# An overview of Egyptian mathematics

Civilisation reached a high level in Egypt at an early period. The country was well suited for the people, with a fertile land thanks to the river Nile yet with a pleasing climate. It was also a country which was easily defended having few natural neighbours to attack it for the surrounding deserts provided a natural barrier to invading forces. As a consequence Egypt enjoyed long periods of peace when society advanced rapidly.

By 3000 BC two earlier nations had joined to form a single Egyptian nation under a single ruler. Agriculture had been developed making heavy use of the regular wet and dry periods of the year. The Nile flooded during the rainy season providing fertile land which complex irrigation systems made fertile for growing crops. Knowing when the rainy season was about to arrive was vital and the study of astronomy developed to provide calendar information. The large area covered by the Egyptian nation required complex administration, a system of taxes, and armies had to be supported. As the society became more complex, records required to be kept, and computations done as the people bartered their goods. A need for counting arose, then writing and numerals were needed to record transactions.

By 3000 BC the Egyptians had already developed their hieroglyphic writing (see our article Egyptian numerals for some more details). This marks the beginning of the Old Kingdom period during which the pyramids were built. For example the Great Pyramid at Giza was built around 2650 BC and it is a remarkable feat of engineering. This provides the clearest of indications that the society of that period had reached a high level of achievement.

Hieroglyphs for writing and counting gave way to a hieratic script for both writing and numerals. Details of the numerals themselves are given in our article Egyptian numerals. Here we are concerned with the arithmetical methods which they devised to work with these numerals

The Egyptian number systems were not well suited for arithmetical calculations. We are still today familiar with Roman numerals and so it is easy to understand that although addition of Roman numerals is quite satisfactory, multiplication and division are essentially impossible. The Egyptian system had similar drawbacks to that of Roman numerals. However, the Egyptians were very practical in their approach to mathematics and their trade required that they could deal in fractions. Trade also required multiplication and division to be possible so they devised remarkable methods to overcome the deficiencies in the number systems with which they had to work. Basically they had to devise methods of multiplication and division which only involved addition.

Early hieroglyphic numerals can be found on temples, stone monuments and vases. They give little knowledge about any mathematical calculations which might have been done with the number systems. While these hieroglyphs were being carved in stone there was no need to develop symbols which could be written more quickly. However, once the Egyptians began to use flattened sheets of the dried papyrus reed as "paper" and the tip of a reed as a "pen" there was reason to develop more rapid means of writing. This prompted the development of hieratic writing and numerals.

There must have been a large number of papyri, many dealing with mathematics in one form or another, but sadly since the material is rather fragile almost all have perished. It is remarkable that any have survived at all, and that they have is a consequence of the dry climatic conditions in Egypt. Two major mathematical documents survive.

You can see an example of Egyptian mathematics written on the Rhind papyrus and another papyrus, the Moscow papyrus, with a translation into hieratic script. It is from these two documents that most of our knowledge of Egyptian mathematics comes and most of the mathematical information in this article is taken from these two ancient documents.

Here is the Rhind papyrus

The Rhind papyrus is named after the Scottish Egyptologist A Henry Rhind, who purchased it in Luxor in 1858. The papyrus, a scroll about 6 metres long and 1/3of a metre wide, was written around 1650 BC by the scribe Ahmes who states that he is copying a document which is 200 years older. The original papyrus on which the Rhind papyrus is based therefore dates from about 1850 BC.

Here is the Moscow papyrus

The Moscow papyrus also dates from this time. It is now becoming more common to call the Rhind papyrus after Ahmes rather than Rhind since it seems much fairer to name it after the scribe than after the man who purchased it comparatively recently. The same is not possible for the Moscow papyrus however, since sadly the scribe who wrote this document has not recorded his name. It is often called the Golenischev papyrus after the man who purchased it. The Moscow papyrus is now in the Museum of Fine Arts in Moscow, while the Rhind papyrus is in the British Museum in London.

