# A brief history of mathematics

« What is incomprehensible is that the world is understandable. « Einstein

Indeed. And to understand it, there is no better tool than mathematics. Because math is everywhere, in the drawing of a spiral galaxy, the flight of the bumblebee, drawing a hive, the path of an apple falling on the heads of scientists …

Knowing mathematics is partially lifted the veil on the mystery of the universe.

### Brief history of mathematics

**The School of Athens by Raphael** – In the center Plato (with Timaeus) index showing the world of ideas and Aristotle, the sensible world – left, Pythagorean theorem writing a Epicurus wearing laurels copying Pythagoras Parmenides Socrates and Parmenides behind – Sitting on the top step, Heraclitus, above Diogenes of Sinope – Right Euclid, leaning over a compass and Ptolemy back in yellow and blue gown

# Little history of mathematics

If we meet one day an alien civilization, we could not communicate with simple words out of our dictionary. Mathematics would be our best and only asset. For two and two make four. Here like in Andromeda or in the depths of the Universe.

One can live without knowing the Critique of Pure Reason by Immanuel Kant, a Bach sonata or the last Stallone; but we would be missing something if we ignored mathematics. Because they are the language in which nature is written. For the Greeks, like Pythagoras, Mathematics are nature.

This modest history of mathematics will, hopefully, make you rediscover this jewel that is mathematics through the great figures who have unveiled their nature.

Maybe spark-so few vocations? In this case, it will have achieved its objective.

But as the Xenon arrow that never reaches its target, the apprentice mathematician must know that he will embark on an endless quest, will flow into a bottomless ocean, towards an unattainable horizon. For mathematics will never be completed. Kurt Gödel in the twentieth century has shown: there will always be true propositions, like 1 + 1 = 2, that will forever unprovable. It’s like that. Weshouldstand outside of the universe for a chance to prove everything. And the only one likely to venture out is God (if he does exist). Good luck to all and to start, good reading.

## Little history of mathematics

### the Paleolithic

In the most ancient archaeological sites, 20 000 years ago, we find the first signs of a thoughtful accounting. Notches on the wood show the first mathematical activity: the first men were, for the first time, able to abstraction, that is to say, to represent an object or an animal by a notch or a symbol.

This invention allowed them to count and keep the memory. The management of a stock became possible, but also trade with neighboring tribes.

## Little history of mathematics

## Mesopotamia and the Indus Valley -5000 to -2000

If Africa was the cradle of man, Mesopotamia was that of civilization. The first cities like Ur and Akkad, agriculture, the wheel, the first administration were born here, between the Tiber and the Euphrates, about 9000 years ago.

Its history is known through the thousands of discovered clay tablets, on the sites of ancient city-states like Sumer or ancient Babylon.

Most of these tablets are books of accounts without interest to the historian.

But they show that civilization and mathematics grew together. An administration worthy of the name can not happen mathematicians.

## Little history of mathematics

**4500-2000 BC – Sumer – the onset of addition and multiplication**

The first simple operations, such as addition and multiplication, are on these clay tablets from the fifth millennium BC; the first traces of writing, meanwhile, appear later, around 3300 BC. The appearance of writing marks the transition between prehistory and history.

The Sumerians were working in base 60. This base is still used: 60 seconds as everyone knows, are, as among the Sumerians, one minute.

The Sumerians knew of advanced concepts such as the square and the cube and logarithms. Tablets showed the Pythagorean Theorem and introduced negative numbers.

A tablet from 1800 BC shows the square of the diagonal and gives an approximation of √2 (the length of the diagonal) with very good accuracy.

## Little history of mathematics

### 2500 BC – Egypt – the development of geometry

The Egyptians built the only wonder of the world still standing: the pyramid of Cheops. This architectural masterpiece of precision demonstrates advanced knowledge of geometry: they knew very probably the golden number .

The golden number (about 14/11 = 1+ √5 / 2) is in fact the relationship between the slant height of the pyramid and its base half.

## Little history of mathematics

### 1900-1600 BC – Babylon – invention of trigonometry

The **Babylonians** have built the Mesopotamian knowledge. They set up the first geometry. They gave an approximate value of Pi = 3 + 1/8 and using the Pythagorean theorem.

They also laid the foundations of trigonometry , including improving their calculations in astronomy.

## Little history of mathematics

## On the side of Egypt – the estimate of Π is refined

Around 1650 BC, a scribe named Ahmose summarized on a papyrus mathematical knowledge of his time.

This papyrus was discovered in 1855 in the region of Thebes and bought a few years later by Alexander Henry **Rhind** in Luxor. He gave his name to the papyrus.

In the Rhind papyrus are solving several problems, a description of multiplication and division, a method of solving equations, calculating areas, including the trapezius and volumes, including pyramids.

There is also a new estimate of Π .

## Little history of mathematics

## 800 BC – Greece and worries of proof

**The demonstration**

The Europe of mathematics was born in Greece. While the Sumerians used the properties they had found, the Greeks wanted to demonstrate them. Euclid was notably the first mathematician who clung to a certain rigor in the demonstration, by systematically seeking to lighten the shadows and find evidence. So he put the fundamental axioms of geometry on which any demonstration should be based.

**The insurmountable obstacles**

The Greeks, despite blazing advances, were hindered in their progress by their intangible dogmas to Pythagoras : numbers and nature were confused. A negative number could therefore not make sense. The Greeks represented 3 apples. In contrast, in Greece nobody had seen -3 apples. Nothing could be measured – 3 units A negative number does not make sense!

