Борис Бурда
Author: Boris Burda
Journalist, writer, bard. Winner of the «Diamond Owl» intellectual game «What? Where? When?»
Liberal Arts
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NON-TRIVIAL SOLUTION: How to make life difficult for forgers of documents?

NON-TRIVIAL SOLUTION: How to make life difficult for forgers of documents?
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Marinus van Reymerswaele. Tax collectors. C. 1540


In the Middle Ages, Arabic numerals appeared in Europe (in fact, Indian, but everyone is already used to calling them that). They are more convenient and simpler, but who knows if there is no harmful witchcraft in them?

In addition, it is much easier to forge them than the usual Roman ones. What to do?

The answer is a little later.




In vain did Lenin say that socialism was accounting. And capitalism, and feudalism, and the slaveholding, and even the primitive communal system without taking into account will not go anywhere – you will die of hunger. And accounting is a quantitative characteristic of a given population, and there is nothing without it.

The wildest gatherer and hunter with a stone ax should have known that three killed rats were more than two, that ten arrowheads were significantly more than five, and that one lion’s skin could decently dress one leader, and not two or three.

The primitive sages thought of making marks very quickly, so as not to forget how many sacks of nuts lie in the corner of the cave. Put, for example, a dash on a piece of bark with a flint and drag one bag to the side. Another dash is another bag. There are as many bags as there are lines.

It was already the first number system – a unit, or unary. It survived the primitive communal system – Robinson Crusoe used it, marking days on a desert island with dashes. And on collective farms, workdays were taken into account with sticks.

NON-TRIVIAL SOLUTION: How to make life difficult for forgers of documents?
Robinson makes marks

What was it good for? You could show Friday, so that he was horrified how long Robinson had lived there all alone. And if Friday forgot or pretended to have forgotten, he would show it to him again. What is written down is saved and can be transferred to someone else.




This system cannot be called convenient. Robinson has lived on the island for a little more than ten thousand days – this is how long Friday will be counting sticks! Now, if some sign immediately meant 10,000, it would be as easy as shelling pears – three such icons, and a trifle side by side, and that’s it!

Such systems, which were later called additive, based on addition, arose among more organized peoples – for example, among the ancient Egyptians. The basis of this system, like many others, was the number 10 – it is the number of fingers on two hands. The standard is always at hand for everyone.

To write numbers up to 10,000,000, the Egyptians needed only 7 different characters: 1, 10, and so on – up to a million. Each sign, if necessary, could be repeated up to nine times. It was enough for a fairly serious arithmetic that helped the Egyptians to build their buildings.

Egyptian mathematicians with the help of these numbers were able to extract integer roots, raise to a power, knew arithmetic and geometric progression, and even solved algebraic equations of the first and the second degrees, calling the unknown not x, but “ahu”, that is, “heap.”




The Egyptians used hieroglyphs – one for each word. But their neighbors, the Phoenicians, have already invented an alphabet that allows you to write down all words not using thousands of hieroglyphs – it was enough to learn a few dozen letters. With words it became easier, but with numbers how?

NON-TRIVIAL SOLUTION: How to make life difficult for forgers of documents?
Egyptian numbers

The Greeks, who had privatized the Phoenician idea quickly, soon realized that the letters of the alphabet could also be used to write numbers. The first letter, alpha, – 1, second, beta, – 2 … tenth, “iota”, – 10, eleventh, “kappa”, – 20 … twenty-first letter – 100 and so on…

However, the letters were not enough – there are 24 of them in the Greek alphabet, but just to get to a thousand like that, you need 27. But the Greeks got out of the situation using the outdated letters “digamma”, “koppa” and “sampi”. In writing of words, they were no longer used, but for numbers – please, you can.

Before 999 everything was simple, but what if you need more? The Greeks quickly found their way: if they put a slash to the left of the number, it was already the number of thousands. The capital “mu” meant 10,000 and was called myriad. Above it, it was possible to write in smaller letters-numbers, how many myriads were there. Later, Diophantus simply suggested separating the myriads with a dot.

Here is how the Greeks solved such problems: “I am a sculpture of gold. The poets brought that gold as a gift: Charisius brought half of the entire sacrifice, Thespia gave part eighth, Solon tenth. Part twenty is the sacrifice of the singer Themison, and the nine who completed the talents are a vow given by Aristonik. How much gold did the poets bring all together as a gift?” Who got 40 – well done!