The Rhind papyrus contains eighty-seven problems while the Moscow papyrus contains twenty-five. The problems are mostly practical but a few are posed to teach manipulation of the number system itself without a practical application in mind. For example the first six problems of the Rhind papyrus ask how to divide nloaves between 10 men where n =1 for Problem 1, n = 2 for Problem 2, n = 6 for Problem 3, n = 7 for Problem 4, n = 8 for Problem 5, and n = 9 for Problem 6. Clearly fractions are involved here and, in fact, 81 of the 87 problems given involve operating with fractions. Rising, in [37], discusses these problems of fair division of loaves which were particularly important in the development of Egyptian mathematics.

Some problems ask for the solution of an equation. For example Problem 26: a quantity added to a quarter of that quantity become 15. What is the quantity? Other problems involve geometric series such as Problem 64: divide 10 hekats of barley among 10 men so that each gets 1/8 of a hekat more than the one before. Some problems involve geometry. For example Problem 50: a round field has diameter 9 khet. What is its area? The Moscow papyrus also contains geometrical problems.

Unlike the Greeks who thought abstractly about mathematical ideas, the Egyptians were only concerned with practical arithmetic. Most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned. To overcome the deficiencies of their system of numerals the Egyptians devised cunning ways round the fact that their numbers were poorly suited for multiplication as is shown in the Rhind papyrus.

We examine in detail the mathematics contained in the Egyptian papyri in a separate article Mathematics in Egyptian Papyri. In this article we next examine some claims regarding mathematical constants used in the construction of the pyramids, in particular the Great Pyramid at Giza which, as we noted above, was built around 2650 BC.

Joseph [8] and many other authors gives some of the measurements of the Great Pyramid which make some people believe that it was built with certain mathematical constants in mind. The angle between the base and one of the faces is 51° 50' 35". The secant of this angle is 1.61806 which is remarkably close to the golden ratio 1.618034. Not that anyone believes that the Egyptians knew of the secant function, but it is of course just the ratio of the height of the sloping face to half the length of the side of the square base. On the other hand the cotangent of the slope angle of 51° 50' 35" is very close to π/4. Again of course nobody believes that the Egyptians had invented the cotangent, but again it is the ratio of the sides which it is believed was made to fit this number. Now the observant reader will have realised that there must be some sort of relationship between the golden ratio and π for these two claims to both be at least numerically accurate. In fact there is a numerical coincidence: the square root of the golden ratio times π is close to 4, in fact this product is 3.996168.

In [38] Robins argues against both the golden ratio or π being deliberately involved in the construction of the pyramid. He claims that the ratio of the vertical rise to the horizontal distance was chosen to be 5 1/2 to 7 and the fact that (11/14) × 4 = 3.1428 and is close to π is nothing more than a coincidence. Similarly Robins claims the way that the golden ratio comes in is also simply a coincidence. Robins claims that certain constructions were made so that the triangle which was formed by the base, height and slope height of the pyramid was a 3, 4, 5 triangle. Certainly it would seem more likely that the engineers would use mathematical knowledge to construct right angles than that they would build in ratios connected with the golden ratio and π.

Finally we examine some details of the ancient Egyptian calendar. As we mentioned above, it was important for the Egyptians to know when the Nile would flood and so this required calendar calculations. The beginning of the year was chosen as the heliacal rising of Sirius, the brightest star in the sky. The heliacal rising is the first appearance of the star after the period when it is too close to the sun to be seen. For Sirius this occurs in July and this was taken to be the start of the year. The Nile flooded shortly after this so it was a natural beginning for the year. The heliacal rising of Sirius would tell people to prepare for the floods. The year was computed to be 365 days long and this was certainly known by 2776 BC and this value was used for a civil calendar for recording dates. Later a more accurate value of 365 1/4 days was worked out for the length of the year but the civil calendar was never changed to take this into account. In fact two calendars ran in parallel, the one which was used for practical purposes of sowing of crops, harvesting crops etc. being based on the lunar month. Eventually the civil year was divided into 12 months, with a 5 day extra period at the end of the year. The Egyptian calendar, although changed much over time, was the basis for the Julian and Gregorian calendars.

