- 21 minutes ago
Why do people in Europe refer to them as "Arabic numerals" when they were invented in India? How could ten tiny symbols conquer the world? And how did zero change the course of history?
Category
🗞
NewsTranscript
00:15The Khajuraho group of monuments in India are famous for their rich iconography.
00:34But they are also home to another, very different treasure, an inscription from the 10th century
00:43featuring early forms of the numerals now used around the world, nine easily combinable
00:51symbols and a tenth, a small circle.
00:59They are used for counting, calculating, measuring, encrypting, or simply unlocking our screens.
01:06First developed in India, they only became universal centuries later.
01:36Europe almost did without these symbols altogether.
01:40Hundreds of years passed before they gained acceptance there, and they were adopted only
01:46after a long and winding journey.
02:15The Khajuraho
02:26Here, beneath the entrance portal, is a number grid that mathematicians now call a magic square.
02:35Like today's sediko, we have equal number in all directions.
02:40So if you will sum up this way, 7, 12, 19, 1, 20, then 14 will be 34, then 14,
02:4811, 25,
02:49then 5, 30, then 34.
02:52From all directions, if you will count, it will be 34.
02:56So then question comes, why they put this kind of sediko on the temple over here?
03:04In Indian culture, astrology is a very important aspect.
03:08So any kid born in this country, it influences with some planet, and all planets have own number.
03:15So if I will manage those numbers, the things will be changed in our life.
03:20So it is a vaniyantra, that giving blessings to this person, whose name is inscribed in the top.
03:32Using numbers to shape one's destiny may seem strange to us today, a form of superstition.
03:50But Professor Rahul Roy from the Indian Statistical Institute in Udelhi points out that this is exactly how modern mathematics
03:58began.
04:04If you want to know your harvest time in an agricultural society, it goes with seasons.
04:10And, you know, the stars in the sky also change according to the seasons.
04:15So astronomy is an important thing just for practical necessity.
04:20But it's also a game associated with astrology, for your date of birth, your time of birth, the sun, the
04:26stars, etc.
04:28So, as a by-product, mathematics also gets studied.
04:34The numerals of ancient India have endured, but have lost some of their mystique.
04:42One, two, three, four, and so on.
04:56With just ten symbols, ten being the number of our fingers, we can write any number, no matter how large.
05:04We combine, shift, replace, and know exactly that a three, like this, counts as three ones.
05:11In this position, three tens, and here, three billion.
05:15And to make sure that 30 really means 30 and not three, the Indians invented a small circle, the symbol
05:22for nothing.
05:23But nevertheless, a number in its own right.
05:25And nothing that counts for a lot and changed everything.
05:30Earlier, we could not do five minus six because the number system just started at one.
05:36So, zero as a number allows us to make the negative numbers and the positive numbers and so-called have
05:44the integer line.
05:46It also allows us to then later talk about fractions in the decimal sense.
05:54So, one-fifth is point two, zero point two.
05:59Zero as a number allowed us to do modern mathematics, literally.
06:06With the invention of zero as a number, India revolutionized arithmetic and science from the 7th century onward.
06:14But how did the Indian numerical system become the worldwide standard?
06:18And why do we call them Arabic numerals?
06:31In South East England, British historian Charles Burnett is researching the miraculous journey of numerals.
06:46Indian numerals, as we should call them, spread all over Asia, Central Asia, Eastern Asia.
06:55But they also came into the Arabic world and the story goes that there was an embassy from India to
07:04Baghdad in the 770s.
07:07And this embassy involved the gifts of books as well as diplomatic exchange.
07:16In particular, a set of astronomical tables.
07:20And the first person to actually take up both the tables and detailed description of how to use the numbers
07:28was Al Khwarezmi.
07:33Al Khwarezmi was born around 780, probably in the oasis of Kiva in what is now Uzbekistan.
07:41Little is known about the life of this Muslim scholar.
07:45But he is believed to have enjoyed a high standing among the caliphs in Baghdad, at the time the intellectual
07:51and cultural center of the Arab world.
07:56Baghdad was very important at this stage because it had just been founded as a new city for a new
08:05dynasty, the dynasty of the Abbasids.