Similarly, for a Greek, a high number represented a square surface. Its cube represented a volume. But what does mean a number raised to the power of 4? A number raised to the power of 4 had no more meaning than a negative number.

**The fundamental figures**

For the Greeks, two figures were special, perfect, almost divine: the circle and the square. Everything had to be « reduced to these two objects ». Any surface should be compared to the square. A rectangle, for example (surface D × d) was reduced to a square of side √ (D × d). We measure also always surfaces in « square ».

**The three Greek puzzles**

Since there were only two basic forms, Greeks only used two tools: non-ruler and compass. Also, three famous problems was raised :

**squaring the circle**, or how to draw a square with the same area as a given circle;**the trisection of an angle**, or how to share a corner three with a ruler and compass;**the duplication of the cube**.

It will take generations of mathematicians to show, 1500 years later, that these three problems are insoluble.

**The mathematical Platonism
**

**Plato** , in the fifth century BC, believed that mathematical objects (numbers, geometrical figures, etc.) had their own existence independent of human activity. Along with the carbon atom has not waited chemists, prime numbers, circles, the Pythagorean theorem were there long before mankind, already organized, already established.

Aristotle quoted his master, in the fourth century BC:

« In addition to the existence of sensible things and ideas, Plato admits that mathematical things [Numbers, lines, surfaces, solids], which are intermediates realities (

metaxu), different,

- firstly, of
sensitive things, in that they are eternal and motionless,- and, secondly, the
ideas, in that they are a plurality of similar copies, while the Idea is in itself a reality, individual and singular. « Aristotle, oral teaching.

According to Plato, mathematicians are only reveal (unveil) on a reality that existed before us.

## Little history of mathematics

### 600 BC – Thales and Theorem

**Thales of Miletus** , the first of the seven sages of Greece, was it also the first mathematician?

**Thales of Miletus** put in place the first elements of geometry. A stay in Egypt allowed him to acquire solid knowledge of science and soak up the Babylonian heritage.

**Thales theorem**

In wanting to measure the height of the pyramids, he discovered the most famous theorem in schoolchildren. OA / OC = AB / CD.

It also showed that a rectangular triangle is inscribed within a circle whose diameter is the hypotenuse. Probably it is also behind other famous theorems schoolchildren as:

- a circle is divided into two halves by any diameter;
- the base angles of an isosceles triangle are equal;
- angles vertically opposite are equal when two lines intersect;
- two triangles are equal if they have two angles and the included side equal.

### 500 BC – Zeno of Elea and the question of the infinite

**Zeno of Elea** , a disciple of Parmenides, lived in the fifth century before Christ. He is known for his paradoxes mathématiques. His life is little known. The only sources come from Parmenides and Plato especially the writings of Diogenes Laertius, the Roman historian who retraced the life and thought of the Greek philosophers Pre-Socratic.

**Arrow paradox**

One of the paradoxes of Zeno : an arrow never reaches its target. Because it must still go through the moist path that remains before reaching the target.

If the distance to the target is 100 meters, while the arrow has to travel 100/2 (half) + 100/4 (fourth) + 100/8 (eighth) + 100/16 (sixteenth) + …. distance. And so on to infinity. It will therefore take an infinity.of time. This paradox will turn the head of the Greeks and their successors. The development of the series will solve the paradox of the arrow and others, such as that of Achilles and the tortoise .

**Infinity Does it exist?**

Unlike the Pythagoreans, the Eleatic Zeno had recourse to infinity to describe the world: a surfaceis necessarily made up of an infinity of points.

Aristotle will be part of this logic in finding that the sequence of integers (1, 2, 3, 4 …) is infinite: we can always imagine an integer greater than a given one.

But it was for Aristotle a potential infinity. He denied the existence of an actual infinite, an object containing an infinity of parts. Archimedes contradict Aristotle instead claiming the existence of the infinite in nature. The debate today is still not decided.

### 500 BC – Pythagoras and irrational numbers

Pythagoras was the founder of a sect that bore his name: the Pythagoreans.

For Pythagoras, everything in nature is made of numbers and even integers.

Also, when one of his disciples discovered the irrationality of √2 (the diagonal of a square of side 1), he put some rocks in his trousers and threw him into the sea.

**Irrational numbers**

But the verse was however in the fruit. The Greeks had discovered a new kind of numbers that could not be put in the form of a ratio (p / q) (the rational numbers like 1/4 or 9/8). These numbers are naturally inclined including the name of irrational numbers.

Pythagoras was also the (re) discoverer of the theorem that still bears his name: the hypotenuse square is equal to the sum of the squares of the other two sides.

His demonstration was more geometric arithmetical: he endeavored, by scientists cutouts to match surfaces, as shown in the figure against.

## Little history of mathematics

**438 BC – The Parthenon and the golden**

Phidias the sculptor was commissioned to build the Parthenon. The proportions he used to draw the facade shows that he knew the golden: Φ = (1+ √5) / 2) = 1.618.

## Little History of Mathematics

### 400 BC – Eudoxus of Cnidus and quadrature

Squaring is the step for any area (eg that of a rectangle surface D × d) determining the square that has the same surface (the square of side √ (D × d)).

**Method of exhaustion**

**Eudoxus of Cnidus** was perhaps the inventor of the « exhaustion » method. This method seeks to determine the surface of a complex area, such as a circle. This surface is approximated by a simple surface, a polygon in our example. The polygon is inscribed inside the circle. The more we increase the number of sides, the more its surface is approaching that of the circle. Archimedes perfected this method, which inspired Fermat, 1500 years later, for his method of tangents, causing the calculus …

**Eudoxus of Cnidus** also attempted to impose in the Greek world the concept of incommensurable numbers, that is to say irrational, timidly opened the door to real numbers that Pythagoras was refused by rejecting the irrationality of √2.