References (43 books/articles)

Other Web sites:

2. Don Allen (Egyptian mathematics)
3. David Eppstein (Egyptian Fractions)
4. Brent Byars
5. R Knot

## 17th Century Math

The Seventeenth-Century Mathematics of British : Hero Ages of Calculus

،ك Characteristic of the Seventeenth-Century Mathematics

Coming upon 1600's, Europe rushed in "Science revolution" together with developing philosophy, astronomy and physics.   In this century, many revelation originality came out in order.
Kepler, Napier, Fermat Descartes, Pascal, Newton and Leibniz reclaimed new fields.
They were geniuses for physics, astronomy and philosophy.
Descartes, a philosopher, wrote "A Discourse on the Method of Rightly Conducting the Reason and Seeking Truth in the Sciences" and he is renowned as a originator of analytical geometry.   He also found the algebraic method considering geometry in connection with algebra.  It has influenced the invention of calculus of Leibniz.
Newton and Leibniz created calculus independently each other and started modern analsis.  And they also influenced physics making rapid progress from geometry and algebra to analysis.
Newton systematized calculus in 1671 and also left the law of gravity and theory of particles of light.
was published in 1687.  The proved to accomplish their achievements each other.
Leibniz left many results of symbolizing of mathematics and also left results of philosophy and the science of law.

،ك British Mahtematics

John Napier and Henry Briggs are representative British mathematicians. These two men's achievement about calculation reflects the way of Englishmen's thinking.
Unexceptionally, mathematicians were professors but mathematician were out of the university until in the middle of the 17th century.
University existed to maintain the dignity of theology.  All kinds of scholarship were exchanged of salon (place for social intercourse of for the aristocracy) Announcements of scholarship were made mostly through letters.
Using numerical formulae, John Wallis (1616~1703) expressed geometrical quadrature which was done intuitionally by Cavalierie and Pascal.
When he was a professor at Oxford in 1649, he treated the conception of infinity analytically from his Influenced by , Newton found clues for differential and integral.
Using the symbol (،ؤ) which means infinity, Newton regarded infinity as a field of mathematics for the first time.
The more favorable political atmosphere of northern Europe and the general conquering of the cold and darkness of the long winter months by advances in heating and lighting probably largely account for the northward shift of mathematical activity in the seventeenth century from Italy to France and England.

،ك The Dawn of Modern Mathematics

،فThe invention of logarithms: Although telescope was invented and astronomy, navigation and thigonomethy developed so rapidly, there was no accurate system of calculation.   By this reason, the logarithms was invented.
Laplace even said "the invention of loganithms saved astronomers a lot of trouble and doubled their lives".
It is true that the invention is one of three inventions which enables people to calculate accurately and is the main prop of 'Hero Ages of (science) in' the 17th century.
Michael Stifel (1486،­1567) explained the logarithm for the first time. He explained the relations between x and 2x as follows.

x¤‎¤‎¤‎-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ¤‎¤‎¤‎
2©ْ¤‎¤‎¤‎¨û, ¨ù, ¨ِ, 1, 2, 4, 8, 16, 32, 64¤‎¤‎¤‎

The sum of upper number corresponds to the multiplication of lower number.
It is the very conception of logarithms today. Using the logarithms,     we can change complex multiplications into simple additions.   A scotch mathematician Napier set to study the logarithm in earnest.
He explained the way of calculation in his book for the first time.   In his book published after two years since he died, the way of calculation for of the table of logarithms was explained.
Logarithm-means ratio number- and numerus were made by Napier.  Henry Briggs, friend of Napier, actually made the table of logaritms which was convenieht to calculate.
Finally the common logarithm was made which have the base of 10.
Using Napier's number 'e', John Speidell made natural logarithm we study in calculus and he announced this earlier than Henry Briggs in 1619.
Briggs wrote a book in which he calculated it to 14 deximals.
Arabic numeration system, the way of expressing deximal and logaritms let the European mathematics get out of traditianal ancient calculation.  Mathematics, therefoe, formulate a system of modern mathematics.

،فHarriot and Oughtred: He improved on Viete's notation for powers
by representing a©÷ by aa, a©ّby aaa, and so forth. He was also the first to use the signs > and < for "is greater than" and " is less than," respectively, but these symbols were not immediately accepted by other writers.
There also appeared the first edition of William Oughtred's popular Clavis mathematicae, a work on arithmetic and algebra that did much toward spreading mathematical knowledge in England. William Oughtred(1574-1660) was one of the most influential of the seventeenth-century English writers on mathematics
The cross (X) for multiplication, the four dots(::)used in a proportion, and our frequently used symbol for difference between(،­).
Invented, about 1622, the straight logarithmic slide rule.