08:08And they plotted this wonderful city, which has a circle, it's entirely circular, so the city itself is a microcosm
08:18of the heavens.
08:18And it became a center of science.
08:23And so we have the foundation in the city of the Bait al-Hikmas, it's called the House of Wisdom,
08:30which was preeminently a place where texts from all kinds of neighboring countries were brought in to be made into
08:40Arabic.
08:41Because it was important that Arabic, because it was important that Arabic should become the universal language.
08:49Until then, numbers in the Arab world were written out using the letters of the alphabet.
08:57Alif meant one, Ba, two, Jim, three.
09:04Ideal for numbering the surahs of the Quran, but not necessarily for complex calculations.
09:09For this reason, Al-Khwarizmi introduced the Indian numerals to complement the Arabic letters.
09:17The Arabs always called them Hindi, Khuruf al-Hindi, and the early Latin translations also called them Indiki, or Indikai,
09:29because they often use the word Figurai, Indikai, so forms that come from the Indians.
09:53In the Middle East and Central Asia, these Indian numerals adapted to the Arabic script are still in use today.
10:00But elsewhere, a different variant prevailed, known as Gobar, Arabic for dust.
10:07It is said to have originated at the other end of the Abbasid Empire, on the Mediterranean coast of North
10:14Africa.
10:35Here in the Maghreb, the Indian numerals introduced from Baghdad evolved into the Arabic numerals we use today.
10:42Professor of Mathematics Jamil Aysani teaches the history of these numerals at the University of Bajaya in Algeria.
10:56What we call Arabic numerals are the numbers used throughout the world today, whose current form developed over time in
11:06the Western Mediterranean,
11:09that is in the Maghreb and Al-Andalus, where several cultures encountered one another, namely Arab, Berber, and Latin culture.
11:26This is where the name Haruf al-Ghobari, numbers from dust, originated.
11:36Because merchants used a kind of board covered with dust or sand for calculations, in which they wrote their numbers
11:43and then wiped them away again.
11:48In the West you use wax tablets.
11:51But when you're talking about North Africa and the Arabian Peninsula, you can't use wax because wax just melts.
11:58So you have dust on a board and you make these symbols, these Indian numerals in the dust.
12:07And a Western form did emerge, and in fact it's the Western form of the Indian numerals which was then
12:16taken over, as you might expect, in the West by Latin scholars.
12:27These Muslim dust boards have long since disappeared, but the form of the numerals that emerged from them is still
12:34used to this day.
12:35But when and how did they spread throughout Europe?
12:38How did Christian countries discover the numerals of the Islamic world?
12:52Northwest of Madrid, at the foot of the Sierra del Guadarrama, is the royal site of San Lorenzo del Escorial.
13:11Here, a 50-meter-long hall houses a library of some 40,000 works.
13:18It's maintained and managed by the conservator, Father José Luis del Valle Merino.
13:29Today, the father wants to show historian Nirea Fernández Cadenas one of his greatest treasures,
13:35a manuscript written at the end of the 10th century in the monastery of Albelda in northern Spain.
13:44These are calendars.
13:48It contains the first Latin text featuring Indian numerals.
13:53They run from right to left, following the Arabic system.
13:57You can clearly make out the one and two, as well as the seven, eight and nine.
14:05The caliphate of Cordoba ruled the Iberian Peninsula at the time,
14:09and was absolutely brilliant when it came to art and culture, as well as astronomy and other sciences.
14:18The neighboring Christian territories benefited from a lively cultural exchange.
14:25Many innovations made their way into Europe from the Muslim parts of the Iberian Peninsula,
14:33including the Arabic numeral system.
14:42As the point of contact between Islam and Christianity,
14:46medieval Spain was a site of both permanent conflict and lively intellectual exchange.
14:50It was there that the first Latin scholars adopted the numerals of their Muslim neighbors,
14:56referring to the dust symbols from the Maghreb simply as Arabic.
15:09These Western forms remained remarkably consistent from the 10th century through,
15:17one could say even through to the present day.
15:19It's only the four and the five which underwent a noticeable development in the 14th century.
15:30The four, the simpler form of the four which were used throughout the West in the early Middle Ages,
15:35is just a loop.
