life,
That’s another two _________ years.
Two hours of study per day
Or works – make _________ years,
Black sorrows, worries –
For the double make _________ years.
For business that we plan
Half an hour, – still _________ years.
Five quarter hours of washing:
Beard et cetera – _________.
Per day for eating and drinking
Two hours is _________.
It carries the memory
Up to ninety-five years.
Still one year to go
What birds do in spring.
Per day man has on Earth
_________ of good times .

Varignon’s theorem (1654-1722)

Consider a quadrilateral ABCD.

We note I,J,K,L the respective middles of the sides [AB],[BC],[CD],[DA].

1. What conjecture can be made about the nature of the quadrilateral IJKL ?
2. Show that the lines (IL) and (BD) are parallel.
3. Show that the lines (JK) and (BD) are parallel.
4. Prove the conjecture of question 1.
5. State Varignon’s theorem.
6. What happens if AC=BD?
7. What happens if the lines (AC) and (BD) are perpendicular?

Who was Varignon?

Pierre Varignon (1654-1722) was a French mathematician. He was
particularly famous in France for having adopted, with the Marquis de
At the Hospital, Isaac Newton’s theory of infinitesimal calculus. He was also
known at the time for his treatise on physical science in which he stated the
rule of composition of concurrent forces. Today, it is not
known in high schools and colleges only for its famous
parallelogram.

The Gauss formula (1777-1855)

It is said that at the age of 10, Gauss determined a hitherto unknown method to calculate very quickly the sum of the first hundred integers: 1+2+3+4+5+6+….+98+99+100.

His method was so successful that Gauss was able to do the calculation faster than his teacher. The purpose of this exercise is to use a spreadsheet to understand Gauss’ trick.

1) In the first column (A) of the spreadsheet, enter the list of the first hundred integers in ascending order.
2) Similarly, in the second column (B), enter the list of the first hundred integers in descending order.
3) In the third column (C), calculate the sum of the two numbers listed side by side on each line. What is remarkable about it?
4) Go to any empty cell not belonging to the first three columns. Calculate the sum of all the numbers in the first column (A).

We will use the formula: =SUM(A:A)
5) In the same way on two other empty cells, calculate the sum of all the numbers on the second column (B), then the sum of all the numbers in the third column (C).
6) Explain why the sum of the numbers in the third column (C) is 101×100.
7) What quick formula can be written to calculate the sum of the first hundred integers?
8) Calculate the sum of the first 1000 integers. Specify the result and the Gaussian formula to quickly calculate this result by hand.

Who was Gauss?

Carl Friedrich Gauss (1777-1855) was a German astronomer, physicist and mathematician. His varied fields of predilection make him an extremely prolific scientist: celestial mechanics, magnetism, optics, number theory, intuition of non-Euclidean geometry, … A child prodigy, he completed his first treatise on arithmetic at the age of 21. He is often considered the greatest mathematician since antiquity. He was particularly fond of mathematics, and used to call it the queen of sciences.

The Sierpinski triangle (1882-1969)

STEP 1: we start with an all-black equilateral triangle.

STEP 2: We construct the white triangle whose vertices are the midpoints of the previous triangle.

STEP 3: we repeat the process. In each black triangle, a white triangle is constructed.

1) Construct the Sierpinski triangle that would be obtained in step 4. We will think of building a sufficiently large initial triangle.
2) At each step, what fraction of area does the black part represent in relation to the total area of the large triangle?

Results are requested as a fraction and then as a percentage.

Complete.

3) Without constructing the following triangles, deduce from the previous results the fraction of area occupied by the black part in step 5 and then in step 10.

Who was Sierpinski?

Waclaw Sierpinski (1882-1969) was a Polish mathematician. He is one of the co-founders of the modern Polish mathematical school. He contributed to the progress of several particular branches of mathematics: set theory, topology, logic. The Sierpinski triangle (see also: the Sierpinski mat) is part of a large family of curves called fractals.

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