Why is zero invented

The difficult birth of zero

The zero was invented long after the other numbers. The Romans did not know it at all, the Babylonians could not deal with it, only the Indians recognized the potential of this bizarre number, which alone is nothing, but can help others to become great.

Half of electronic data processing and thus a large part of our modern life consists of zeros. Representing the alphabet and digits as combinations of 0 and 1 has proven to be best for the mentally limited computer: the machine can digest the simple yes or no of the binary number system without much knowledge. The fact that even a short text turns into an almost endless worm of zeros and ones is compensated by the computer with work anger.

The layperson is not aware of the digital omnipresence of zero hidden in machine language. However, we also encounter zero uncovered in everyday life. A condominium costs 630,000 francs; A Greek road project is being discussed in Brussels for EUR 7,500,000. A power plant has an output of 1,200,000,000 watts; the virus made visible in the electron microscope is 0.000000025 meters long. By evaporating nine zeros into “Giga” or “Nano”, for example, the hustle and bustle can be made more legible.

The fact that an excess of zeros can be harmful is shown by the "zero attack" diagnosed in Germany in the twenties of the last century, a disease that occurred during hyperinflation: the need to leaf out hundreds of billions of marks and postage stamps at the baker's for a bread sticking with hosts of zeros made many citizens lose their psychological balance. A frequent symptom of the "zero patients" was the compulsion to write endless rows of zeros.

With moderate use, however, we perceive the zero as a thoroughly decent member of the digit family. The zero is anything but ordinary. Even the simplest arithmetic operations reveal their bizarre nature. If you add or subtract zero to any number, nothing changes at all. But woe to multiply by zero. Now every number, no matter how proud, perishes in one fell swoop and becomes zero itself.

And anyone who thinks of dividing a number by zero will be declared crazy by the mathematicians' guild. Because the result of dividing by zero should be a number that, multiplied by zero, in turn gives the starting number. However, since any number, no matter how exotic, multiplied by zero can only deliver zero and never a number different from zero, the mathematicians have forbidden dividing by zero without further ado. A shrewd person might object that one could at least allow zero to be divided by zero, because then there would be the number 23, for example, which, multiplied by zero, actually leads back to the initial number zero. "All well and good, but mathematically indecent nonetheless," replies the numbers man. Because every number multiplied by zero results in zero, a permitted division of zero by zero would result in the entire cosmos of numbers - an arbitrariness that for the righteous mathematician is probably even worse than the lack of a solution.

As a computing partner, zero can be ineffective as well as destructive and produce the impossible or anything. The empty round becomes Venus with almost limitless fertility when it is attached to the end of a number. 37 becomes 370 and soon 370,000,000,000. The property of being worthless in and of itself, but turning a modest number into a respectable size and even a giant by appearing in a certain place is probably the greatest Virtue of zero.

As natural as such computational use may seem to us today - the path from zero into the world was anything but easy. Not only did calculating with “nothing” cause philosophical worries, the changeable face of zero aroused suspicion in some places. Even in the third millennium AD, my bank still demands that I also write out the 5030 francs on the check as "thirty-five thousand".

The Babylonian mathematicians created a forerunner of zero. Here people already knew at the beginning of the 2nd millennium BC A numerical script made up of digits, the value of which depended on the position within the number shown. A convention that we also know from our system of tens, where 324 of course means 3 hundreds, 2 tens and 4 ones - and not about 3 plus 2 plus 4, i.e. 9 Hundreds with CCC, 2 tens with XX and 4 ones with IV. So Roman C always meant 100 wherever the character appeared in the number. For large numbers, however, the Roman number system became cumbersome. So one made do with additional characters, in which a horizontal line above the number meant a thousand times and a frame around the number that was open at the bottom meant a hundred thousand times.

The Babylonians got by with just two digits: a vertical nail meant one; an angle open to the right stood for ten. So you combined up to 5 angles with up to 9 nails and came to 59. If you wanted to write 60, you wrote a single nail again and meant a "sixties". And if you had 59 sixties with 5 angles and 9 nails, the number 3600 (60 times 60) was again the individual nail. The Babylonians had thus invented a position system that gave the digits a certain order depending on their position within the number: from the right as the 1st order, the units as the 2nd, the sixties, the third order the sixty-sixty, etc.