،فGalileo and Kepler: Two outstanding astronomers contributed notably to mathematics in the early part of the seventeenth century : the Italian, Galileo Galilei, and the German, Johann Kepler.
Galileo was a scientist and mathematician.   Aristotle was in the position of ontology and mataphysics whereas Galileo, phenomenalism.
Galileo urged phenomena were closely related to mathematical (geometrical) conception so phenomenal truth was the truth of mathematical law . Kepler found three laws about the movement of planet.
Newton invented celestial mechanics when he was making efforts to prove the movement of planet.   By above reason, the law about the movement of planet was recorded as a remarkable event in astronomy and mathematics.
Kepler, also thought of the conception of successive movement in which circle moves to ellipse, ellipse to parabola, parabola to hyperbola, hypebola to straight line.
He thought that a cirle was made up of many narrow sectors. Which could be regarded as isosceles triangles.   He thinks about the conception of infinity that all isosceles triangles have the same height and the sum of bases (radius) equals circumference.

،فCavalieri's Method of Indivisibles: , published in its first form in 1635, devoted to the precalculus method of indivisibles. This thesis is about infinitesimal. Area is made up of segments which are indivisibles and solid consists area which is indivisibles too. So the indivisibles becomes the base of the conception of definite integral.

،ك Development of Projective Geometry

With Renaissant fine arts and architecture for a background, practical Projective geometry, different from Euclid geometry appeared. Projective geometry is a kind of geometry with which mathematicians study unchanging property except measurable elements such as length and angle to be drawn by projection.
For example, a circle is changed by projection into conic sections (ellipse and hyperbola) but mathemcticians don't need to distinguish these curves.
Pascal is famous for his 'mystic hexagram the orem' in which he said "if a hexagon be inscribed in a conic, then the points of intersection of the three pairs of opposite sides are collinear, and conversely"
Desargues wrote a book on conic sections that marks him the most original contributor to synthetic geometry in the 17th century.
But the projection geometry was so difficult and different nature from former geometrical conceptions.   Because it was the precocious baby of mathematics, it wasn't acknowedged for 150 years.   Poncelet systematize the projective geometry by using Monge's <descriptive geometry, 1795>

،ك Begnning of Probability

It is generally agreed that the one problem to which can be credited the origin of the science of probability is the so-called problem of the points. This problem requires the determination of the division of the stakes of an interupted game of chance between two players of supposedly equal skills, knowing the scores of the players at the time of interruption and the number of points needed to win the game. Mathematicians were interested in distributing gambling money.
In 1654, to Pascal by the Chevalier de Mere, an able and experienced gambler whose theoretical reasoning on the problem did not agree with his observations. Pascal became interested in the problem and communicated it to Fermat. In which the problem was correctly but differently solved by each. In their correspondence, then, Pascal and Fermat laid the foundations of the science of probability.

،ك Appearance of Analytic Geometry

While Desargues and Pascal were opening the new field of projictive geometry, Descartes and Fermat were conceiving ideas of modern analytic geometry. The projective geometry is a 'branch' of geometry whereas the analytic geometry is a 'method' of geometry.
The essence of the idea, as applied to the plane, it will be recalled, is the establishment of a correspondence between points in the plane and ordered pairs of real numbers, thereby making possible a correspondence between curves in the plane and equations in two variables, so that for each curve in the plane there is a definite equationf(x,y)=0,and for each such equation there is a definite curve, or set of points, in the plane.
Analytic geometry combines algebra with geomety by introducing coordinates.
The real essence of the subject lies in the transference of a geometric investigation into a corresponding algebraic inverstigation.
French mathematician Descartes and Fermat launched modern analytic gometry. They contributed so much to the mathematics in the 17th century. Thus, where to a large extent Descartes began with a locus and then found it equation, Fermat started with the equation and then studied the locus.
That is to say, Descartes' analytic geometry is the algebraic geometry but Fermat's is geometrical algebra. Fermat copied after analytic geometry but he failed to generalize it. Euclidean geomery is static algebra which adopts triangle as a basic figure. But analytic geometry is dynamic geometry which adopts algebraic figure of segment.
Analytic geometry is regarded as function y for variable x namely y is regarded whole action of the change of x. The variation of y forms figures.
Dynamic thought, forming figure by moving points, was inpossible in Greek mathematics which sticked to static thought.