## Little history of mathematics

### 300 BC – EUCLID – arithmetic and geometry

**Euclidean geometry**

Euclid was probably the greatest in the field of arithmetic. Euclid has left us an encyclopedia of numbers in 13 volumes: the **Elements**. He gathered all the knowledge of his time and the foundation of plane geometry, one where the right angle is 90 °: Euclidean geometry. These main axioms are:

- there is always a line through two points of the plan;
- any segment can be extended in its direction in an infinite line;
- from a segment, there is a circle whose center is one of the points of the segment and whose radius is the length of segment;
- all right angles are equal;
- given a point and a line not passing through this point, there is only one line passing through that point and parallel to the first.

**The last axiom**

The last axiom is still not proven. 2000 years later, seeking to fill this gap, a Russian mathematician **Lobatchevsk** will put in place a new geometry, non-Euclidean geometry, one where the parallel can intersect and where the angles are not necessarily 90 °.

**fundamental theorem of arithmetic**

Euclid also enunciated that any number is divisible by a prime number which allowed the nineteenth century Carl Friedrich Gauss to prove the fundamental theorem of arithmetic :

Any positive integer can be written as a product of primes in a unique way, to order nearly factors.

Carl Friedrich Gauss

## Little history of mathematics

### 220 BC – Archimedes and the calculation of Pi

**Archimedes** improved the method of exhaustion of **Eudoxus of Cnidus** to approach π : he can not accurately measure the perimeter of the circle, he drew a first polygon inside the circle and a second outside. He framed as the value of π between 22/7 and 223/71.

**Exhibitors**

In his **Arénaire***, * **Archimedes** became attached to a rather worrying problem: calculate the number of grains of sand in the universe! Of course, he was confronted with pharaonic numbers. To get by, he therefore developed a new notation numbers by introducing exhibitors: he wrote 1000 in the form: 10³ and 100: 10². So instead multiply 1000 by 100, multiplied it by 10³ 10² or 10³ × 10³ 10² = + ² = **10 ^{5}** , 10 followed by 5 zeros, 100 000.

**Archimedes** laid the foundations of the logarithmic calculation , we meet again 1,000 years later with **John Napier** ! **Archimedes** had replaced the multiplication by addition.

## Little History of Mathematics

*Second century before Christ – Diophantus the real father of algebra*

*Second century before Christ – Diophantus the real father of algebra*

**Diophantine equations**

**Diophantus**** ** was born somewhere at a certain time. We do not know really not much of one who gave his name to a form of very special equations Diophantine equations.

The **Diophantine equations** are equations with several unknowns, in whole or rational coefficients and whose solutions are **whole or sound** , for example:

**x² + y ² = z², with x, y and z whole**

These equations have interested the greatest mathematicians for centuries to come: Fermat, Euler, Gauss and others. The Fermat’s last theorem is based on a diophantine equation. Of the three books he has authored two have reached us:

- polygonal numbers;
- the Arithmetic.

In this second book, Diophantus was interested including quadratic equations: ax² + bx + c = 0. But He was still steeped in Greek philosophy and its prohibitions. He refused to acknowledge the existence of negative numbers or irrational numbers.

** The birth of algebra**

However Diophante innovations. Unlike the Greeks, he ignored geometry to solve these equations. Diophantus used numerical methods, by introducing into the equations, a letter to denote **the unknown** : algebra was born.

He also proposed exceed the power of 3 major innovation in the Greek world.

## Little History of Mathematics

*II century BC – Eratosthenes calculated the earth’s meridian*

*II century BC – Eratosthenes calculated the earth’s meridian*

When calculated the perimeter of the earth, he knew circular, **Eratosthenes** made a mistake under 100 km He was also the author of the screen that bears his name and which gives a method for finding prime numbers.

**190 BC – Nicaea – Hipparchus developed trigonometry**

**the trigonometry**

**Hipparchus of Nicea** was an astronomer. He had the idea of introducing the concept of rope, ancestor of the sinus to track the position of the planets.

It was connected by a line segment between the intersection:

- the angular sector described by a planet
- the circle of radius 1.

The length of this line segment representing the rope. Hipparchus was able to build **strings tables**. Knowing a rope, Hipparchus obtained by seeking in its tables, the angular sector. The rope, as given by Hipparchus, represented what is now called the sinus, or, to be more exact, would correspond to twice our sinuses.

## Little history of mathematics

### Seventh century – BRAHMA-Gupta, India and birth zero

Since the second century BC, the Indians had abandoned the hexadecimal system for more convenient decimal system, for in the image of the 10 fingers of hands. Their numbers will be adopted by the Arabs and then be imported into Italy by Fibonacci we meet in the twelfth century.

**The invention of zero**

To separate hundreds of units, Indians began by putting spaces and a point and finally a round: the ancestor of our zero was born.

**BRAHMA-GUPTA** went further theorizing the new number meaning « nothingness », vacuum. 0 was defined as the result of the subtraction of an integer by itself a-a = 0. **BRAHMA-GUPTA** also gave the following results a + 0 = a and a × x 0 = 0, which was right, but wandered giving a / 0 = 0 then it is known today that the result is infinite.

**Negative numbers**

He introduced, is quite new, negative numbers considered here as accounting losses: 5-7 = -2.

