Leonhard Euler (1707 – 1783)

From `A Short Account of the History of Mathematics’ (4th edition, 1908) by W. W. Rouse Ball.

Leonhard Euler was born at Bâle on April 15, 1707, and died at St. Petersburg on September 7, 1783. he was the son of a Lutheran minister who had settled at Bâle, and was educated in his native town under the direction of John Bernoulli, with whose sons Daniel and Nicholas he formed a lifelong friendship. When, in 1725, the younger Bernoullis went to Russia, on the invitation of the empress, they procured a place there for Euler, which in 1733 he exchanged for the chair of mathematics, then vacated by Daniel Bernoulli. The severity of the climate affected his eyesight, and in 1735 he lost the use of one eye completely. In 1741 he moved to Berlin at the request, or rather command, of Frederick the Great; here he stayed till 1766, when he returned to Russia, and was succeeded at Berlin by Lagrange. Within two or three years of his going back to St. Petersburg he became blind; but in spite of this, and although his house, together with many of his papers, were burnt in 1771, he recast and improved most of his earlier works. He died of apoplexy in 1783. He was married twice.

I think we may sum up Euler’s work by saying that he created a good deal of analysis, and revised almost all the branches of pure mathematics which were then known, filling up the details, adding proofs, and arranging the whole in a consistent form. Such work is very important, and it is fortunate for science when it fall into hands as competent as those of Euler.

A history of Zero

One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.

The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name “zero” derives ultimately from the Arabic sifr which also gives us the word “cipher”.)

Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as ‘real’ problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 “things” and then to the abstract idea of “five”. If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.

One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 ” 6.

The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 ” 6 we never find 216 ”. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.

If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer “It’s three fifty” means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.

We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.

Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid’s Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.

Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely “ouden”. Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number – it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for “obol”, a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.

Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.

The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee in [6] claims:-

… the mathematical conception of zero … was also present in the spiritual form from 17 000 years back in India.

What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.

In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word “kha” for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.

We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.

We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.

Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-

The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.

Subtraction is a little harder:-

A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.

Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-

A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.

In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta’s book. He correctly states that:-

… a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.

However his attempts to improve on Brahmagupta’s statements on dividing by zero seem to lead him into error. He writes:-

A number remains unchanged when divided by zero.

Since this is clearly incorrect my use of the words “seem to lead him into error” might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.

Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-

A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 0² = 0, and √0 = 0.

Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.

You can see a separate article about Mayan mathematics.

The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmi wrote Al’Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra, in the 12^th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Number describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12^th century al-Samawal was writing:-

If we subtract a positive number from zero the same negative number remains. … if we subtract a negative number from zero the same positive number remains.

The Indian ideas spread east to China as well as west to the Islamic countries. In 1247 the Chinese mathematician Ch’in Chiu-Shao wrote Mathematical treatise in nine sections which uses the symbol O for zero. A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.

Fibonacci was one of the main people to bring these new ideas about the number system to Europe. As the authors of [12] write:-

An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.

In Liber Abaci he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that. It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the “sign” zero while the other symbols he speaks of as numbers. Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.

One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on. However, this was far from the case. Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500’s so much easier if he had had a zero but it was not part of his mathematics. By the 1600’s zero began to come into widespread use but still only after encountering a lot of resistance.

Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21^st century begin on 1 January 2001. Zero is still causing problems!

References (14 books/articles)

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Astroseti (A Spanish translation of this article)

Article by: J J O’Connor and E F Robertson

Fibonacci biography

Leonardo Pisano is better known by his nickname Fibonacci. He was the son of Guilielmo and a member of the Bonacci family. Fibonacci himself sometimes used the name Bigollo, which may mean good-for-nothing or a traveller. As stated in [1]:-

Did his countrymen wish to express by this epithet their disdain for a man who concerned himself with questions of no practical value, or does the word in the Tuscan dialect mean a much-travelled man, which he was?