15:37And if you cut the loop, as it were, you can see how that can develop into the figure of
15:43a four.
15:44The five was like a cup with an extension.
15:48And maybe if you turn that, you can get to the modern five.
15:55The Chronicle of Al-Belda, completed around 980,
16:00bears witness to the journey the numerals made in less than 200 years.
16:05From India, via Baghdad, across North Africa to the Maghreb,
16:10and onward to the Caliphate of Cordoba,
16:12where the numerals of the decimal system appeared as early as the 10th century.
16:18Now they only had to cross the Pyrenees,
16:21but it would take another 500 years before they finally spread throughout the rest of Europe.
16:31One reason for the slow adoption of Indian numerals
16:37was simply that they were strange shapes and not like anything that the Latins knew.
16:46And so they were rather suspicious of these shapes.
16:50In Europe, the Roman numeral system was used throughout the Middle Ages.
16:57It was the system of power, that of the rulers in Rome,
17:01and it had spread throughout the Roman Empire during its conquests.
17:08And since medieval society sought to emulate Rome,
17:12it retained the Roman numeric system.
17:24Roman numerals have simple, straight forms,
17:27because they were originally used mainly as inscriptions carved into marble or stone.
17:33To this day, they are used to indicate historical dates or the names of popes and kings,
17:37but neither in antiquity nor in the Middle Ages were they especially well suited to doing calculations.
17:51Most people today know Roman numerals and can work with them.
17:57But doing calculations with them is something entirely different.
18:01It was extremely complicated.
18:04A multiplication could take hours.
18:07That is why the Roman numeral system was reserved for the elite,
18:11mostly members of the clergy, known as calculators,
18:15who studied the system for many years before mastering it.
18:24Despite their disadvantages, Roman numerals prevailed in Europe throughout the Middle Ages.
18:37At the time, calculations were done using tablets made of stone, wax, or wood.
18:43Martin Hellman, a mathematics teacher at a high school in the town of Wertheim, Germany,
18:47has recreated a historical counting board.
18:52Today, his students are learning mathematics as it was taught in the Middle Ages.
19:00This is a counting board used for calculations in the decimal system,
19:06with every three columns grouped together by an arch.
19:12It starts in the far right.
19:13S stands for the units.
19:18Latin, singularis.
19:20Next to it, D stands for de zeni, the tens.
19:25And C, centeni, the hundreds.
19:30They are the value headers.
19:32M would be billions for us.
19:34And the next level would be our trillions.
19:37That is, a thousand times, a thousand times, a thousand times, a thousand.
19:45In Latin, this counting board was called an abacus.
19:48The Romans were already using it as a pocket calculator,
19:52placing pebbles on the fields known as calculi.
19:55This was simple and inexpensive, but it quickly became confusing.
20:01And so, in the 10th century, a visionary scholar came up with the idea of replacing the pebbles with counting
20:07stones,
20:08marked with Indian Arabic numerals.
20:16They look quite similar to ours.
20:21They are Arabic numerals in an older form.
20:26The one is easy to recognize, it's a line.
20:30And the two also looks like our two when seen like this.
20:36But in manuscripts, it's always represented flipped down like this.
20:45How exactly were you supposed to know which way round the counting stone should be?
20:50We live with some misinterpretations to this day.
20:56Our modern two is indeed an Arabic two, but upside down.
21:01But at least the new system made the old Roman abacus much more efficient.
21:12Atzin, can you calculate something for us?
21:15Please, namely 98,709 plus 12,345.
21:37So here we have nine and five, which makes 14.
21:41And then we have a carryover, which we put here.
21:43And four and one makes five.
21:46Then we have seven and three, which makes 10.
21:50Nothing goes here.
21:51And we have another carryover, eight, two, one, makes 11.
21:55And another carryover, nine plus one, plus one, also makes 11.
22:00So we have 11,054.
22:05Perfect. Yes, that's the correct result.
22:09Atzin not only placed all the counting stones the right way up,
22:13but also once again demonstrated the ingenuity of the system.
22:16It's attributed to a monk from Auvergne named Gerbert of Aurillac, who was just 20 years old.
