Al-Khwarizmi About Indian Numerals

[vak]

Зантересовался я происхождением самой концепции алгоритмов. Вроде как всё пошло от труда Мухамеда ал-Хорезми про индийскую арифметику. Оригинал утерян, но сохранился перевод на латынь: "Algoritmi De Numero Indorum" (PDF). Предполагается, что перевод с арабского сделал Аделард Батский около 1126 года, хотя прямых доказательств нет. Я оцифровал это дело в текстовый файл (здесь).

Гуглопереводчик отлично понимает латынь. Текст на английском получается связный, но трудный для понимания. К счастью, спасает искусственный интеллект. Gemini умудряется вытащить и изложить смысл, при этом более-менее сохраняя авторскую структуру текста. Вот первые четыре абзаца.

Al-Khwarizmi said: "Let's first give thanks to God, our almighty protector, deserving of all praise. We should acknowledge our debt to the Indians, who developed a clever system using just 9 digits. They did this to simplify calculations and make math easier for everyone, from beginners to experts. This system allows us to perform all arithmetic operations, including multiplication, division, addition, subtraction, and more."

Al-Khwarizmi said: "I studied the Indian system of 9 numerals and their arrangement. Seeing its potential, I wanted to develop a way to use these digits for calculations, making math easier to learn, God willing. The Indians likely had their own reasons for this system, which are clear to me. But God also guided me to see its potential for simplifying math. If there are other reasons, beyond what I've explained, for the Indian system, they can surely be discovered by those who diligently study it."

The Indians created a system of 9 numerals with specific shapes (although there might be slight variations in how some people write the fifth, sixth, seventh, and eighth digits). This variation doesn't cause any problems because there are additional symbols that clarify the value of each digit, regardless of their specific shape. In my previous works on algebra ("restoration and opposition"), I explained that any number can be built from smaller numbers, ultimately coming down to the concept of "one." This concept of "one" is included in the universal system of numerals, as discussed in another book on arithmetic. "One" is the foundation of all numbers. It exists outside of any specific number itself but acts as the root from which all numbers grow. You can find "one" on its own, without needing any other number. However, you cannot create other numbers like "two" or "three" without first having "one." Numbers are essentially collections of ones. Think of it this way: you can't say "two" or "three" unless you already understand "one." This applies to the concept itself, not just the spoken word. Two objects wouldn't exist as "two" if you removed one of them. However, "one" can exist by itself without needing a second or third object. Following this logic, "two" is simply one being doubled, and "three" is one being tripled. You can understand the rest of the numbers in the same way. Now, let's return to the main topic of the book.

In my studies, I have explored the ways in which numbers can be expressed. Here, I describe a system for representing any number greater than one and less than 9. We begin with one, and by doubling it we reach two, and tripling it we reach three. This process continues in this manner for all numbers up to 9. Once we reach 9, a new concept is introduced: tens. The symbol for 10 takes the place of one, and just as we did with single digits, we can double or triple 10 to create numbers like 20 (doubling ten) and 30 (tripling ten). This process extends further. For numbers from one hundred to 900, the symbol for 100 takes the place of one. We can again double or triple one hundred to create 200 and 300, respectively. This method can be applied infinitely, using thousands, ten thousands, and so on. Now, let us turn to the notation used in this system. We utilize the concept of "places" to represent units, tens, hundreds, and so on. These places progress from right to left as the numbers increase. An empty place is signified by a small circle, resembling the letter O. For instance, a circle in the units place indicates that there are no units in the number. It's important to note that the symbol for 10 can appear similar to the symbol for 1. To differentiate between them based on their place value, a small circle might be used with ten to show it occupies the tens place. Specific symbols exist for each tens digit (10, 20, 30, and so on). Understanding this system requires recognizing the place value associated with each symbol. For example, a circle in the first place signifies "units," whereas a circle in the third place signifies "hundreds." Now, let us return to the main discussion of this book.

Comments

[juan_gandhi]

Сурово там было с арифметикой. Конечно, позиционная система - гениальное изобретение.

[spamsink]

Можно ли считать аль-Хорезми первым реверс-инженером?

[vak]

Я так понимаю, аль-Хорезми подсмотрел у индусов только поразрядную форму записи чисел справа налево, от единиц к десяткам, сотням и дальше. И идею кружочка как нулевого разряда. Вероятнее всего, это просто был способ записи результата вычислений на абаке.

Производить операции сложения, вычитания и прочие над такой формой записи вроде аль-Хорезми сам придумал. Доперевожу текст, посмотрим.