،ك The Exploitation of the Calculus

Unquestionably, the most remakable mathematical achievement of the period was the invention of the calculus, toward the end of the century, by lsaac Newton and Gottfried Wilhelm Leibniz. With this invention, creative mathematics passed to an advanced level, and the history of elementary mathematics essentially terminated.
It is interesting that, contray to the customary order of presentation found in our beginning college courses, where we start with differentiation and later consider integration, the ideas of the integral calculus developed historically before those of the differential calculus. The idea of integration first arose in its role of a summation process in connection with the finding of certain areas, volumes, and are lengths. Some time later, differentiation was created in connection with problems on tangents to curves and with questions about maxima and minima of functions. And still later it was observed that integration and differentiation are related to each other as inverse operations.  Newton and Leibniz devoloped ،¸differential and integral calcuius،¹ by their own way.  Newton found 'differential and integral calcus' first and then Leibniz announced the outcome of it.
English Wallis and Barrow had influence on the Newton's 'differential and integral calculus' so much. Whereas Wallis' chief contribution to the development of the calculus lay in the theory of integration, Issac Barrow's most important contributions were perhaps those connected with the theory of differentiation.
In 1671, Newton created method of fluxions; as he called what today is known as differential calculus. In this work,Newton considered a curve as generated by the continuous motion of a point.
Under this conception, a changing quantity is called a 'fluent' (a flowing quantity), and its rate of change is called the 'fluxion' of the fluent, this constant rate of increase of some fluent is called the principal fluxion, which he calls the 'moment' of a fluent; it is the infinitely small amount by which a flueut increases in an infinitely small interval of time 0.
Differentiation is found by flaxion from fluent but to the contrary Integration is found by fluent from fluxion.
N ewton made numerous and remarkable applications of his method of fluxious. He determined maxima and minima, tangents to curves, curvature of curves, points of inflection, and convexity and concavity of curves, and he applied his theory to numerous quadratures and to the rectification of curves.
Leibniz, Newton's rival in the invention of the calculus, invented his calculus sometime between 1673 and 1676.
He first used the modern integral sign, as a long, letter Sderived frome the first letter of the Latin word 'summa'(sum), to indicate the sum of Cavalieris's indivisibles.
His first published paper on differential calculus did not appear until 1684. In this paper, he introduces dx as an arbitrary finite interval and then defines dy by the proportion.
dy:dx=y:subtangent.
Most symbols of the calcus used today are handed down from Leibniz.

،ك Summary of Major Achievements In the 17th Cenutry
The seventeenth century is outstandingly conspicuous in the history of mathematics. Early in the century, Napier revealed his invention of logarithms, Harriot and Oughtred contributed to the notation and codification of algebra, Galileo founded the science of dynamics, and Kepler announced his laws of planetary metion.   Later in the century, Desargues and Pascal opened a new field of pure geometry, Descartes launched modern analytic geometry, Fermat laid the foundations of modern number theory, and Huygens made distinguished contributions to the theory of probability and other fields.   Then, toward the end of the century, after a host of seventeenth-century mathematicians had prepared the way, the epoch-making creation of the calculus was made by Newton and Leibniz.   We can see that many new and vast fields were opened up for mathenatical investigation during the seventeenth century.

# عشرة استراتيجيات لتتحكم في غضبك

في بعض الأحيان، تبدأ بالشجار مع زوجتك أو أصدقائك او زملائك في العمل، وقد يكون معك الحق، لكن تصرفك وطريقة تعاملك مع الآخرين تضيع عليك هذا الحق، بل وتجعلك عرضة للوم.

إليك عشرة طرق لتتحكم في الغضب ولا تجعله يتمكن منك ويجعلك آداة لتحطيم نفسك والآخرين:

1- توقف قليلا لالتقاط الأنفاس:

أن تقوم بالعد من رقم 1 حتى رقم 10 ليست حيلة للأطفال فقط لتعليمهم الهدوء، فلا تقم باتخاذ ردود الأفعال فورا وأنت غاضب.. تمهل قليلا حتى تهدأ، ثم فكر في الأمر وكيف ستعالجه.

