So the ancients believed. 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 of it, multiply in a column, move all members with the unknown to one side. It's so easy! In fact, these are the huge intellectual gains of humanity, which were often inaccessible to the smartest people of the past. I am going to (if I have the patience and time) write a few notes about how they believed in the past. In this I will talk about how the Egyptians believed.


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

Numbers and numbers

The Egyptians used the non-decimal decimal system. The numbers looked something like this:



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



A very clear system, although not too concise. It’s easy to learn, but the numbers aren’t very convenient. At first glance it is difficult to grasp the exact meaning of the number. The Egyptians wrote in different directions, but here I am writing as usual from left to right.
Now about the fractions. For three fractions, there were special icons:



All other fractions with a unit in the numerator were denoted by the denominator and the 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 are sometimes easier to compare the size of fractions. For example, if someone wants to know if there is more 4/5 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, for the sake of some task of the school course, summed up the ranks in his mind. I am a humanist and I especially don’t know how to count, but it seems to me much easier to compare ordinary fractions in their normal notation than to translate them into Egyptian form. 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. What did the Egyptians think? Addition and subtraction of integers in them happened the same as ours, and maybe even easier, because they just needed to combine the hieroglyphs and take into account the change of digits. And what about multiplication and division? In the ancient Egyptian world, this was not a trivial task.
The Egyptians used such an algorithm for multiplication. Numbers were written in two columns. The first column began with one, and the second with multiplicable. Then each number in the column was doubled until a factor could be added from some numbers in the first column. Did you understand? The example is better. For example, 7 on 22

1	22
2	44
4	88

8 is already greater than 7, so the plate ends in 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 easier, because multiplying by 2 in the Egyptian record is very simple.
Another example, a bit more complicated: 13 times 57

1*	57
2	114
4*	228
8*	456

1 + 4 + 8 = 13 and 57 + 228 + 456 = 741
Sometimes, to speed up the process, they resorted to multiplying by 10. The
question may arise, is it possible to always imagine the factor 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 i.e. 1 + 100 + 1000 = 1101
Division was performed using a similar algorithm. Divide 238 by 17:
Again we make a plate on one side, which costs 17 on the other unit. The doubling process stops at a number that, when doubling, will be more than the dividend.

1	17
2	34
4	68
8	136

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

1	8
2 *	16
4	32
8*	64
16*	128

Here we stop, because 128 is 2 = 256, and this is more than 213. 128 + 64 <213. 128 + 64 + 32 is again more. Not suitable. 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 we had to divide 213-208 = 5
We divide the divider by gender, using the already familiar table. Fortunately 5 is 1 + 4.

1/2 *	4
1/4	2
1/8 *	1

Thus, 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 does not always happen.