Such a sixty system instead of the familiar system of ten may seem strange. But we also maintain the Babylonian heritage when we write 5 hours, 32 minutes and 19 seconds - a total of 19,939 seconds, represented as 1st, 2nd and 3rd order in the sixties system. And even when measuring angles, we act Babylonian.

If a Babylonian dealer now had to write 62, he scratched a nail for 60 in the soft clay of the writing board and to the right of it another two nails for 2. To show what was 60 and what was 2, he left between the first and the other two Nail a small gap. But if he meant 3602, the distance had to be even wider, because now it had to be made clear that there was no sixties and that the first nail on the far left rather meant 60 by 60.

The misunderstanding was programmed. Scattered or careless writers often forgot a space in between. And if you had to mark two or more missing orders with a correspondingly wide gap, misinterpretations were almost inevitable. The Babylonian scholars countered ambiguities with additional comments or recognized the magnitude of the number from the context. This was easy in the sixties system thanks to the relatively large leap from one order to the next. It was clear to the clerical accountant that the farmer donated 5 to 5 times 60 sheep for the temple festival.

It took almost another 2,000 years before an unknown, bright head in Mesopotamia came up with the idea that one could mark the lack of an order within the number with a special sign: On an astronomical tablet from Uruk there is where between 2 times 3600 the 3rd order and the 15 of the 1st order a 2nd order is missing, as a placeholder two diagonally raised small nails, similar to an apostrophe. It was clear that 7215 (2 times 3600 plus 15) and not about 135 (2 times 60 plus 15) was meant - Babylon had invented the "zero". The plaque with the original zero is now part of the collection of the Louvre in Paris.

Now with the zero there was a symbol for the space, a mark for a lack of order within the number. Mathematicians did not understand this early zero as an empty set or as a "number zero" that could also be calculated. The Babylonian accountant was still at a loss when he had to subtract two equal amounts from each other. And he noted: “20 minus 20. . . you know." Another writer withdrew from the affair with the remark "The grain has run out" when the result of the settlement of a grain distribution had become zero.

India gave the world zero in all its mathematical wealth. In northern India, in the third century BC, scholars developed A system of ten, whereby abstract, graphic signs were created for the digits 1 to 9. Our modern basic numbers can already be seen in this. Our digits are therefore wrongly called “Arabic”. The Arabs only got to know and appreciate their number system and algebra as trading partners of the Indians and finally passed them on to the Christian West. However, the Arabs enriched Indian knowledge with numerous own findings; With his books on Indian mathematics, Mohammed Ibn Musa al-Charismi became a pioneer of modern mathematics in Europe around 800 AD. The author's name became a synonym for the new arithmetic as alchoarismi and later algorithms.

The Indians created different series of characters for the digits 1 to 9, depending on whether they denoted ones, tens, hundreds, thousands or tens of thousands. This allowed numbers up to 99,999 to be displayed - not enough for the astronomers with their passion for large numbers. The sky-gazers therefore resorted to Sanskrit, the language of scholars. They gave names to the basic digits (1, 2, 3, 4... Eka, dvi, tri, catur...) And also assigned Sanskrit words to the powers of ten (10, 100, 1000... Dasa, sata, sahasra...) .) and expanded the number system almost arbitrarily with terms for very high potencies (e.g. padma for 1 000 000 000).

A number was now expressed by simply stringing the names together, starting with the smallest order: dvi eka sata ca tri sahasra (two, one hundred and three thousand). In the 5th century AD, the Indian mathematicians had the brilliant idea of ​​greatly simplifying their number system by foregoing mentioning the powers. But this could only work if you indicated missing powers with your own word. With sunya for “emptiness” the Indians created their mathematical zero. With dvi sunya eka tri two ones, no tens, hundreds and three thousand were meant, so 3102. With that, the number friends on the Ganges had invented a position system as well as the zero - a progress that was made with Buddhism and the spice and Ivory trade was carried rapidly to Cambodia to the Khmer and as far as Java.

