So believed the ancients. Egypt

Few people think that the techniques that we use for writing and counting have been formed over many thousands of years. They seem obvious to us, well, think, multiply into a column, transfer all terms from the unknown to one side. It's so easy! In fact, these are huge intellectual conquests of humanity, which were often inaccessible to the smartest people of the past. I am going (if I have enough patience and time) to write a few notes about how they felt in the past. In this I will tell about how the Egyptians believed.


I was always a little interested in ancient Egypt. Well, first of all, Egypt is one of the first states about which we know a lot, and besides, this is a very great state that left a huge legacy. I do not mean the huge size of the pyramids. Even our writing is both Latin and Cyrillic, dating back to ancient Egypt. I also always liked Egyptian sculpture, and fashion to shave the head in women and men. It seems very modern. But this article is not about artistic culture. So let's get started.

Numbers and numbers
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The Egyptians used non-positional decimal notation. The numbers looked like this:



These figures refer to the so-called. hieroglyphic letter, which was later replaced by hieratic. I love hieratic writing. It looks very stylish. But here I will use the hieroglyphic outline.
All integers were formed by repeating the characters given above (and some others for even higher digits). For example, 3215 would be:



A very clear system, although not too concise. It is easy to learn, but the numbers are not very comfortable. At first glance it is difficult to catch the exact value of the number. The Egyptians wrote in different directions, but I am writing here as customary to us left to right.
Now about fractions. For three fractions there were special badges:



All other fractions that had a unit in the numerator were denoted by a denominator and an eye-like icon on top. For example, below I wrote 1/14

All correct fractions were written as the sum of such fractions. For example:

On one site I read that “in some cases” Egyptian fractions are “better than ours”. And even in the English wiki, there is such a wonderful example: “Egyptian fractions sometimes make it easier to compare the size of fractions. For example, if someone wants to know if 4/5 is more than ¾ he can turn them into Egyptian fractions:
4/5 = 1/2 + 1/4 + 1/20
3/4 = 1/2 +1/4 "
This “easy way” reminds me of a joke about Feynman, who summed up the series in his mind for the sake of some task of the school course. I’m a humanist and I don’t even know how to count, but it’s much easier to compare ordinary fractions in their normal notation than to translate them into an Egyptian look. Perhaps for the Egyptians comparisons of this kind were more convenient, since they did not know our fractions.

Addition and multiplication

Well, here we go to the main thing. How did the Egyptians count? Addition and subtraction of integers happened in them as well as in our country, and maybe even easier, because they just needed to unite the hieroglyphs and take into account the change of digits. And what about multiplication and division? In the ancient Egyptian world this was not at all a trivial task.
The Egyptians used such an algorithm for multiplication. Numbers were written in two columns. The first column begins with one, and the second with multiplicand. Then each number in the column was doubled until some of the numbers in the first column could add up the multiplier. Did you understand? The example is better understood. For example, 7 to 22

one	22
2	44
four	88

8 is already greater than 7, so the plate ends with four. Now 1 + 2 + 4 = 7 means 22 + 44 + 88 = 154 . Believe it or not, 154 is the right answer. Of course, in the Egyptian record (I don’t know how it looked exactly) such calculations were simpler, because multiplying by 2 in the Egyptian record is very simple.
Another example, a little harder: 13 times 57

one*	57
2	114
four*	228
eight*	456

1 + 4 + 8 = 13 and 57 + 228 + 456 = 741
Sometimes, to speed up the process resorted to multiplying by 10.
The question may arise, is it always possible to imagine a multiplier in this form? Yes, in fact, we are actually dealing with a binary number system: 1 * 2 ⁰ + 0 * 2 ¹ + 1 * 2 ² + 1 * 2 ³ ie 1 + 100 + 1000 = 1101
The division was performed using a similar algorithm. Divide 238 by 17:
Again we draw up a sign on the one hand, which costs 17 on the other. The doubling process stops on the number that doubles will be more divisible.

one	17
2	34
four	68
eight	136

Here you need to make the number 238 of the numbers in the second column, starting from the end. 136 + 68 + 34 = 238 , so we need the numbers 8 + 4 + 2 = 14 . So, 238/17 = 14
Unfortunately, division does not always result in an integer. In some cases it was quite difficult. I will show a simple example borrowed by me from one book.
Divide 213 by 8
At first, everything is as usual.

one	eight
2 *	sixteen
four	32
eight*	64
sixteen*	128

Here we will stop, because 128 is 2 = 256, which is more than 213. 128 + 64 <213. 128 + 64 + 32 is already bigger again. Does not fit. 128 + 64 + 16 <213 So far, everything is OK. 128 + 64 + 16 + 8 is already more. So we were able to dial only 208 = 128 + 64 + 16 of 213. And it remains for us to divide 213-208 = 5
We divide the divider by sex, using the already familiar table. Fortunately, 5 is 1 + 4.

1/2 *	four
1/4	2
1/8 *	one

So the final result will be
213/8 = 2 + 8 + 16 + 1/2 + 1/8 = 26 + 1/2 + 1/8
Now we have a good case, but this is not always the case.