**Quadratic equation**

He was the inventor of algebra, with **Diophantus Greek**. This discipline was to solve equations where hiding is **unknown**, as there is today « x »: he gave the solutions of quadratic equations of Type: ax² + bx + c = 0.

## Little history of mathematics

### IX century – AL Khawarizmi – Persia and the (re) birth of algebra

The arithmetic was to make calculations, certainly more complicated, but still calculations. With Algebra (Al-Jabr in Arabic), there appeared in the equations, one or more unknown: the thing to discover. **AL Khawarizmi**, following BRAHMA-Diophantus and GUPTA, wrote in 813, the first algebra treated: The Compendious Book on Calculation by Completion and Balancing .

We had seen with the Greeks, and especially with the Greek astronomers, born the trigonometry , the art of measuring the angles in the triangle.

AL Khawarizmi developed this art and especially introduced the concept of sinus (Latin fold), probably from the oldest Indian works.

## Little history of mathematics

### Christianity – black period for mathematics

Europe, with its conversion to Christianity in the fourth century, had seriously behind. Greek knowledge was lost and science in general and mathematics were quarantined.

For a thousand years the theological questions became priorities, as the divinity of Christ or the nature of the host. The works of Archimedes, Euclid and others were forgotten.

The Greek works we finally saved by Arabs, who had made systematic translations and by the Byzantines who had recovered from Alexandria scrolls. It was not until the Italian Renaissance so that Europe began again at work.

## Little history of mathematics

### XIII century – FIBONACCI matter zero and the Hindu-Arabic numerals

Fibonacci trader had followed his father in North Africa. He imported in Europe:

- the decimal system;
- the zero ;
- negative numbers;
- Indo-Arabic numerals

The inconvenient Roman notation for calculations disappeared: Try to multiply by MMXCVI CCMXVI!

Fibonnacci left us a famous result that bears his name, which allows to draw beautiful spirals. Starting from 0 and 1, this suite is built by adding the previous two numbers to get the next one. (0 + 1 = 1) ( 1 + 1 = 2) (1 + 2 = 3 = … What gives 0, 1, 1, 2, 3, 5, 8, 13, 21 … Fibonacci discovered that the ratio of two successive members of the sequence converges to a particular value: the number of gold Ψ : the divine proportion.

This golden number [(1 + √5) / 2] was already known in architecture as the « divine proportion », especially among the Egyptians and Greeks.

This proportion, used by Leonardo among others and always in vogue today, represented the aesthetic perfection.

Surprisingly, the number of gold , like Pi , found everywhere in nature. The artichoke flower or sunflower, an atmospheric depression, the petals of a rose, the shell of a snail, quasi-crystals, take a spiral shape built from the Fibonacci sequence .

# Brief history of mathematics

### 1540 – Francois Vieta invented « x »: the unknown

Francois Vieta, a Protestant lawyer, was one of the father of algebra. His mathematician talents made him the advisor to King Henry IV of France with a mission to decipher the coded missives of Spanish Catholics.

**The European algebra**

Good King Henry, who were advanced that there were no mathematicians in France, said, *« Let me go get Vieta. » *In the midst of religious wars, Viete was the first (in Europe) note the unknown in an equation with a letter.

But let Jean Rond d’Alembert to his slightly modernized praise:

« His first achievement was to introduce into the equations letters to denote the unknown; his second was to have imagined transformations to simplify the equations; the third was his method to determine the number of roots of an equation according to its degree; the fourth was his contribution in solving the equations of the fourth degree; the fifth is the formation of equations composed by their simple roots, when they are all positive, or the determination of all parts of each of the coefficients of these equations;

[….]

Jean Rond d’Alembert, encyclopedic.

## Little history of mathematics

**1581 – Rafael Bombelli invented imaginary numbers**

**Imaginary numbers**

Following **Giordano Cardano** , **Rafael Bombelli,** seeking to solve the equations of the third degree, invented the imaginary number i = √-1. This number will be a huge posterity. Besides facilitating many calculations, it will be an indispensable tool in physics. As already seen, the Indians had taken a step in appealing to negative numbers the Greeks refused. By inventing the imaginary number , Bombelli detached a little more math to reality. A basket of apples -5 had no sense. Another containing i bananas, even less!

Mathematics borrowed paths increasingly abstract.

## Little history of mathematics

### 1550 – John Napier invented logarithms

**logarithms**

**John Napier** was a Scottish astronomer faced with the multiplication of large numbers, as Archimedes was with sand. Napier had the idea, as the Greek scholar before him, to replace this complicated operation with a more simple addition. What was it ? An example: 4 × 8 = 32. We can also write 2² 2³ × = 32. Or 2² + ³ + = 32, that is to say 2 ^{5} = 32. The multiplication of 4 by 8 has been replaced by the addition of 2 + 3!

John Napier then constructed tables of logarithms with a rule:

**log (a × b) = log (a) + log (b)**

In our example, we have for a log base 2:

- log (4) = 2 and log (8) = 3.
- log (4 × 8) = log (4) + log (8) = 2 + 3 = 5
- The table Napier said log (32) = 5

Napier table showed that the number whose logarithm is 32 is 5. The multiplication became useless. A simple table reading allowed to say that 4 × 8 = 32.

## Little history of mathematics

**1629 – Albert Girard announces the fundamental theorem of algebra**

Albert Girard, the inventor of the brackets and the notation « sin » for « sine » function, confirmed in 1629 an intuition of Francois Viete, which foreshadowed the fundamental theorem of algebra, which will be demonstrated by Gauss 200 years later:

All algebraic equations receive as many solutions as the name of the highest amount demonstrates.Albert Girard.