Kievan Rus borrowed from the Greeks not only religion, but also many cultural attributes. They did not disdain the number system either: the idea is the same, only the letters are slightly different. “Az” – 1, “lead” – 2… in general, and so on. From 1 to 999 – practically no difference, everything is the same…

An additional convenience was that the numbers could be immediately distinguished from the letters – a special “titlo” sign, something like a zigzag, was placed above them. Either they put such a sign above each letter, or they covered the whole number with one long “title” at once.

NON-TRIVIAL SOLUTION: How to make life difficult for forgers of documents?
Slavic numbers on the clock in Suzdal

A thousand was denoted in almost the same way as the Greeks – with a slash to the left of the number, only they put two more small dashes across. And to write the number 10,000, the letter “az” was circled and called “darkness” – not in the sense of “gloom”, but in the sense of “gloomy darkness”.

There were numbers and more – “az” in a circle of dots was 10 times more than “darkness” and was called “legion”, in a circle of rays it was already a leodr – 10 legions, a raven was depicted in a circle of oblique crosses – 10 leodrs, and in brackets above and below – 10 ravens, or a deck.

But it was a “small count”, and there was also a “great count”. In it, 1,000,000 were already called darkness, the darkness was the legion, the legion of legions was the leodrome, the leodr of leodrs was a raven, and 10 ravens were still a “deck”, but it was already 1049 – the Earth has about the same number of atoms.

NON-TRIVIAL SOLUTION: How to make life difficult for forgers of documents?
Small and great account




The most influential power of antiquity was still Ancient Rome, and therefore it was the Roman system of recording numbers that became the most common throughout medieval Europe. It was simple: only 7 numbers were designated by letters: I – 1, V – 5, X – 10, L – 50, C – 100, D – 500, M – 1000.

If it seems to you that it is difficult to remember, here is an assistant phrase for you: “We Give Advice Only to Well-Mannered Individuals” – all these 7 letters-numbers in order, you will never confuse. Just remember that you cannot use more than three identical letters in a number – it is not accepted.

How to represent, for example, 4? It is very simple – IV, if the smaller number is to the left of the larger one, it is not added, but subtracted. Using this method, the Romans depicted all numbers from 1 to 3999 without difficulty. And then they didn’t really need it – well, they were not mathematicians…

Moreover, when the ancient Roman accountants were impatient to write down larger numbers (I suppose they had stolen too much in another war), there were no problems. The numbers above which there is a line are thousands, two lines are millions. Just think – it could have been written more…

Roman numerals are still not forgotten. The number of the century and volume, the ordinal number of the monarch, the valence of the chemical elements, the numbers on the dials of “antique” watches, the number of an important event, the number of the derivative in calculus – everything is traditionally written in Roman numerals.


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But all these systems have at least one irreparable flaw – it is not convenient to count with their help, adding and subtracting is troublesome, multiplying is difficult, and dividing is practically unthinkable. If only scientists needed arithmetic – Jupiter is with them! But what are the merchants to do?

How can a merchant pay for 728 bottles of the rose oil Rhodes, if you have to pay 7 sesterces for each? And how much will it be in denarii, if one denarius is four sesterces? And how to pay with copper, if in one sestertia there are two and a half assa?

For it, multiplication would be enough – I generally keep quiet about division. Even in modern times it was considered not a science, but an art. The simplest was considered the method of “dividing into a galley”, proposed by Magnitsky’s Arithmetic, by which Lomonosov studied. Few mastered it.

And the worst of all was not that algorithms were not found – with additive number systems, like Latin, Greek and Slavic, it was generally impossible. A completely new idea was required that would allow everyone to count easier.




But even in Ancient Babylon, before the Egyptians, they came up with the first positional number system, in which the meaning of a digit depends not only on the outline, but also on its place in the number. However, the Babylonian system was sexagesimal – was it really necessary to remember 60 digits?

No, the Babylonians decided it simply, with just two cuneiform signs – a straight wedge for units and a recumbent wedge for tens. But with the appearance of a new straight wedge after the recumbent wedge (from right to left), a new sexagesimal discharge began. Its value has already been multiplied by 60.

NON-TRIVIAL SOLUTION: How to make life difficult for forgers of documents?
Babylonian numerals

The Babylonians almost got to the most important invention, which could become revolutionary – they came up with a sign to denote the missing digits. But for some reason they were afraid to put it in the very first category, which led to ambiguity.

