Why are symbols not written in words?

We could have written = as "equals", + as "plus", $\exists$ as "thereExists" and so on. Supplemented with some brackets everything would be just as precise.

$$\exists x,y,z,n \in \mathbb{N}: n>2 \land x^n+y^n=z^n$$

could equally be written as:

ThereExists x,y,z,n from theNaturalNumbers suchThat 
     n isGreaterThan 2 and x toThePower n plus y toThePower n equals z toThePower n

What is the reason that we write these words as symbols (almost like a Chinese word system?)

Is it for brevity? Clarity? Can our visual system process it better?

Because not only do we have to learn the symbols, in order to understand it we have to say the real meaning in our heads.

If algebra and logic had been invented in Japan or China, might the symbols actually have just been the words themselves?

It almost seems like for each symbol there should be an equivalent word-phrase that it corresponds to that is accepted.

"Integral From a x squared Plus b Plus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Plus One RightParenthesis Differential x IsGreater Integral From a x squared Plus b Minus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Minus One RightParenthesis Differential x Implies Integral From a x squared Minus b Plus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Plus One RightParenthesis Differential x IsGreater Integral From a x squared Plus b Minus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Minus One RightParenthesis Differential x" ?

Quiz:

Do you recognize this one ?

Summation On j From 1 Upto N Of y Index k Times the Product On k From 1 And k NotEqual to j Upto N Of x Minus x Index k Over x Index j Minus x Index k.

Consider this problem, taken from The Evolution of Algebra in Science, vol. 18, no. 452 (Oct 2, 1891) pp. 183-187 (taken from JSTOR, itself translated from work of Nesselman on a problem by Mohammed ibn Musa):

A square and ten of its roots are equal to nine and thirty units, that is, if you add ten roots to one square, the sum is equal to nine and thirty. The solution is as follows: halve the number of roots, that is, in this case, five; then multiply this by itself, and the result is five and twenty. Add this to the nine and thirty, which gives four and sixty; take the square root, or eight, and subtract from it half the number of roots, namely five, and there remains three: this is the root of the square which was required and square itself is nine.

This is how algebra used to be done; you have similar descriptions in Babylonian scribe tablets, Egyptian papyrii, Middle Age textbooks, etc.

Using symbols, the problem becomes, first, to solve $x^2+10x = 39$. The process is to complete the square: $$\begin{align*} x^2 + 10x &= 39\\ x^2 + 10x + 25 &= 64\\ (x+5)^2 &= 64\\ x+5 &= 8\\ x &= 3 \end{align*}$$ Something that is much easier to do without too much thought, and certainly much less effort, than the decription. Also, the idea of completing the square is much simpler to explain in symbols than it is to do so rhetorically.

Others have already answered on why one should use symbols. I want to add that one shouldn't overuse symbols, as people sometimes do.

With too many symbols, statements get clustered and confusing. Out of lazyness, many like to write $\exists$ quantors in the middle of a sentence. Others overuse e.g. $\land$, etc. ($\land$ is not a synonym for "and"!)

So basically, one shouldn't overuse symbols.

Quoting Robert Recorde, inventor of the equals sign:

"And to avoid the tedious repetition of these words: is equal to: I will set as I do often in work use, a pair of parallels, or Gemowe lines of one length, thus: =, because no 2 things, can be more equal." (