وكذا عندما تكون في موقف من شأنه ان يخرج منك طاقات غاضبة قد تؤذي من حولك بالقول أو الفعل، فعليك أن تستأذن من حولك في الانزواء بنفسك لفترة وجيزة حتى تستجمع أعصابك وتستعيد هدوءك، بدلا من أن تبدي أي رد فعل يؤثر عليك وعلى المحيطين بك مستقبلا.

2- عبر عن غضبك.. ولكن بعد أن تهدأ:

ما ستقوله وأنت غاضب قد يجعلك تندم لاحقا، لكن ما ستقوله أو تفعله بعد أن هدأت سيكون بالتأكيد هو التصرف الصحيح لأنه ناتج عن تفكير عقلي منطقي وليس فعلا أخرق تحرقه العاطفة.

نظم أفكارك واجعلها واضحة وانتقي كلماتك، فالهدف ليس أن تبدأ شجارا ولكن ان تعبر عن غضبك حيال ما حدث أو قيل.

3- تمارين الاسترخاء:

بعض التمارين الخاصة بالاسترخاء تساعد على تهدئة الأعصاب.

غير أن ممارسة الرياضة بصفة عامة تصفي الذهن وتفرغ الطاقات السلبية.

لذلك احرص على ادخال بعض التمارين والأنشطة الرياضية ضمن روتين حياتك اليومية.

4- الحكمة:

عندما تواجهك مشكلة أو أزمة معينة، لا تبدأ أبدا باتخاذ ردود الأفعال الفورية.. التزم التفكير الهادئ والبحث عن حلول مختلفة وبدائل للخروج من هذه الأزمة، ثم اعرض هذه الأفكار للمحيطين بك.

5- توجيه الاهتمام بشكل إيجابي:

مرن نفسك على استيعاب المشكلات والبحث عن أسبابها وسبل حلها، بدلا من اهدار طاقتك في التعبير عن غضبك بصور مختلفة.

لا تتصرف كالأطفال وتسعى للانتقام ممن أغضبك أو رد الصفعة له بأي شكل من الأشكال. بل تأكد أن عدم التفاتك لمحاولات استفزاز الآخرين هو النجاح بعينه بالنسبة لك.

6- لغة الخطاب:

هناك عبارات معينة اذا استخدمتها ستزيد المشكلة تعقيدا، فمثلا اذا كنت تتحدث إلى زوجتك، لا تنتقدها بكلمات مثل: "أنت تتعمدين... أنت بدأتِ"، واذا كنت تتحدث لزميلك في العمل، لا تقل له: "أنت لا تؤدي المطلوب منك.. أنت تتقاعس"، بل تحدث بصيغة المتكلم، كأن تقول: "ما فهمته أنك تقصدين كذا وكذا.. هل هذا صحيح؟" أو تقول لزميلك "أعتقد أن هذا الأمر يمكن علاجه بطريقة كذا وكذا".

7- لا تختزن ما يحدث في ذهنك وتحوله لضغائن متراكمة تفجرها في كل شجار جديد:

عندما تنتهي المشكلة، انس كل شئ، ولا تجعل أي جملة سيئة أو تصرف خاطئ يعلق بذاكرتك، حتى اذا ما حدثت مشكلة جديدة لا تبدأ بلوم من أمامك وتذكره بما فعله أو قاله في المرة السابقة، لأن هذا يزيدك غضبا ويزيد الأمر تعقيدا.

8- هدئ الجو بمزحة أو دعابة طريفة:

اذا شعرت أن هناك حالة من التوتر قد انتشرت بين المتواجدين في هذا المكان، وأن الأمر سيصل إلى مشاجرة بلا شك، فقم بتلطيف الجو قليلا بنكتة او مزحة او دعابة، بشرط ألا تتضمن هذه الدعابة آية إهانة لأي شخص من الموجودين.

9- اكسر روتين حياتك المتكرر وتعلم مهارات التواصل الفعال مع الآخرين:

أحيانا يتسبب الملل في خلق حالة من الضجر ينتج عنها نوع من الغضب والعصبية يتم افراغه في المحيطين بك دون ان تشعر أنت بذلك.