22:25Gerbert of Aurillac was born in southern France.
22:32What was remarkable about his education was that he went to Spain, to Catalonia, to Barcelona.
22:45The Caliphate of Cordoba was not far away.
22:49This is where he learned about the new way of calculating using numerals.
22:55The idea of putting this into practice in the counting board was brilliant.
23:04Back in France, Gerbert of Aurillac embarked on an extraordinary career.
23:10He became secretary to King Hugo Capet of France and was later elected pope, taking the name Sylvester II.
23:17Did all of Christendom now embrace the new pope's numerals?
23:21Well, it wasn't quite that simple.
23:26Since the Arabic system is much more practical, you would think that it soon replaced the Roman system.
23:33But the opposite was true.
23:35The old system disappeared only very slowly.
23:39That's because calculating with Roman numerals was reserved for a small elite of clergy.
23:45They did not want to lose their privileges and their livelihoods.
23:51They claimed that if the Arabic numeral system was so simple and calculations with it so quick, then the devil
23:58in witchcraft must be involved.
24:02That frightened people off.
24:07They were always regarded as being a little bit strange, and perhaps for specialists only,
24:13and therefore were not generally known or generally used by the general public.
24:20And maybe for this reason, too, you often find them used in magical texts, almost as ciphers in the English
24:27senses,
24:29as symbols of unknown quantities, you know.
24:33And only the magician would know what they stood for.
24:40No sooner had they been introduced in Europe that the new numerals were banished from the Christian world.
24:47And Gerber of Aurillac, who had been Pope Sylvester II since 999, was accused of being in league with the
24:54devil.
25:09But the majority of the rural population in the Middle Ages could neither read nor write.
25:15Whether the elite used Arabic or Roman numerals made no difference to them.
25:23For ordinary farmers, the Roman system of calculation was far too complicated.
25:30But they had to keep records of their herds, milk yields, stocks, trade transactions and so on.
25:38In other words, they needed a numerical system.
25:41And so they carved notches into pieces of wood or stone tablets, materials that were readily available in everyday life
25:48and cost little.
25:49In this way, they counted without any arithmetic at all.
25:56For those who could neither read nor write, there was still the good old tally system from the Stone Age.
26:02The fact that these medieval notches look confusingly similar to Roman numerals is a coincidence.
26:08The large V for five is simply derived from the shape of our fingers.
26:21Humans can only perceive quantities of up to four at a glance.
26:26Beyond that, we must count one by one.
26:29That is why the human hand is a practical counting tool.
26:32We have four vertical strokes, one, two, three, four.
26:38From five onward, the shape changes to a V, just as it did with the Romans and the medieval farmers.
26:44In both cases, the symbols were carved into solid materials such as wood or stone.
26:51Hence the straight lines like the X, the V and the vertical stroke.
27:02X, V and the stroke can be found in both numerical systems.
27:07Sylvester II, the Pope of Arabic numerals, left a legacy that remained largely overlooked for nearly two centuries.
27:29As the Christian kingdoms expanded their influence on the Iberian Peninsula in the 12th century, they discovered and translated Arabic
27:37texts, including the works of the scholar Al-Khwarizmi.
27:46Al-Khwarizmi wrote two major works.
27:49The first is his book on algebra, Kitab Al-Jabbar Wal-Mujaba.
27:56It would later form the foundation for this discipline and give it its name.
28:02Al-Jabbar became algebra, and Al-Khwarizmi's name gave rise to the term algorithm.
28:10The second is his book on the Indian numerical system and Indian methods of calculation.
28:18The Arabic original is considered lost, but it was translated several times, particularly in Al-Andalus.
28:28There are four Latin translations, and it's thanks to these that the contents of the book have been passed down
28:34to us.
28:43One of the four translations of Al-Khwarizmi's work is housed in a library in Cambridge, England. In it, Charles
28:50Burnett made an interesting discovery.
28:52Could I ask, could I ask, could I just try to find the right page? Here it is, yes.
28:59Dixit Al-Ghwarizmi. And this is where he says, we see that the Indians invented nine symbols which stood for
29:12every number.
29:13And so we see, and so we see, and so we see, and so we see .