## Little history of mathematics

### 1637 – Descartes connects algebra and geometry

**The Cartesian coordinate system**

- an abscissa axis (Ox) and
- a vertical axis (Oy).

A point can be placed in this frame. It is defined by its Cartesian coordinates:

- the x-coordinate and
- the y.

A line can be drawn in this benchmark. It is defined by the equation y = a x + b. For example **y = 2x + 1** is a line through the point M (2.5). Indeed, when x = 2, then y = 2 × 2 + 1 = 5.

**y = -x + 4.**

- y = 2x + 1
- y = -x + 4

It is an algebraic system of two equations with two unknowns. We just found his solution by geometry alone. Descartes has connected geometry and algebra. His Cartesian coordinate system would also give rise to functions, a powerful tool used in the branch of the analysis.

Descartes also worked on optics. He defines the principle of inertia which will be taken by Galileo and the law on the conservation of momentum.

## Little history of mathematics

### 1648 – Pierre Fermat and method of the tangent

**Differential calculus**

Mathematician, he was undoubtedly the true inventor of calculus, no offense to the English (who prefer Newton) and Germans (who prefer him Leibniz).

Indeed, his method called « tangent » was indeed the ancestor of differential calculus .

Fermat also worked on prime numbers. He left a theorem, called the **great theorem of Fermat**, based on ubne Diophantine equation, demonstrated in 1998 by the English Wiles.

There is no positive integers

x,y,zsatisfying the equation

x^{n}+ y^{n}= z^{n}when

nis an integer strictly greater than 2.Great Fermat’s theorem

Fermat left a note in the margin that seemed to suggest that he had a demonstration:

»

Rather, it is impossible to share either a cube into two cubes, or a bicarré two biquadrates or in general any power higher than the square into two powers of the same degree: I have discovered a truly marvelous proof that margin is too narrow to contain.« Pierre Fermat.

Given the power of Wilesdemonstration and tools he used, that did not exist at the time, the demonstration of Fermat is doubtful.

## Little history of mathematics

### 1650 – Gregory St. Vincent, hyperbole and the constant **e of natural logarithm said « Napierian ».**

Georges St. Vincent, in 1650, became interested in the area under the curve of the hyperbola y = 1 / x (shown in the graph below).

He noticed that the areas under the curve are constant when the progress of the x-axis is geometrically (1, 2, 4, 8, 16 …).

If one is interested in the area from the horizontal axis 1, the increase in area is arithmetic:

**Area (a × b) = area of (a) + area of (b). And area (1) = 0.**

He had to show that the integral of the function f (x) = 1 / x is the natural logarithm (base e).

If the x an equilateral hyperbole grow in geometric progression, the areas of the cut surfaces between hyperbole and its asymptote by the corresponding ordinate lines increase in arithmetical progression.

Gregory St. Vincent

## Little history of mathematics

### 1664 – **Nicholas MECATOR** invents infinite series

**Infinite series**

**Nicholas MECATOR **published *Hypothesis astronomica nova*. MERCATOR worked on theories developed by Kepler concerning the elliptical orbits of the planets. MERCATOR was also interested in the area under the hyperbola y = 1 / x. According to d’Alembert, he was the inventor of infinite series: he changed a little the hyperbole to set up the function y = 1/1 + x. By division, he obtained the following infinite sequence:

**1 / (1 + x) = 1- x + x ^{2} – x ^{3} + x ^{4} + ….**

## Little history of mathematics

### 1650 – 1700 – Bernoulli family

**The infinite series (continued)**

Bernoulli family left Basel to settle in Italy. She gave 9 world-renowned mathematicians, including Nicolas, Jean, Jacques and Daniel.

Jacques, son of Nicolas, published major treatises on infinite series initiated by MERCATOR, of the form:

**1/2 + 1/3 + 1/4 + 1/5 + … + 1 / n = Σ1 / n.**

Jacques tried to demonstrate that this amount differed. He was also interested in a more interesting sum:

**1 / + 1 2² / 3² + 1 / + 1 4² / 5² + … + 1 / n ² = Σ1 / n = Σ1 / n ².**

It seemed indeed converge, despite its infinite terms. This will show that Euler in that:

**1 + 1/4 + 1/9 + 1/16 + .. 1 / n ² = Π / 6.**

A strange result! What comes to Π, the constant circle, in the middle of that sum?

Another surprising result. Working on interest rates, Jacques Benoulli discovered:

**Σ (1 + 1 / n) ^{n } = e**

**(e here is the constant of the natural logarithm ln verifies that ln (e) = 1).**

That came to this constant in such a sum?

**The exponential function e **^{x}

^{x}

Jean Bernoulli introduced the exponential function in the great ball of mathematics:

- e
^{x}is the reciprocal of the function ln x - It is denoted e
^{x } - There was thus: e
^{lnx}= x - e
^{x }is the only function that is its own derivative

## Little history of mathematics

#### XVII century – Leibniz and Newton the authorship of the calculus

**Differential calculus**

The first, a German to whom we owe the term function, and the second, an Englishman who was responsible for the mechanical bearing his name, disputed the invention Duke calculus .

Of course, the true discoverer is Fermat, a French.

The Cartesian coordinate system now used to draw curves. The temperature was plotted sunshine, the size of an individual was based on his age, the position of a ball function of time.