Fibonacci was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father's job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia. Bejaia is a Mediterranean port in northeastern Algeria. The town lies at the mouth of the Wadi Soummam near Mount Gouraya and Cape Carbon. Fibonacci was taught mathematics in Bugia and travelled widely with his father and recognised the enormous advantages of the mathematical systems used in the countries they visited. Fibonacci writes in his famous book Liber abaci (1202):-

When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.

Fibonacci ended his travels around the year 1200 and at that time he returned to Pisa. There he wrote a number of important texts which played an important role in reviving ancient mathematical skills and he made significant contributions of his own. Fibonacci lived in the days before printing, so his books were hand written and the only way to have a copy of one of his books was to have another hand-written copy made. Of his books we still have copies of Liber abaci (1202), Practica geometriae (1220), Flos (1225), and Liber quadratorum. Given that relatively few hand-made copies would ever have been produced, we are fortunate to have access to his writing in these works. However, we know that he wrote some other texts which, unfortunately, are lost. His book on commercial arithmetic Di minor guisa is lost as is his commentary on Book X of Euclid's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view.

One might have thought that at a time when Europe was little interested in scholarship, Fibonacci would have been largely ignored. This, however, is not so and widespread interest in his work undoubtedly contributed strongly to his importance. Fibonacci was a contemporary of Jordanus but he was a far more sophisticated mathematician and his achievements were clearly recognised, although it was the practical applications rather than the abstract theorems that made him famous to his contemporaries.

The Holy Roman emperor was Frederick II. He had been crowned king of Germany in 1212 and then crowned Holy Roman emperor by the Pope in St Peter's Church in Rome in November 1220. Frederick II supported Pisa in its conflicts with Genoa at sea and with Lucca and Florence on land, and he spent the years up to 1227 consolidating his power in Italy. State control was introduced on trade and manufacture, and civil servants to oversee this monopoly were trained at the University of Naples which Frederick founded for this purpose in 1224.

Frederick became aware of Fibonacci's work through the scholars at his court who had corresponded with Fibonacci since his return to Pisa around 1200. These scholars included Michael Scotus who was the court astrologer, Theodorus Physicus the court philosopher and Dominicus Hispanus who suggested to Frederick that he meet Fibonacci when Frederick's court met in Pisa around 1225.

Johannes of Palermo, another member of Frederick II's court, presented a number of problems as challenges to the great mathematician Fibonacci. Three of these problems were solved by Fibonacci and he gives solutions in Flos which he sent to Frederick II. We give some details of one of these problems below.

After 1228 there is only one known document which refers to Fibonacci. This is a decree made by the Republic of Pisa in 1240 in which a salary is awarded to:-

... the serious and learned Master Leonardo Bigollo ....

This salary was given to Fibonacci in recognition for the services that he had given to the city, advising on matters of accounting and teaching the citizens.

Liber abaci, published in 1202 after Fibonacci's return to Italy, was dedicated to Scotus. The book was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe. Indeed, although mainly a book about the use of Arab numerals, which became known as algorism, simultaneous linear equations are also studied in this work. Certainly many of the problems that Fibonacci considers in Liber abaci were similar to those appearing in Arab sources.

The second section of Liber abaci contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China.

A problem in the third section of Liber abaci led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today:-

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The resulting sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... (Fibonacci omitted the first term in Liber abaci). This sequence, in which each number is the sum of the two preceding numbers, has proved extremely fruitful and appears in many different areas of mathematics and science. The Fibonacci Quarterly is a modern journal devoted to studying mathematics related to this sequence.

Many other problems are given in this third section, including these types, and many many more:

A spider climbs so many feet up a wall each day and slips back a fixed number each night, how many days does it take him to climb the wall.
A hound whose speed increases arithmetically chases a hare whose speed also increases arithmetically, how far do they travel before the hound catches the hare.
Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given.

There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series.

Fibonacci treats numbers such as √10 in the fourth section, both with rational approximations and with geometric constructions.