29:20Unfortunately, though this text is explicitly about the Hindu-Arabic numerals, the scribe has omitted every single Hindu-Arabic numeral,
29:32just left gaps.
29:33And so we have these phrases, and they look like this, and then nothing.
29:44Arabic studies sure can be challenging. The original manuscripts disappear, and the few surviving translations you're left with are incomplete.
29:57So I think either the scribe was loath to introduce symbols which people just didn't understand, or, and this is
30:08a more positive interpretation,
30:12he wanted another scribe who could use red ink, or he wanted to find some red ink in order to
30:20write the numerals in what we call rubric.
30:24And that is a completely different thing, because that would emphasize these new symbols, and so that they are really
30:32very important.
30:36It's from the Iberian Peninsula, still riven by interreligious warfare, that translators of Al Khwarezmi's works brought the new numerals
30:44into Europe.
30:46But with Roman numerals still dominating the continent, their influence remained limited.
30:55On the other side of the Mediterranean, the use of Arabic numerals and written calculations had become part of everyday
31:02life for many.
31:04In the late 12th century, a young man from Pisa arrived in Bejaia, Algeria.
31:10His name was Leonardo Fibonacci, and he would change mathematics forever.
31:28Leonardo Fibonacci was one of the most important mathematicians in history.
31:33He came to Bejaia in the late 12th century.
31:36His father worked here as an official, and had noticed that the local merchants and fishermen used a method of
31:43doing calculations that was different than the one common in Europe.
31:53So, the father decided to bring his son to him, so that Leonardo could learn the numeral system, methods of
32:00calculation, and trading techniques of the Islamic world.
32:22From Bejaia, Fibonacci continued his studies, traveling to Egypt, Syria, Constantinople, Greece, and Sicily.
32:32From Bejaia, Fibonacci continued his studies, traveling to Egypt, Syria, Constantinople, Greece, and Sicily.
32:34He returned to Pisa, where he wrote his famous work, Liber Abaci, the Book of Calculation.
32:47Novem figure indorum e sunt. Novem, otto, septem, sex, quinque, quattvor, tria, duo, unum.
32:58Philologist Paolo D'Alessandro and his team recently republished the seminal text.
33:08He writes,
33:10The nine symbols of the Indian numerals are as follows.
33:15With these nine, together with another symbol, the zefirum, or zero,
33:22any number can be written, as shown here.
33:32Fibonacci first introduces the Indian numerals and explains how the reader can remember the numerals when calculating.
33:42There is a symbol for one, a symbol for two, for three, and so on, for one hundred, for one
33:48thousand.
33:51These are all memory aids.
34:00He explains the methods of calculation.
34:04Then he describes in detail how they can be used for tasks in everyday life and in trade.
34:15One has to bear in mind the following.
34:18In the Middle Ages, every country had its own currency, its own units of measurement, units of weight and length.
34:30Whenever merchants exchanged goods, they had to convert everything.
34:37Sometimes this involved doing very complicated calculations.
34:45Fibonacci explains how to do these in the first part of the book.
35:00The Liber Abaci is both a theoretical and practical handbook, making Fibonacci the patron saint of Arabic numerals in Europe.
35:11According to historian Rafael Edanna, Fibonacci was the first to address directly all those who still counted on their fingers.
35:21The text of Fibonacci survives probably because it was so exceptional.
35:26It is clearly an exceptional work.
35:28It is definitely, if not the most important, one of the most important mathematical texts produced in Europe in the
35:38entire 13th century.
35:40But at the same time, this work is interesting because it includes a breadth of examples in which the mathematics,
35:49basically Arabic arithmetic, is used and applied to solve commercial and practical problems.
35:56So from this point of view, it provided a sort of treasure trove of solutions to problems that merchants could
36:07have been facing in their everyday business.
36:13In Fibonacci's time, the merchants of his home city of Pisa were at the height of their power.
36:19They built ever more magnificent buildings, built bridges, ports and caves where their ships could dock.
36:37Pisa, along with Venice, Genoa and Amalfi, were the great winners of the Italian commercial revolution that occurred at the
36:44beginning of the 13th century.