Gunners His Majesty felt that the accuracy of shelling was random (also expensive). His majesty gave Newton the task of solving this problem. NEWTON sought to know at every moment, the speed of the shell. For this, he must have known, for each tiny time interval (denoted dt), the small change in ball position (denoted dx).

The instantaneous speed (v (x)) was then equal to the ratio between this small change in position and the time interval.

**dx**, the infinitely small change in position;**dt**, that of time;

**V (x) = dx / dt** is then the instantaneous speed.

We say that V (x) is the derivative over time of the position of the shell.

## Little history of mathematics

#### NEWTON 1694- establishes the fundamental theorem of calculus

#### fundamental theorem of calculus

In his Principia Mathematica, Newton establishes the * fundamental theorem of analysis* . He completed for this, the first drafts of James Gregory. This theorem allowed to link functions « derivative » and « integral » functions.

For several years :

- is »
**drifting**» functions (e.g. of the shell position) to calculate their speed at each time interval and - one »
**integrated**» other functions to calculate surfaces.

At first, there is no relation between the two calculations. Newton demonstrated the opposite. The fundamental theorem of calculus specified that the derivative was the reciprocal of the integral.

## Little history of mathematics

**1712 – Brook Taylor connecting functions and series**

We knew that any algebraic function could be expressed as a combination of simple functions like:

**f (x) = ax + b**

For example, an algebraic quadratic function can be written

**f (x) = (a _{1 }x + b _{1 }) (a _{2 }x + b _{2 })**

And more generally, for the upper degrees f (x) = (a _{1} x + b _{1} ) (a _{2} x + b _{2} ) (a _{3} x + b _{3} ) … (a _{n} x + b _{n} ). Taylor had the idea to calculate the successive derivatives:

**f (x) = f (a) + f ‘(a) _{ }(x) ^{1} /1! + f »(a) _{ }(xa) ^{2} /2! + .. + f ^{n} (a) (xa) ^{n } / n! _{ + …}**

Colin Maclaurin, in 1742, put **a = 0.** Laurin got what we know from under the name Taylor series :

f (x) = f (0) + f ‘(0)_{ }x + f » (0)_{ }(x)^{2}/2! + .. + F^{n}(0) (x)^{n }/ n!_{ }

This gave to the usual functions:

- sin (x) = x¹ / 1! – x³ / 3! + x
^{5}/5 Stars! – x7 / 7 …! - cos (x) = 1 – x
^{2}/2! + x^{4}/4! – x^{6}/6! + x^{8}/8! … - e (x) = 1 + x + x
^{2}/2! + x^{3}/3 ! + X^{4}/4! …

The functions and infinite series were well linked.

## Little history of mathematics

**1748 –** EULER and exponential

**Exponential**

Euler was probably the greatest of all time. He worked on all branches of mathematics, particularly arithmetic. It thus establishes a relationship between integers and prime numbers .

This formula is remarkable: it demonstrates that the prime numbers are not randomly distributed. They are the backbone of the whole numbers, the primary colors from which all others can be reconstructed.

Euler was, following Napier , one of the developer of logarithms .

He is responsible, with John Bernoulli, the calculation of the **constant e** (called Euler’s constant) which is involved in the exponential function reciprocal of the logarithm function.

It is the source of the most beautiful formula of all time that can be found from the Taylor series :

**e ^{iΠ} – 1 = 0**

- cos x + i sin x = 1 + ix / 1! + (ix) ² / 2! + (ix) ³ / 3! + …
- e
^{i }^{x}= 1 + ix / 1! + (ix) ² / 2! + (ix) ³ / 3! + … - So e
^{i }^{x}= cos x + i sin x (in putting x = Π):

# e ^{iΠ} – 1 = 0

This synthetic formula 7 connects the most famous symbols constants of the universe:

- e, the base of the natural logarithm;
- i, the imaginary number of Bombelli;
- Π, the constant circle, geometric essence;
- 1, the neutral element of the multiplication;
- 0, the neutral element of the addition.

## Little history of mathematics

### 1800 – Pierre Simon Laplace simplifies branch

Perhaps the invention must be returned to Euler. Yet it was popularized by Count Laplace, chief mathematician of the Emperor. This is of course of the **Laplace transform,** which allowed to substitute for the difficult operations that are the derivation and integration in simple operation: addition and multiplication. We can say that the Laplace transform is the derivation that the logarithm is multiplication.

And the emperor asked Laplace:

- where is God in your world?
- I have not needed that hypothesis, Sire! then replied the count.

## Little history of mathematics

### 1830 – Carl Friedrich Gauss opens the door to non-Euclidean geometry

Gauss was the most brilliant mathematician of the first part of the eighteenth century. He worked as Fermat and Euler before him, on the theory of numbers.

**The fundamental theorem of arithmetic**

Euclid, as we have seen, in his Elements, showed that every integer can be divided by a prime number.

A thousand years later, Gauss showed in his « Disquisitiones Arithmeticae » the fundamental theorem of arithmetic :

Any natural number n> 1 can be written as a product of primes, and this representation is unique, the order of the prime factors closely. Or in other words:

q n =

_{1 }^{a1}× q_{2 }^{a 2}× q_{3 }^{a3}× … × .. q_{r }^{ar}; with distinct prime qi and have positive integers. Carl Friedrich Gauss

With the Gauss theorem, confirmed that the primes are the basic building blocks from which all integers are built.

**The density of primes**

We owe him a curious relationship that gives the density of primes less than some integer n:

**Π (n) ≅ I (n) / n**

**The method of least squares**

He is responsible for the invention of the least squares method which allowed, then, to find a celestial object (Ceres) exactly where he had calculated through this method. This method allows very widespread today, as part of experiments to reduce the effect of measurement error.