وللقضاء على هذه الحالة، لا تستسلم لروتين حياتك اليومي، خذ اجازة من عملك كل فترة واذهب إلى مكان جديد ومختلف، مارس أنشطة اجتماعية غير معتادة بالنسبة لك.

من مهارات الاتصال أن تستمتع ولا تتكلم فقط، فلا تنفرد بالمحادثة وتوجه الآخرين ليسمعوا ما تقوله، ولا تجبرهم أن يقتنعوا بوجهة نظرك، ولا تعلن للجميع أن رأيك هو الصواب.. لأنه ببساطة قد لا يكون رأيا صحيحا أصلا.. حاول الوصول إلى حلول عملية بدلا من المباهاة بالكلمات.

10- اطلب مساعدة المتخصصين:

أحيانا يكون غضبك من الصعب السيطرة عليه، ولا يعيبك أبدا أن تلجأ لمساعدة اخصائيين نفسيين واجتماعيين لمساعدتك على تخطي هذه المشكلة ببعض التمارين المعينة.

 م اسم عضو هيئة التدريس ( من ثلاثة مقاطع ) رقم السجل المدني / الاقامة الجنسية رقم المستخدم على بوابة النظام الأكاديمي اسم المقرر الذي يدرسه رقم ورمز المقرر رقم الشعبة متفرغ / متعاون ملاحظات 1 محمد محمد خلف حسين مصري التحليل الحقيقي 1 382 MAT 261-262 متفرغ 2 محمد محمد خلف حسين مصري التحليل الحقيقي 2 483 MAT 291-292 متفرغ 3 محمد محمد خلف حسين مصري مقدمة في التوبولوجي 373 MAT 279-280 متفرغ 4

### ت المكتبية

 اليوم 10-8 12-10 الأحد 10-8 11-10 الأثنين 8 -10 12-10

### حكمة الأسبوع

فى دراسة مثيرة أجريت مؤخرا على بلدان العالم المختلفة، قام بها فريق تابع لمنظمة التعاون الاقتصادى والتنمية، عن علاقة قدرات الشباب العلمية من معارف ومهارات وبين وفرة الموارد الطبيعية (نفط وماس وغيرها).

أثبتت الدراسة ان:-

1. البلدان التى لا تتمتع بموارد طبيعية كبيرة مثل اليابان وكوريا الجنوبية وفنلندا وسنغافورة، حقق طلابها اعلى الدرجات فى المعارفوالمهارات حيث يتوفر الحافز لشحذ همتهم واستنفارهم.

2. البلدان الغنيه بالموارد الطبيعية مثل قطر وقازاخستان، حاز طلابها ادنى الدرجات حيث لا يتوفر الدافع القوى للتفوق.

3. البلدان محدودة الموارد نسبيا مثل لبنان والأردن وتركيا كانت نتائج طلابها أفضل من طلاب البلدان ذات الموارد المرتفعة نسبيا مثل السعودية والكويت وعمان والبحرين.

4. بعض البلدان الغنية بالموارد مثل كندا والنرويج واستراليا حصل طلابها على درجات عالية، وذلك لان تلك البلاد وضعت سياسات متوازنة لادخار واستثمار عوائد هذه الموارد ولم تكتف باستهلاكها.

الخلاصة

1. أنك إذا أردت أن تتعرف على مستقبل أى بلد فى القرن الواحد والعشرين فلا تحسب احتياطياته من النفط أو الذهب،

ولكن انظر إلى مكانة مدارسه وكفاءة مدرسيه العالية وقوة مناهجه التعليمية وإيمان آبائه والتزام طلابه.

ذلك أن المعارف والمهارات هى التى ستحدد موقع كل بلد فى خريطة المستقبل.

وهو ما لابد أن يصدم كثيرين فى بلادنا ممن تصوروا أن غاية المراد أن يفوز فريق البلد القومى بكأس فى مباريات كرة القدم، أو أن يتفوق بعض أبنائه فى برنامج «ستار أكاديمى».

ــ فهل نحن نحث السير على طريق الندامة؟

- أم أننا نسير في طريق البناء والتطوير والتعمير؟.

للتواصل

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