Nevertheless, even with such an incompletely completed number system, they created one of the greatest astronomy of those times, and it did not happen without the ability to calculate. It also helped the Babylonian architects – the tower in the Bible is not just Babylonian, but not any other.




Those who, for some reason, envy the Hindus and wish to belittle their great invention, with a clear conscience can declare that this invention is the absolute zero, and no one will even be offended. They are right, but nevertheless those who introduced zero into mathematics are great people!

The world’s first record of a number in the decimal positional system dates back to 595 AD – not so long ago! Zero did not exist then; instead of it, an empty space was left in this category. By the way, “empty space” in Arabic is “syfr”.

A little later, around the 9th century, the Indians introduced a special sign “shunya” in the form of a ring – it is zero. Shortly thereafter, the great Al-Khwarizmi, who invented the science of algebra, wrote a book about Indian counting, in which he popularized Indian numbers and suggested to others.

NON-TRIVIAL SOLUTION: How to make life difficult for forgers of documents?
First Indian and Arabic numerals

The Arabs honestly called them “ar-kam Hindiya”, “Indian numbers”, but the Europeans learned about them from the Arabs. The first mention of them in Europe is the “Vigilan Codex”, Spain of the 10th century, still without zero. And these figures were already quite familiar to the advent of book printing in Europe.




Around the same time, economic contacts were revived, and economic growth was very difficult without banks. Banks were an old invention, they appeared in the 8th century BC. in Ancient Babylon. And the very word “bank” was born at the beginning of the II millennium, among the Italian money changers.

They did not have offices – only a bench on which they sat astride and laid out the money in front of them for exchange. Bench in Italian is “banco”, hence the word “bank”. If the money changer bargained, he was deprived of the right to work, breaking his bench (sometimes on the ridge). In Italian, “broken bench” is “banco rotto”, hence the word “bankrupt”.

For a long time, Florence has been the capital of Italian banking, a fundamentally plebeian city that does not respect the grandees, that is, the nobility. Up to the point that people there were recorded in grants for life for some crimes, which deprived them of almost all civil rights.

For a long time, Florence has been ruled by the Medici – rich merchants without a drop of noble blood. Later in France, the proud Mirabeau, boasting about the nobility of the family, said, “There was only one mesalliance in my family- the Medici”. The Florentine banks under the Medici were the first in the world.

NON-TRIVIAL SOLUTION: How to make life difficult for forgers of documents?
Medici coat of arms

It is clear that the Florentine bankers liked the convenient Arabic numerals. But is there no evil witchcraft in the invention of the wicked Mohammedans? Not to mention that these numbers are easy to fake – I has added a tail to a zero, and it is already a nine! How to protect yourself?




The numbers are actually convenient, and the Florentine bankers exterminated the possibility of forging them in 1299!

In addition to numbers, they simply began to write AMOUNT in WORDS in the totals column. It was impossible to forge such a document, or, in any case, it was very difficult.




There are few inventions, the success of which would be so indisputable and unconditional. Such a simple thing, but reliably protected the document from erasures, forgeries and other meanings. The numerical record was not canceled in words, but simply supplemented and certified.

There were even quite definite rules on how exactly the amount should be written in words. In particular, it was agreed that it must be done with a capital letter. 

It is specially indicated that not only numbers are written – the names of all digits of a number must certainly be written in words. But if the currency has, as often happens, signs of lesser value (like we have a penny), you can write their number in numbers, and the name – in abbreviated form.

There are many programs for the correct spelling of the amount in words in electronic payment documents. Well, everyone knows that the sum is written in words in parentheses. I will add only one thing – writing the amount in words, signing for the receipt of money, is very nice!




Robinson knew how to write and count, right? And on the island he managed with chopsticks – why? Because he’s gone wild. Society is unthinkable without counting, and on a desert island it was not.

You can’t think of a more convenient basis than 10 for human counting – we all begin to count on our fingers. The Babylonians had a progressive and sensible idea of ​​positional numbering, but the monstrous foundation of the 60 system ruined everything.

Egyptian numbers were difficult enough to use – only scribes, specially trained people, knew how to deal with them. It was not good for Egyptian business.

The numbers were invented by the Indians, and they called them Arab – from the one who introduced, and not from the one who invented. So it is with many disputes about priority: who has showed us is the author.

The amount in words does not add anything new to the document, but it makes the old more reliable. Better to spend money on reliability by writing a few more words than to let such a question take its course. Try to make it difficult to distort your words and deeds – otherwise you will regret!


All illustrations are from open sources

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