**Non-Euclidean geometry**

He also envisaged a non-Euclidean geometry, that is to say, a geometry that does not meet the fifth axiom of Euclid. The sum of the angles of the triangle in this geometry can be less or greater than 180 °!

**The fundamental theorem of algebra**

All polynomials with complex coefficients has at least one root. Accordingly, any polynomial with integer, rational or real coefficients has at least one complex root, as these numbers are also complex.

Gauss proposed four different demonstrations.

## Little history of mathematics

### 1830 – In the middle of July Revolution, Cauchy developed complex analysis

**Cauchy, Augustin Louis** was a French mathematician born in the middle of Revolution (August 26, 1789). He developed complex analysis. A second revolution (1830) led him to exile in Switzerland. He refused indeed to swear allegiance to the **King of the French Louis Philippe.**

The complex analysis is simply the study of complex variable functions, such as f (z) = 1 / z, where z and f (z) of complex numbers (z = x + iy). The fundamental difference with the conventional analysis is:

- the domain of definition of the function, that has two dimensions (the complex plane);
- and the field « image » of the function, too, has two dimensions.

**Cauchy** defines the differentiability of complex variable functions. A function is differentiable (holomorphic) at z0 if f ‘(z0) exists:

**Cauchy-Riemann**

- Take a complex number:
**z = x + i** - f (z) can be written f (x + iy)
**= u (x, y) + iv (x, y)**where u and v are two functions for real variables.

Then f is differentiable if;

- ðu / dx = dV / dy and
- dV / dx = – ðu / Dy.

In short, we returned to the conventional analysis.

**Line integral and Cauchy’s integral theorem**

The integration also exists in the imaginary world, but most upright along an axis (real), as in classical analysis, but on a domain belonging to two-dimensional complex plane C. The integration is along a path or curve where the line integral name.

Cauchy showed that if the starting point and the ending point of the path are identical, then the integral is zero (Cauchy’s integral theorem).

**Residue theorem**

With the theorem of residues, Cauchy gave a simple method to calculate line integrals around a singularity; a singularity being a point in the complex plane where the function values differ; for f (z) = 1 / z, the singularity (or pole of the function), for example, point 0 (where f (z) is ∞).

## Little history of mathematics

### 1850 – Bernhard Riemann and the surface of the same name

**Riemann surface**

Bernhard Riemann, following Cauchy, if attacked, a problem unanswered: multi-valued functions.

Indeed, a worthy function of this name has a single image. For all x, for example, was one x² or one cos x or one ln x.

In complex analysis: z can have two images, even infinite. This is the case of log (z) and root functions.

**For example log (z)**

We know that :

^{z = r x eiθ = r x ei(θ + 2kπ)
}

Indeed, we can do tricks (2Π) in the complex plane and return to the starting point as many times as you want!

Then calculate Log (z)

- Log (z) = Log (r xe
^{iθ}) - = ln r + iθ x ln e
**= ln r + iθ**

Or

- Log z = Log (r x e
^{i(θ + 2kπ)}) **= ln r + iθ +2ikπ**

although two different expressions were for log z! What is forbidden for a function! Fortunately, Riemann found a trick. Each turn, z has to climb stairs and end up on the top floor!

Thus z = r x e^{iθ} ≠ r x e^{i(θ + 2kπ} These forms weird took the name of Riemann surfaces.

## Little history of mathematics

### 1880 – Georg Cantor founded set theory and measure infinity

Attacked on all sides by his contemporaries, Georg Cantor ended his days in a deep depression. These attacks are explained by the deeply innovative nature of its work: Leibniz was already the victim of sarcasm. Descartes eproached him his calculations on the infinitely small quantities.

He defines, at first, the concept of sets: A set is a consistent and coherent collection of objects. For example, the set of natural numbers (1, 2, 3, 4, …). If we add two natural numbers, we get a third who is also in the set.

We can have finite sets (for example all the plates in a cupboard), but also infinite (eg the complex numbers C).

Cantor went further by comparing infinite-between them: he showed and the infinity of all N numbers of natural numbers (1, 2, 3,4 …) is of the same order of magnitude (cardinal in his vocabulary ) that the infinite number of the set of numbers even integers (2, 4, 6, 8 …); what seems a priori surprising; Yet each integer, we may well involve a single even number (multiplying by two) and vice versa. There is a bijection between these two sets which therefore are of the same cardinal

**The diagonal CANTOR**

He showed, however, that the infinity of the set R of real numbers is greater than the infinity of the set N of integers, because such a bijection can not be built: take for example the real between 0 and 1 . then Suppose we have such a bijection.

In the natural number 1 is assigned a random real between 0 and 1. A 2 also and so on.

1 → 0 **1** 12345678899123 ….

2 → 4 **6** 46,578,421,254,427 …

3 → 0.14 **5** 7121512144712 ..

4 → 0567 **2** 434 131 215 454 ..

etc.

now building a real number X from this result, as follows: one seeks:

- the first decimal of the real first: 1, and 1 = 2;
- the second decimal of the real second 6, and 1 = 7;
- the third decimal place of the third real: 5, and 1 = 6;
- the
**fourth**decimal of the**fourth**real: 2, and 1 = 3; - etc.

X = 0.2763.

It is obvious that X does not belong to the list that we have built. It is not related to a natural number. In the vocabulary of CANTOR, it just proved that R is « uncountable » N and is greater than he. Some infinite so are larger than others!