The scholars of India were fond of poetry, and astronomers dressed their numbers in gorgeous verse form. The rhythm of words and verses may have been useful to scholars for memorizing numerical or astronomical tables, but with the best will in the world, the poems could not be reckoned with. The Indian trader used the abacus, the abacus, for mundane mathematical activities. Pebbles or marks were arranged in columns and shifted according to the four basic arithmetic operations. If the tens, hundreds, etc. were full in a column, a transfer was made to the next higher column. You can still admire in the Far East or in Russia that you can calculate quickly with larger numbers.

The Indians used a board sprinkled with fine sand as an abacus. Vertical lines were drawn to delimit the columns, and numbers 1 to 9 were written in the sand. When calculating, you kept wiping away the old digits and writing the new result between the lines. And if one of the columns was zero, the place simply remained empty.

At the beginning of the 6th century AD, merchants in northern India came up with the idea of ​​using the basic concept of the scholars' poetry of numbers - the position system and the zero - for daily business. In the case of the abacus, the columns were dispensed with and the digits were given the corresponding power of ten depending on their position within the number representation. And where “emptiness” was to be marked, you wrote a period and later a small circle: the number zero, as we still know it today, was born. The beauty of the new number delighted the poet, and Biharilal wrote to a woman: "The point on your forehead increases your beauty tenfold - like the zero point tenfold a number."

In principle, the Babylonians had come that far. But while in Mesopotamia from then on the use of the zero as a placeholder within the representation of numbers was limited, the Indians quickly recognized the enormous potential of the zero as a number and empty set. As early as 628 AD, the astronomer Brahmagupta presented in his mathematical work how to apply the five basic operations of addition, subtraction, multiplication, division and exponentiation not only to "goods" (positive numbers), but also to "debts" (negative numbers). and applies to "nothing" (the number zero). So, in strict logic, a debt, subtracted from nothing, became a credit, or a credit, subtracted from nothing, became a debt. Brahmagupta even dared to divide by zero with cosmic generosity: "If you divide any number by nothing, it becomes infinity." India had given algebra to the world. The way was clear for further generalizations of the concept of number; Science and technology had a solid mathematical basis. Anyone who thinks that Eastern wisdom was greeted with joy in the Old World does not know the minds of the Middle Ages. It is true that the French monk Gerbert d’Aurillac learned Indian numerals and the corresponding arithmetic from Arabic teachers on an educational leave in Andalusia around the year 970. But when trying to introduce the ingenious method in Christian Europe as well, the cleric met with unanimous resistance. It is true that instead of the individual pebbles, arithmetic marks made of horn with the “Arabic” numbers 1 to 9 were accepted on the abacus. But one did not want to know anything about the zero or a calculation with it. Archconservatives prefer to use the Roman numerals I to IX on the new horn plates, so as not to come into contact with the "diabolical symbols of the Arabs".

It was not until the mathematician Leonardo Fibonacci from Pisa, who had traveled to Muslim North Africa as a school child with his trading father, that merchants were convinced of the great benefits of Indo-Arabic arithmetic in the 13th century. Because with the negative numbers and the zero, debts and losses could finally be calculated mathematically in the books. In Florence, however, they still didn't trust the thing. The use of Arabic numbers in contracts and official documents was banned by law in 1299. After all, there was a solid reason for the measure: In the time before the art of printing was invented, there was great fear of forgeries. As practical as Arabic numerals were for arithmetic, it was easy to add another digit to the number, or forgers quickly turned the 0 into a 6 or 9.

It was also Fibonacci who created the Latin name zefirum from the Arabic word as-sifr (the void), which eventually resulted in zero. The Latin cifra and later the French cipher and our numeral emerged from as-sifr, whereby the original term for zero was generally extended to all Arabic numerals.

As useful as Indo-Arabic mathematics may have been for Europe's merchants, the Church opposed its widespread introduction. Because the algorithm, writing Arabic numerals with a pen, could even be learned by the common people, while the use of the abacus was reserved for the professional calculator, often a clergyman. For centuries an ideological struggle raged between the «algorists» and the «abakists». The Indian invention only achieved its definitive breakthrough with the French Revolution, when the way for "democratic" arithmetic was finally cleared with the banishment of the abacus from schools and administration.

Herbert Cerutti is the science editor for the NZZ.

This article comes from the February 2002 NZZ Folio magazine on the subject of "Total Digital". You can order this issue or subscribe to the NZZ Folio.