A second edition of Liber abaci was produced by Fibonacci in 1228 with a preface, typical of so many second editions of books, stating that:-

... new material has been added [to the book] from which superfluous had been removed...

Another of Fibonacci's books is Practica geometriae written in 1220 which is dedicated to Dominicus Hispanus whom we mentioned above. It contains a large collection of geometry problems arranged into eight chapters with theorems based on Euclid's Elements and Euclid's On Divisions. In addition to geometrical theorems with precise proofs, the book includes practical information for surveyors, including a chapter on how to calculate the height of tall objects using similar triangles. The final chapter presents what Fibonacci called geometrical subtleties [1]:-

Among those included is the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles; the inverse calculation is also given, as well as that of the sides from the surfaces. ... to complete the section on equilateral triangles, a rectangle and a square are inscribed in such a triangle and their sides are algebraically calculated ...

In Flos Fibonacci gives an accurate approximation to a root of 10x + 2x2 + x3 = 20, one of the problems that he was challenged to solve by Johannes of Palermo. This problem was not made up by Johannes of Palermo, rather he took it from Omar Khayyam's algebra book where it is solved by means of the intersection of a circle and a hyperbola. Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction. He then continues:-

And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.

Without explaining his methods, Fibonacci then gives the approximate solution in sexagesimal notation as 1.22.7.42.33.4.40 (this is written to base 60, so it is 1 + 22/60 + 7/602 + 42/603 + ...). This converts to the decimal 1.3688081075 which is correct to nine decimal places, a remarkable achievement.

Liber quadratorum, written in 1225, is Fibonacci's most impressive piece of work, although not the work for which he is most famous. The book's name means the book of squares and it is a number theory book which, among other things, examines methods to find Pythogorean triples. Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n2 + (2n+1) = (n+1)2. Fibonacci writes:-

I thought about the origin of all square numbers and discovered that they arose from the regular ascent of odd numbers. For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.

To construct the Pythogorean triples, Fibonacci proceeds as follows:-

Thus when I wish to find two square numbers whose addition produces a square number, I take any odd square number as one of the two square numbers and I find the other square number by the addition of all the odd numbers from unity up to but excluding the odd square number. For example, I take 9 as one of the two squares mentioned; the remaining square will be obtained by the addition of all the odd numbers below 9, namely 1, 3, 5, 7, whose sum is 16, a square number, which when added to 9 gives 25, a square number.

Fibonacci also proves many interesting number theory results such as:

there is no x, y such that x2 + y2 and x2 - y2 are both squares.

and x4 - y4 cannot be a square.

He defined the concept of a congruum, a number of the form ab(a + b)(a - b), if a + b is even, and 4 times this if a + b is odd. Fibonacci proved that a congruum must be divisible by 24 and he also showed that for x, c such that x2 + c and x2 - c are both squares, then c is a congruum. He also proved that a square cannot be a congruum.

As stated in [2]:-

... the Liber quadratorum alone ranks Fibonacci as the major contributor to number theory between Diophantus and the 17th-century French mathematician Pierre de Fermat.

Fibonacci's influence was more limited than one might have hoped and apart from his role in spreading the use of the Hindu-Arabic numerals and his rabbit problem, Fibonacci's contribution to mathematics has been largely overlooked. As explained in [1]:-

Direct influence was exerted only by those portions of the "Liber abaci" and of the "Practica" that served to introduce Indian-Arabic numerals and methods and contributed to the mastering of the problems of daily life. Here Fibonacci became the teacher of the masters of computation and of the surveyors, as one learns from the "Summa" of Luca Pacioli ... Fibonacci was also the teacher of the "Cossists", who took their name from the word 'causa' which was first used in the West by Fibonacci in place of 'res' or 'radix'. His alphabetic designation for the general number or coefficient was first improved by Viète ...

Fibonacci's work in number theory was almost wholly ignored and virtually unknown during the Middle ages. Three hundred years later we find the same results appearing in the work of Maurolico.

The portrait above is from a modern engraving and is believed to not be based on authentic sources.

