Famous mathematicians and their problems

Famous mathematicians and their problems

I. The Ishango bone:

The Ishango bone which was found at Lake Edward in Zaire, close the Ugandan border, in 1960 is about 20.000 years old.

Not only the age of the bone is remarkable but also the numbers on this bone illustrated as notches.

At one place we see the numbers 3-6, 4-8 and 10-5. These are obviously the representations of the doubling and the bisection of numbers. Another place on the bone is even more remarkable. We find the numbers 11-13-17-19, all the prime numbers between 10 and 20.

So far nobody really knows what these prime numbers were used for 20.000 years ago.

II. The calculating system of the Babylonians

The first sensible number was used by the Babylonians 4000 years ago.

The system was developed in Mesopotamia which is Iraq nowadays.

The Babylonians had something similar to our decimal system, but for them not 10 was the decisive number, but 60.

This means they had a positional notation system to the base of 60.

Some relics of the Babylonian number system are still relevant today. Thus one hour consists of 60 minutes and 60 minutes consist of 60 seconds, so that an hour consists of 60 x 60 = 3600 seconds.

A circle consists of 360 (6 x 60) angular degrees.

All this has survived for 4000 years.

A number system to the base of 60 has the following consequences:

In this system the numbers 1 to 59 are used.

The value of a number is dependent on its position.

The number 5 in final position (unit position) also has a value of 5.

The last but one position is not, as in our decimal system, the number 10, but the number 60.

A 2 in this position leads to a value of 120.

The third last position in the decimal system is the 100 position.

In the Babylonian number system to the base of 60 the third last position is 3600.

A 3 in this position would have a value of 3 x 3600 = 10800.

With this system the Babylonians could add, subtract, multiply and divide. There was just one main thing missing: cipher

III. The unsolvable mathematical problems of the ancient world:

The doubling of the cube:

A cube has the (side) length of 1 and thus also the volume of 1. The task is to construct a cube with double the volume. This cube would have to have edges, the lengths of which are the third root of 2. The question is whether it is possible to construct this length (third root of 2) by only using compass and ruler.

The angle trisection:

Is it possible to construct for each random angle an angle which is exactly one third of the original one? For instance in order to trisect the angle of 45 degrees we would have to construct an angle with 15 degrees. This could relatively easily be done because we would only have to bisect the angle of 60 degrees twice. But does it work for all angles?

Squaring the circle:

The problem of whether it is possible to square the circle has been around for more than 2000 years.

Euclid mentions this problem in his book "Elements" and he postulates that only ruler and compass should be allowed to solve the problem.

In the year 1882 the mathematician Ferdinand Lindemann claimed that he had found a solution to this problem, though only a negative one. In the sense that the problem of squaring the circle is not solvable.

IV. The five Platonic solids:

(1) Tetrahedron:

number of surfaces: 4

number of angles: 4

ancient element: fire

(2) Hexahedron:

number of surfaces: 6

number of angles: 8

ancient element: earth

(3) Octahedron:

number of surfaces: 8

number of angles: 6

ancient element: air

(4) Icosahedron:

number of surfaces: 20

number of angles: 12

ancient element: water

(5) Dodecahedron:

number of surfaces: 12

number of angles: 20

ancient element: universe

V. The Pythagoreans: Amicable numbers

The Pythagoreans were fascinated by amicable numbers like 220 and 284. In this pair of numbers 220 can be divided by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, leading to a sum of 284. 284 can be divided by 1, 2, 4, 71 and 142 leading to a sum of 220. Until the year 1747 the Swiss mathematician and physicist, Leonhard Euler, reported that there is a limited number of only 30 amicable pairs. Today with the help of modern computers mathematicians have found more than 11 million amicable pairs, but in only 5001 pairs both numbers are smaller than 3,06 x 10^11. In the year 850 the Arabian astronomer and mathematician Thabit ibn Qurra presented a formula according to which amicable numbers can be calculated: If p = 3 x 2^ (n-1) – 1, and q = 3 x 2^ (n) – 1, and r = 9 x 2^ (2n-1) – 1 for a number n > 1

→ It follows that if p, q and r are prime numbers:

The numbers 2^ (n) pq and 2^ (n) r are amicable numbers.

VI. Pythagorean triples:

In the sequence of uneven whole numbers 1,3,5,7,9,11, the sum of succeeding elements starting from 1 results in a square number:

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Pythagorean triples:

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VII. Euclid of Alexandria on perfect numbers:

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VIII. Euclid’s Golden Ratio:

In the 6th book of the "Elements" by Euclid, we find a definition of a particular type of partition of a line segment in two uneven parts. According to Euclid a line segment AB can be divided by an interior point C. If this is the case, then the line segment AB is equicontinuously divided if the quotient AC/CB is equal to the quotient AB/AC. Since the 19th century this relation has been called the "golden ratio" or "golden section". The value of the golden ratio is (1 + √5)/2 = 1,6180339887 …

IX. The Sand Calculator:

Apart from his ideas on many mathematical problems, Archimedes of Syracuse is also known for calculating with enormously big numbers. In his treatise “The Sand Calculator” Archimedes calculated how many grains of sand would be needed to fill the universe. Archimedes calculated that approximately 8 x 10^63 grains of sand would be needed to fill the universe.

X. The Bakhshali-Manuscript:

The Bakhshali-Manuscript is a remarkable collection of mathematical scripts written in the 3rd century A.D. It describes methods and rules to solve mathematical problems and presents exercises for the user. One of these exercises reads as follows: We have a group of altogether twenty men, women and children. Together they earn twenty coins. Each man earns three coins, each woman one and a half and each child half a coin. Then the author of the exercise formulates his question: How many men, women and children do we find in this group? The author then presents two formulas as a hint:

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There is only one solution for this problem: We have two men, five women and thirteen children in the group.

XI. The Indians and the number cipher:

Since when has the number cypher been in use? The number cypher was definitely invented in India. The first well-documented Indian cypher can be found in a Vishnu temple in Gwalior, 400 kilometers south of Delhi. On a stone tablet marked with the year 876 the number cypher is used to illustrate the numbers 50 and 270. The Arabs adopted the number cypher from the Indians and with the expansion of the Islam the cypher also came to Europe. In the year 1202 at the latest the number cypher had arrived in Western Europe. In this year the book "Liber abaci" by the arithmetician Leonardo of Pisa was published. Leonardo of Pisa later on became famous under the name of Fibonacci. Fibonacci wrote in his book: The nine Indian numbers are 9 8 7 6 5 4 3 2 1. With these nine numbers and the number 0, which the Arabs call "Zephirum" any number whatsoever can be written.

XII. Indian-Arabian number system:

We all can be glad that we use Indian-Arabian numbers nowadays which were mainly developed in the 6th and 7th century.

We all remember the Roman numbers:

I: 1

V: 5

X: 10

L: 50

C: 100

D: 500

M: 1000

Thus 1492 would be MCDXCII.

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