36:51It seems that the first people who adopted these numbers were the protagonists of the commercial revolution.
36:59We see for the first time people using contracts that are the ancestors to insurance contracts.
37:06We see people, merchants, using for the first time advanced bookkeeping methods that are similar to double entry bookkeeping and
37:16what a lot of people would consider modern bookkeeping methods.
37:21And in all of these calculations, when you are exchanging a quantity for another, when you are trying to estimate
37:30the odds of something happening, or when you are also converting currencies,
37:37you are always dealing with multiplications and divisions.
37:41That is really the key element that made the Hindu Arabic notation important and necessary for these people.
37:52Commercially, Fibonacci's work could not have come at a better time.
37:56Above all, however, it promoted written calculation in Europe, at the very time that a new writing medium was spreading
38:03across the continent.
38:09By producing pulp from old rags, Europeans were adopting an ancient process they'd first learned from the Arabs.
38:16Arabs. The Arabs, in turn, had learned it from the Chinese during the time of al-Khwarizmi.
38:31The product in question, paper. Its production spread throughout Europe in the 13th century.
38:41Well, it is a strange coincidence, as it were, that paper came into the Islamic world just at the same
38:49time as Hindu Arabic numerals arrived.
38:51And on the other hand, in Europe, paper started to be made locally in the early 13th century, at the
38:59same time as Arabic numerals started to be popular.
39:03But I think the introduction of paper is very important, because, you know, you were doing so many more calculations
39:12on paper, rather than using the abacus.
39:14Medieval counting boards allow someone to arrive at a result quickly, even with Gerber of Orillac's counting stones.
39:21The problem is, that result is gone just as quickly.
39:28On a sheet of paper, by contrast, every step of the calculation remains visible for verification. It makes cheating harder.
39:39It's no surprise, then, that, from the late 13th century onwards, Italian merchants became ambassadors of written calculations.
39:50Once it had become necessary for these commercial classes to transmit across generations this knowledge that was first acquired informally
40:02and first transmitted informally.
40:04Once it reached a certain diffusion, a certain stage, it became expedient to sort of formalize the transmission of this
40:16knowledge, the social transmission of this knowledge.
40:21So we see first manuals appearing in most central, northern, but also a few southern Italian cities.
40:29And then we see texts appearing in the southern part of Spain, in the southern part of France, with Montpellier
40:38and Lyon as very important centers.
40:41And then we also see the spread towards Germany. We see first texts appearing in the southern part of Germany,
40:48and then slowly these texts move towards north.
40:56A mid-15th century invention from Germany helped spread the art of calculation even more.
41:04The temperature is just right.
41:08Johannes Gutenberg developed movable metal type to print letters on paper.
41:16This is what is known as a hand mold.
41:20From around 1450, in the 15th century, it was used to cast the lead type.
41:28Letters and words, but also numbers.
41:32This is a six.
41:35I'll break off the casting sprue, and can then use this for printing.
41:51Cast in standardized forms and infinitely reproducible, without the risk of faulty or imaginative variations, Arabic numerals increasingly acquired a
42:00uniform appearance.
42:10Gradually, thanks to Gutenberg, they came to look the same across the entire continent.
42:26Here in Leipzig, the new printing presses produced arithmetic textbooks on a large scale for young people in economic and
42:34commercial centers.
42:44There was a certain pressure because merchants naturally had a need to publish books of calculations.
42:50This growing interest among merchants in mathematics led to the emergence of a new profession, namely that of the master
42:57calculator.
42:58This person taught people how to perform specialized calculations, including with Arabic numerals.
43:05Master calculators had nothing to do with universities.
43:08There was the profession of university mathematician and the separate profession of master calculator.
43:21In 1489, a graduate of the University of Leipzig went one step further and published an arithmetic textbook specifically for
43:29merchants, a genuine bestseller that was reprinted many times.
43:38Here's an example involving saffron.
43:42There are lots of examples involving spices, almonds, and raisins.
43:48Johannes Wittmann not only compiled knowledge, he also laid the foundations for a new language.
43:56Wittmann's manual was the first to contain a multiplication table, which we can see here, where the numerals are multiplied
44:03by one another.
44:052 times 2 equals 4, or 9 times 9 equals 81.