Article by: J J O'Connor and E F Robertson

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AQUITA RICKO M NIM 08/269841/PA/12099 Asal Sekolah SMU Negeri 7, Yogyakarta

YOSEPHINE KURNIA SW NIM 08/269842/PA/12100 Asal Sekolah SMU Negeri 8, Yogyakarta

DWI PURNAMA NIM 08/269845/PA/12102 Asal Sekolah SMU Negeri 1, Wonosari

NUR AENI NIM 08/269848/PA/12105 Asal Sekolah SMU Negeri 1, Salaman

QORIATUN MARYAMAH NIM 08/269914/PA/12125 Asal Sekolah SMU Negeri 5, Purwokerto

ENDANG TRI HASTUTI NIM 08/269937/PA/12136 Asal Sekolah SMU Negeri 7, Purworejo

ADHITYA EKA PERMANA NIM 08/269974/PA/12155 Asal Sekolah SMU Negeri 5, Tangerang

SUBROTO NIM 08/269995/PA/12164 Asal Sekolah SMU Negeri 5, Madiun

DIAN ERAWATI NIM 08/270026/PA/12181 Asal Sekolah SMU Negeri 1, Wates

RAHMAT ROBBI BINUR NIM 08/270035/PA/12182 Asal Sekolah SMTA Lain-lain,

TYAS YULIVIANI NIM 08/270036/PA/12183 Asal Sekolah SMU Negeri 1, Karanganom

FITRIATI SOLIKHAH NIM 08/270051/PA/12191 Asal Sekolah SMU Negeri 1, Muntilan

DESY NUGRAHAENI NIM 08/270065/PA/12197 Asal Sekolah SMU Negeri 1, Wates

DEWY AYU LIKAWATI NIM 08/270083/PA/12204 Asal Sekolah SMU Negeri 3, Mojokerto

AJENG RIESMITASARI NIM 08/270124/PA/12218 Asal Sekolah SMU Negeri 78, Jakarta

MIFTAHU RAHMAH W.E NIM 08/270249/PA/12267 Asal Sekolah SMU PIRI 1, Yogyakarta

HAKIKI RADHIETYAWATI NIM 08/270269/PA/12272 Asal Sekolah SMU Negeri 6, Jakarta

SITI MUANAH NIM 08/270275/PA/12274 Asal Sekolah SMU Negeri 1, Magelang

ROMADZON SYAIFUL HAQ NIM 08/272722/PA/12313 Asal Sekolah SMU Negeri 8, Yogyakarta

IKA SUSANTI NIM 08/272846/PA/12321 Asal Sekolah SMU Negeri 1, Purwakarta

M IRSYAD ISMI NIM 08/272906/PA/12327 Asal Sekolah SMU Negeri 1, Muntilan

CARA YANG BENAR DALAM BELAJAR MATEMATIKA

Banyak orang yang takut dengan pelajaran matematika. Ada juga orang yang sangat benci dengan matematika. Padahal pelajaran ini benar-benar berguna bagi kehidupant kita sehari hari, bahkan bagi orang biasa sekalipun. Matematika adalah kunci dari semua pelajaran sains, baik itu Fisika, Ekonomi, Akuntansi dan Kimia karena pelajaran tersebut tidak akan dapat kita pahami tanpa mempelajari terlebih dahulu dasarnya yaitu matematika. Ada orang yang bilang " ilmu lain tidak bisa berkembang tanpa matematika, tetapi matematika bisa berkembang tanpa ilmu lain". Ada juga yang bilang " kepanjangan dari matematika adalah makin tekun makin tidak karuan". Namun yang jadi permasalahan sekarang adalah, bagaimana cara belajar yang baik agar kita dapat menguasai ilmu matematika ini?

