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In a scenario where scientists decipher a Math book of an alien culture, the last thing to expect is equations presented the way we know them.The way we write mathematical equations, is mostly taken for granted, as we know of one source. Early developments by the Arabs started with Algebra. The Renaissance in Europe was another hallmark in the evolution of equations, as new calculation methods were developed by Leibnitz, Newton, Euler, and so on until today.

In a parallel "world", the Chinese, Mayans, Aztecs and Incas have developed some type of technology, which may have required some advanced mathematics beyond arithmetic. They may have (or have not) developed some mathematical equations, which may have been written using different symbols, but not only that. All those equations may have their own writing system, as long as they were developed before first contact with the Europeans.

To clarify my point, take a look at the formula editor in this article. The editor converts a string of ASC-][ characters from a keyboard, and converts it into a visual representation which is easier to read than the string of ASC-][ characters. This representation is called a typeset representation and an example is shown at the upper-right of the article page.

The string of characters is rather limited in the number of ways we can represent a single equation, and it does not matter what symbols different cultures used for each mathematical operation. That is not the point. The point is how different cultures write their version of typeset representation before the computer era? It may be an outdated representation from an old Earth culture, or alien origin. Whatever it is, it must have been independently developed and without the influence of European contact.

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    $\begingroup$ This sound more like a question on the history of mathematics. $\endgroup$ – L.Dutch - Reinstate Monica Sep 26 '18 at 8:56
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    $\begingroup$ You might get some great answers if you reworded your question to something like 'historical examples of alternative typeset representations', leave out the alien bit and ask the question on math.se $\endgroup$ – Douwe Sep 26 '18 at 8:58
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    $\begingroup$ Are you asking about a different typeset (I've seen six in the last month, not including hand-written, all showing the same equation) or the use of different symbols (say using Chinese or Japanese characters instead of Greek/Babylonian/Aramaic like the ancient Chinese actually did)? I could create either answer but, really, how ancient cultures developed printing (versus just writing things down) has little to do with math. $\endgroup$ – LinkBerest Sep 26 '18 at 11:12
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    $\begingroup$ You are using very non-standard terminology. I understand that by "ASC-][" you mean ASCII. By "typeset representation" you mean mathematical notation, for example $x_{1,2}^y$, is this true? And you are you asking for examples of mathematical notation different from the modern conventions, is this true? The question as asked, how was mathematical notation written before TeX, is easily answered: it was written as such. It is still written as such when writing longhand. Highly trained compositors were able to take a mathematical manuscript and compose it (by hand) with metal types for printing. $\endgroup$ – AlexP Sep 26 '18 at 11:36
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    $\begingroup$ Most of our mathematical symbols like Summation, Integration (another type of summation), epsilon, derive from the Greek alphabet, which had nothing to do with Europeans. And if you read translations of Plato or Archimedes, you may find diagrams (which would be common and similar amongst all languages) and words, in many cases they didn't use any symbols at all, just complex verbal arguments (later translated to symbols). Even our word "sum" is from Latin "summa", meaning "top", as the Romans summed a list of numbers from bottom to top, writing the answer at the top of the list. $\endgroup$ – Amadeus-Reinstate-Monica Sep 26 '18 at 13:33
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Just Make It Up.

Most of our mathematical symbols like Summation, Integration (another type of summation), epsilon, derive from the Greek alphabet, which had nothing to do with Europeans. And if you read translations of Plato or Archimedes, you may find diagrams (which would be common and similar amongst all languages) and words, in many cases they didn't use any symbols at all, just complex verbal arguments (later translated to symbols). Even our word "sum" is from Latin "summa", meaning "top", as the Romans summed a list of numbers from bottom to top, writing the answer at the top of the list.

In fact, if you look up the etymology of the equals sign "=", you will find it only dates to 1557; and was explicitly devised to replace the words "is equalle to", two equal length parallel lines "as nothing could be more equal". The word equal is derived from Latin, "æqualis", meaning "identical". See Equals Sign.

Alternatives in wide use were "||", "æ" and "œ" (to imply the Latin word). The latter are very language specific, thus completely random if you don't know the language, completely random symbols.

The same goes for all the Greek letters, Arabic letters, Latin words and everything else. The Greek letter $\Sigma$ is used because "Sum", derived from a Latin word, begins with S.

Likewise, the Integral symbol $\int$ was chosen by Leibniz to look like an "S" because he thought of the integral as an infinite sum of infinitesimal parts. But, if his language or alphabet were different, we'd just have some other symbol the represented the first letter or sound of his word "sum". The letter "S" itself was supposedly chosen to look like a snake, an earth animal.

Our current notations are not at all "sensible" or "logical", they are just like a language, completely arbitrary symbols derived from sounds we humans can make or graphology we use, to which WE have attached meanings. Just like words, what we call a "bird" in English is wildly different sounds in Spanish, German, Swedish, Chinese, etc. The English sound is arbitrary and only means something to an English speaker because we are taught the association.

So just make it up. Even the ordering of equations is arbitrary, as the Romans proved by adding from the bottom to the top, and other languages prove by writing from right to left, or bottom to top. The symbols themselves are derived from a hodge-podge of languages, usually chosen to remind people of a word that describes their operation.

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  • $\begingroup$ Although the exact form of the signs are arbitrary, any logical system would require some sign (independent of form) to represent identity, or it couldn't be logical. $\endgroup$ – tbrookside Sep 26 '18 at 14:17
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    $\begingroup$ @tbrookside I don't argue that. You probably need most of our signs; but there could be others. For example, I don't think we have a special sign to indicate a prime, or unfactorability of an equation, etc. I can imagine many other symbols I haven't seen in mathematics that aliens might find useful, because even our history of mathematical investigation is somewhat arbitrary, predicated on various problems that happened to be popular at different times. $\endgroup$ – Amadeus-Reinstate-Monica Sep 26 '18 at 14:33
  • $\begingroup$ That's a good point. Symbolic systems will cluster around the most common used concepts that require symbolization. Totally agreed. $\endgroup$ – tbrookside Sep 26 '18 at 15:51
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    $\begingroup$ @tbrookside As well as our less-than-ideal positional notations, like superscripts to indicate exponentiation (like $x^2$, but sometimes superscript is used for other purposes), and subscripts to indicate numbered versions ($X_2$), or putting one number on top of another like $\frac{1}{2}$, or indicating a different but related variable with bars, dots, or hats like $\hat{x}$. Aliens might do that kind of thing with colors, or size, or width of the character, or extra symbols on the left or right of the character. Weird stuff we don't do. $\endgroup$ – Amadeus-Reinstate-Monica Sep 26 '18 at 16:14
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Ancient cultures would write their math the way we wrote our math before computers: by hand. I remember seeing academic articles that were typed on a typewriter, but formulas were drawn by hand.

Understanding their math notation will be hard, but looking at plots should help (e.g. Pythagoras theorem or definition of derivative), or finding a math textbook.

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Just to offer one of the stranger answers, consider the Quipu. These were knotted strings which could represent numbers. This kind of representation was very important for collecting taxes.

It would be trivial to add new knots to their repertoire. Indeed, they have a unique advantage that the knots themselves provide structure to the equation so there's no need for orders of operations, or subscripts or exponents. Everything is just "follow the ropes."

Example of Quipu from http://tuteja.info/inequalities-quipu/

As an added bonus, integration, being related to repeated summation might happen to look like a Rodin coil. That ought to keep the conspiracy theorists in your world bubbling with excitement for decades!

Rodin Coil

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