44:14What makes the book special is that for the first time in a book of arithmetic, we find symbols for
44:20addition, namely a plus sign as well as a minus sign.
44:25It's the first arithmetic book of this kind.
44:31Have you ever wondered where these practical symbols actually come from?
44:35At the end of the 15th century, German master calculators developed a new mathematical notation system for algebra, in which
44:43abbreviations and symbols increasingly replaced words.
44:49In Germany, we refer to this movement as the German Kurs.
44:53It took place between 1460 and 1550.
44:57Its aim was to develop a system in which mathematical concepts are no longer written out as words, but abbreviated
45:03using exponential notation and calculation symbols for basic arithmetic.
45:10Before the introduction of symbols, such as plus, minus, and equals, mathematicians would write down equations word by word.
45:22So there was no way to represent an equation with the aspect that we are used to, right?
45:29So equations look like sentences.
45:32And in doing this, this German tradition was pushing farther what had already been introduced with Arabic numerals themselves.
45:49The algebra we know today emerged with the development of these first mathematical symbols.
45:54The new notation system marked the full arrival of the Indian Arabic numerals in Europe.
46:00It's the end of a long journey.
46:09This story is interesting because it shows an incremental piecemeal and collective development of what would become a new mathematical
46:20language, a new way of doing mathematics.
46:27The language of numbers emancipated itself from the language of words, even inspiring other forms of expression.
46:43In Nuremberg, not too far from the center of algebra in Leipzig, Albrecht Dürer was one of the first to
46:49formulate art in mathematical terms.
46:59In 1509, he moved into the former home of a famous local astronomer.
47:06Christina Demmeler runs the archive of the German Renaissance master.
47:15Dürer went down in history not only as an artist, but also as a mathematician.
47:21He achieved so much in this field that he is still considered one of the most important mathematicians of his
47:28time.
47:29For example, he wrote the first German language geometry textbook.
47:38Like his contemporary Leonardo da Vinci in Italy, Dürer used mathematics in his artistic and spiritual quest.
47:46Both men blurred the boundaries between art and science and treated numbers as aesthetic or philosophical objects.
47:59This is probably the first printed form of a magic square, the so-called Jupiter square.
48:06It consists of a total of 16 squares.
48:10So the numbers 1 to 16 are arranged on it in such a way that the sum of each row
48:16and column, as well as the diagonals, is always 34.
48:20And Dürer arranged it so artistically that the date, 1514, can be seen in the middle of the bottom row.
48:28And to the left and right of it, there are also the numbers representing his initials, A and D, that
48:35is 1 and 4.
48:41Dürer's engraving Melancholia from the early 16th century is considered one of the most mysterious works of the Renaissance.
48:48What does its wealth of symbols mean?
48:51What does the table of numbers on the wall signify?
48:57Experts continue to debate these questions.
49:00The engraver, Zovia Frankel, who works in the master's former studio, suspects that it contains a very personal mathematical code.
49:12I've always been interested in what Albrecht Dürer would have considered important in the work.
49:20The last numbers, here at the bottom, are the date of his mother's death.
49:28Dürer created Melancholia two months later.
49:32This may well have something to do with the fact that Dürer loved his mother very much.
49:42I think Melancholia might just be a tribute to his mother. Maybe.
49:51One thing's certain. Dürer never visited India or the temples of Kajuraho.
49:59And yet his magic square bears a striking resemblance to the one in the Pajvanatha temple.
50:06This similarity is a fitting symbol for the 8th century long journey of Indo-Arabic numerals from medieval India to
50:13Renaissance Europe.
50:14It took some time for the Indo-Arabic numerals to become established.
50:19But once they were established, they were essential.
50:24And from then on, nobody could do any commerce, as it were, without using the Indo-Arabic numerals.
50:34Indo-Arabic numerals revolutionized the world, affecting trade, banking, science and technology.
50:41During the Renaissance, Europeans were still learning to master the art of calculations.
50:47But they paved the way for a time when all the world could be expressed with numbers.
51:11.
51:14Well ...
51:16.
51:31.
51:39.
51:41.
51:41.
51:41.
51:41.
51:41You