Harus diingat bahwa tidak ada cara mudah dan cepat untuk menguasai matematika ini. Yang ada adalah Cara yang benar dalam belajar matematika. Dibutuhkan kesabaran dan kegigihan yang tinggi untuk berusaha, tapi dengan niat yang kuat saya yakin kita bisa menguasai pelajaran matematika. Ada beberapa tips yang bisa kita tempuh agar kia bisa menguasai Matematika:


1. Luruskan Niat
Hal pertama yang harus kita lakukan adalah "Meluruskan Niat" dalam belajar matematika, janganlah kita belajar matematika hanya untuk mendapatkan nilai yang bagus sebagai syarat lulus mata ujian Matematika.Kalau cuma mau dapat nilai itu mudah tinggal nyontek aja kan bisa. Ingat tujuan kita adalah mencari ilmu, bukan mencari nilai. Kebanyakan dari kita jika telah melewati ujian/test, maka kita akan meninggalkan dan melupakan materi yang telah kita pelajari tersebut. Maka dari itu niatkan belajar matematika untuk menambah pengetahuan kita. Karena dengan belajar matematika, daya nalar otak kita akan terasah dengan baik sehingga mudah untuk menerima pelajaran yang lainnya. Ingat sekali lagi, jangan hanya berorientasi kepada Hasil ujian, tapi berorientasilah pada Proses belajarnya.

2. Kenali,lalu Cintai matematika
Point ini merupakan poin yg paling penting dalam belajar matematika. Kita akan sangat mudah mempelajari sesuatu jika kita mencintainya terlebih dahulu. Bagaimana mau mencintai matematika jika kita tidak mengenalnya? maka langkah pertama adalah kita harus mengenal dulu atau istilah anak muda PDKT dulu. Kita harur mengenal apa itu matematika?, apa fungsi matematika bagi kehidupan sehari hari?. Memang mengenal itu sulit, tapi kalau sudah niat anda pasti bisa. Jika kamu sudah mengenalnya, maka kamu akan tahu bahwa matematika sangatlah dibutuhkan dalam kehidupan sehari hari. Contoh sederhananya adalah setiap orang pasti perlu menghitung uang. Sungguh tak mungkin kita bisa hidup jauh dari matematika. Maka Tanamkanlah dalam pikiran kita bahwa matematika itu sesuatu yang berguna, indah, menarik dan sebagai teka-teki yang menyenangkan untuk dipecahkan. jika kita sudah kenal maka cintailah matematika. Jika kita telah mencintainya, Semua rumus yang kelihatannya rumit tiba-tiba akan menjadi mudah untuk dipelajari. Begitulah kekuatan cinta, kalau sudah cinta kita pasti rela memberikan segalanya demi yang kita cintai.

3. Berdoa
Sebelum kita memulai belajar matematika, ada baiknya kita berdoa agar Allah SWT memberi kemudahan bagi kita untuk memecahkan setiap persoalan yang terdapat di materi yang kita pelajari. Allah SWT itu kan Maha Pintar, maka mintalah kepada-NYA agar kita bisa memahami materi yang kita pelajari. Selain itu agar kita tetap konsisten dalam belajar dan gigih dalam berusaha, serta tidak mudah putus asa dalam belajar. Jadi selain berusaha kita juga harus berdoa.

4. Banyak Latihan dan Belajar
3 point diatas akan sangat tidak berguna jika ujung ujungnya kamu tidak mengambil langkah untuk segera belajar dan banyak latihan dengan rajin dan konsisten. Terkadang ada masanya kita semangat sekali untuk belajar, namun ada juga masa-masa ketika malas sekali untuk belajar. Maka disini butuh kedisiplinan serta kekonsistenan dalam mempelajari matematika. Dalam 1 hari Tidak perlu meluangkan terlalu banyak untuk belajar, cukup sedikit waktu namun tetap kontinyu dan konsisten. Matematika adalah ilmu hitung, tentu akan semakin baik belajar ilmu hitung dengan berlatih menghitung dengan rajin. banyakin latihan membahas/mengerjakan soal-soal, karena jika kita sudah terbiasa, maka akan mudah bagi kita untuk menyelesaikan soal yang sama dikemudian hari. Selain itu hal tersebut juga bisa membuat pemahaman kita kepada matematika semakin mendalam.
Setidaknya ada 6 tahap cara belajar yang baik:

a. Pahami Materi dengan rumus rumusnya
b. kelompokan rumus rumus yang ada
c. mulai mengerjakan soal-soal yang ada pembahasannya.
d. kerjakan soal tadi tanpa liat pembahasan.
e. kerjakan soal lain yang tipenya sama.
f. Terus berlatih soal-soal yang lain.
g. jangan hanya belajar dari satu buku, karena biasanya ada buku yang tidak menjelaskan persamaan secara detail sehingga susah untuk dipelajari. Jadi disarankan agar mencari buku referensi yang lain agar semakin mudah dalam mempelajari.


5. Tiada kata "Aku Tak Bisa" dan "Putus Asa"
Putus Asa merupakan penyakit yang paling sering ditemui setiap orang ketika berusaha untuk mendapatkan sesuatu. Ketika kita belajar matematika, hindarilah sejauh mungkin kata putus asa, ketika kita menemukan soal yang rumit,maka segera minta bantuan ke guru matematika atau ke teman yang sudah memahami. sebisa mungkin jauhkan diri dari mengucapkan kata "Aku Tak Bisa" karena hal tersebut hanya memperburuk keadaan, ketika kamu merasa bahwa kamu tidak bisa mengerjakannya, maka katakanlah "Aku Pasti Bisa"!! Berilah semangat motivasi untuk diri sendiri, karena setiap permasalahan pasti ada pemecahannya.. Ingat AKU PASTI BISA.....

6. Sabar.
Sabar dalam belajar, sabar dalam memecahkan persoalan, sabar dalam melaksanankan segala sesuatu, Ingat orang sabar disayang Tuhan.

Semoga tips-tips diatas bermanfaat. Amiin.

Pembelaan Seorang Matematikawan

Tahukah anda kalau bagi matematikawan, semakin tidak berguna suatu teori matematika di kehidupan sehari-hari, semakin bernilai-lah teori tersebut? Tahukah anda bahwa matematikawan berkutat dengan matematika yang berbeda dengan yang umumnya dipelajari di sekolah-sekolah? Kenalan lebih dekat dengan matematika yang sebenarnya yuk!

Kenalan di sini, bukan berarti ini adalah artikel Matematika. Ini adalah artikel “Tentang” Matematika. Supaya kita dapat gambaran yang lebih jelas, sebenarnya makhluk apa sih Matematika itu. Untuk lebih detail-nya, secara definitif, anda bisa lihat di wikipedia: http://en.wikipedia.org/wiki/Mathematics. Saya hanya akan mengupas satu aspek fundamental yang mestinya jadi pengetahuan populer (umum) tentang apa itu Matematika.

Banyak jurusan Matematika di perguruan tinggi yang mengeluh bahwa siswa yang masuk ke jurusan mereka tidak punya bekal yang cukup dalam matematika. Padahal, kita semua dari sekolah dasar hingga masuk kuliah sudah sepuluh tahun lebih belajar matematika, dan mungkin punya nilai tinggi bahkan sempurna. Bukan cuma di Indonesia, Amerika yang maju juga mengalami hal yang sama. Problemnya adalah, selama di sekolah, kita semua tidak pernah belajar dan fokus secara formal tentang “proof”. Inti dari matematika yang sebenarnya. Kerangka dari seluruh sistem Matematika. Di sekolah, kita hanya belajar MENGGUNAKAN matematika. Hal yang bagi matematikawan adalah “trivial and dull”.

“Proof” juga yang membuat matematikawan Yunani (dipelopori oleh Euclid) dianggap sebagai peletak dasar kerangka berpikir bagi matematika, bukannya matematikawan Persia, Cina, India, dan lain-lain yang mungkin sudah muncul ratusan tahun sebelum matematikawan Yunani bermunculan. Ini ya karena matematikawan Yunani-lah yang muncul dengan metode “proof” yang formal (“rigor”) untuk melahirkan teorema-teorema.