CANTOR showed that an infinite number of infinite sets of sizes and that all the sub-parts of an infinite set A is greater than A.

**The continuum hypothesis**

CANTOR put the hypothesis that, in set theory:

There is no set whose cardinality is strictly between the cardinality of the set of natural numbers and that of all real numbers. Georg Cantor

It was not until 1963 that a mathematician to show that such a statement was free because unprovable (within the meaning of the term Gödel happens just after)!

## Little history of mathematics

### 1905 – Poincaré implements topology

The highlight of abstract mathematics, topology is to study the forms, to classify, to find matches and thus deduce properties that apply to one, be transposed to another.

## Little history of mathematics

## 1933 – Gödel demonstrates that we can not show everything!

Greatest logician since Aristotle, **Kurt Gödel** showed that within a logical system, such as arithmetic, it is impossible to tell if some statement is true or false, « they are unsolvable. » This was of course a major achievement that showed the pathetic character consisting of the quest (as was David Hilbert) to complete math. That gave a negative answer to the second problem Hilbert (see below the 23 issues): Can we prove the consistency of arithmetic?

## Little history of mathematics

### And now ?

There is more nowadays mathematicians who can boast of having the entire knowledge. The best specialists are at best a branch of mathematics. They devote their lives to a theorem, an equation, a tiny detail in the ocean of numbers.

Mathematics is increasingly abstract. It does not hesitate to speak of dimensions of space 4, 5, or n or even an infinite number of dimensions, which, of course, has no physical meaning, but works fine in the world of mathematics.

Similarly, the spaces are not rectilinear, as in Euclid time. The spaces may be curved. In this non-Euclidean geometry, the angles are not 90 and parallels can be cut! Yet the mathematical point of view, it makes sense.

We are working on strange objects such as matrices, formal groups of Evariste Galois, Cauchy symmetric functions, vectors, and thus away more and more of the good old arithmetic problems that had given birth yet the mathematics.

## Little history of mathematics

### The 23 problems Hilbert

Mathematics is not completed. They never will. In the early twentieth century, David Hilbert presented a list of 23 problems to be solved, he believed, complete math. Here are some:

### Any subset of actual infinity can be put in bijection with the set of natural numbers or the set of real itself.

The set of real comprises the set of integers (1, 2, 3 …), the integers (negative integers), rational numbers (which are written in the form p / q as ¼), numbers Pi as irrational or √ 2 (which can not be written in the form p / q). There would bijection if a set of numbers, included in the number of actual, was connected with the set of integers by one relationship: a bijection. Please note, we are talking about overall infinite. Cantor had shown that they could build such a relationship between whole numbers and even numbers: simply multiply the whole (eg 3) by 2 and is indeed obtained an even number (6 in this example). Each integer can be connected to an even number and there is indeed a bijection and there is so much that whole even numbers !!!

However, in 1963, it was shown that the proposed Hilbert was undecidable. It was Gödel with his incompleteness theorem of 1931, which showed indeed that in any logical system as arithmetic, we can not say anything certain proposal or whether they are true or if it is false.

### Can we prove the consistency of arithmetic?

Are there non-removable axioms? Or rather undecidable, which we can not say if it is true or false. Again, Gödel has taught us that this is the case.

### Demonstrate the transcendence of numbers ab, with a different algebraic 0 and 1, and irrational algebraic b.

A transcendental number is a number that is not the solution of an algebraic equation in the case of Π, for example. With this proposal, 2 ^{√2} , no example is transcendent because √2 is irrational. The demonstration was made in 1934 by Schneider and Gelfond.

### Demonstrate three guesses:

- – The Riemann hypothesis;
- – Goldbach’s conjecture;
- – The conjecture of twin primes.

**The Riemann hypothesis**

The zeta function RIEMAN is perhaps the most mysterious of mathematics: it takes the form s with a complex number.

Riemann’s hypothesis is that the zeros of the function « control » the distribution (in fact their oscillation about an expected position) prime numbers. The zeros, as Riemann, are, in the complex plane located:

- symmetrically about the axis
*s*= ½ +*it*; - within a critical band between 0 and 1 for their real part.

Despite numerous attempts, the conjecture has not been proven. It’s a shame because it would probably the secret of the distribution of primes.

**Goldbach’s conjecture **

*All even integer greater than 3 can be written as the sum of two primes.*

Examples: 8 + 3 = 5 or 5 + 5 = 10.

For fans, the conjecture still resists!

**The conjecture of twin primes.**

Twin primes are prime numbers separated by a single integer; Examples 5 and 7 to 11 and 13. It would be an infinite number of twin primes. It remains to demonstrate! With the power of modern computers, we find such couples, more gigantic, even Pharaonic, containing more than 200,000 numbers! But as for conjecture Golbach, twins resist. However, advance: the idea is to show that there exists an infinity of couples whose difference is less than a given number. This gap is now 6 …

### Find a key algorithm if a Diophantine equation has solutions

What is a diophantine equation must be the concept to Diophantus, a Greek mathematician. The equations that bear names are algebraic equations whose roots are whole. For example x-1 = 0 has the solution x 1. Fermat’s theorem is another example: i l are no nonzero integers *x* , *y* and *z* such that *x ^{n} + y ^{n} = z ^{n}* , if

*n*> 2.

No solution to this problem HILBERT. In 1970, **Matiassevitch** showed that this algorithm can not be written.

## Little history of mathematics

### For the future

Most problems HILBERT therefore always resists. They often open doors to more complex problems, underground even darker, deeper, more branched, foreshadowing work for another thousand years …