Matematikawan bekerja dengan tujuan membuat teorema baru. Jadi, produknya adalah teorema. Teorema didapat melalui proses “proof” yang formal (“rigor”) terhadap suatu pernyataan (“conjecture”) dengan menggunakan teorema (dan aksioma) yang sudah ada sebelumnya. Nanti kemudian, teorema baru tersebut akan digunakan untuk “proof” teorema selanjutnya. Begitu seterusnya.

Matematika berkutat dengan teorema. Tidak memikirkan apakah teorema tersebut bisa dipakai di dunia nyata untuk memecahkan masalah sehari-hari atau tidak. Nilai suatu teorema tidak dinilai dari seberapa aplikatif teorema tersebut untuk dipakai di dunia nyata, tapi dari seberapa jauh (seberapa banyak) teorema tersebut bisa dipakai untuk “proof” teorema-teorema lain.

Jadi, Jika ada dua teorema, A dan B. Teorema A sangat aplikatif dan dapat dipakai untuk memecahkan masalah sehari-hari, tapi tidak banyak dipakai untuk “proof” teorema baru. Teorema B, sama sekali tidak berguna di dunia sehari-hari, tapi begitu banyak teorema yang bisa di “proof” dengannya. Maka, bagi matematikawan, teorema B jauh lebih bernilai. Malah, secara ekstrim, melalui essay yang menjadi klasik bagi matematikawan di dunia, “A Mathematician’s Apology”, G.H. Hardy mengatakan: “Semakin tidak berguna suatu Matematika di kehidupan sehari-hari, semakin besar-lah nilai Matematika tersebut. Matematika yang dapat dipakai di kehidupan sehari-hari adalah ‘trivial and dull’ “. Memang ini pernyataan bombastis, tetapi pesannya adalah, matematikawan itu memberi nilai pada matematika sebagai matematika dan untuk matematika. Bukan penerapannya. Orang menyebutnya sebagai “Matematika Murni”.

Di sisi lain, bidang Matematika yang berurusan dengan penerapan teori Matematika di berbagai bidang lain, seperti Fisika, Ekonomi, Engineering, dll disebut “Matematika Terapan”. Jadi, apakah para Fisikawan, Ekonom, Insinyur, dll menekuni Matematika Terapan? BELUM TENTU. Sebagian besar dari mereka hanya “using” Matematika Terapan, bukan “doing” Matematika terapan. Yang dimaksud “doing” Matematika Terapan adalah menekuni teori-teori Matematika mana yang bisa diimplementasikan bidang lainnya dan bagaimana teori tersebut bisa digunakan. Sekali diketahui bagaimana suatu teori bisa diimplementasikan, matematikawan yang “doing” matematika terapan akan “move on” untuk implementasi teori lainnya. Sedangkan orang yang menekuni Fisika, Ekonomi, Engineering, dll, yang hanya menekuni penggunaan suatu teori matematika yang sudah diketahui implementasinya, hanyalah “using” Matematika Terapan. Saya harap anda bisa membedakan antara “doing” dan “using”.

Jadi, intinya adalah, matematikawan (murni, bukan yang terapan), “Doing Mathematics for the shake of Mathematics”. Mereka menekuni Matematika untuk Matematika, karena bagi mereka Matematika adalah hal yang indah yang bisa ditekuni untuk Matematika itu sendiri. Mungkin aneh bagi kebanyakan orang, tapi bagi matematikawan, mereka melihat matematika itu seperti halnya puisi ataupun karya seni yang mesti dihargai keindahannya sebagai karya seni itu sendiri, bukan apakah bisa diterapkan atau tidak.

“Mathematics is one of essential emanations of human spirit, a thing to be valued in and for itself, like art or poetry” – Oswald Veblen.

“Beauty is the first test: there is no permanent place in the world for ugly mathematics” – G.H. Hardy.

sumber:www.wikimu.com