Does math have a convention for multiplying percentages like we do with other units?

Question

We know that $10 \text{ cm} \times 10 \text{ cm} = 100 \text{ cm}^2$ , where we can write $\text{ cm}^2$ which is a Square Unit.

Does it fit mathematical convention to write $\%^2$ with the Square Symbol like this?

$10\% \times 10\% = 100\%^2 = \frac{100}{10,000} = \frac{1}{100} = 1\%$.

If not, how does mathematics typically express this multiplication of percentages?
For instance, does it fit convention instead to convert immediately to a fraction/decimal?

$10\% \times 10\% = \frac{10}{100} \times \frac{10}{100} = \frac{100}{10,000} = \frac{1}{100} = 1\%$

Edit: In addition to meaning "as a fraction out of 100", some say that % is also a unit, so I wonder how much we can treat it like other units.

[ https://math.stackexchange.com/questions/3122554/is-the-percentage-symbol-a-constant ]

Answer 1

(A) Per Cent indicates "Per Every 100" & the Symbol $\unicode{37}$ always indicates a multiplication by $1/100$

Hence when "$2\unicode{37}$ of the Population has COVID" , then a City with $100,000$ Citizens will have $100,000 \times 2 \times 1/100 = 2,000$ COVID Cases.

(B) When we multiply "Per Cent" values eg "$5\unicode{37}$ of the COVID Cases require treatment" , the over-all multiplication factor is $(2/100)\times(5/100)=10/10000$.
Incidentally , that has a Symbol : $\unicode{8241}$ : that indicates a multiplication by $1/10,000$ , with the rarely used name "Per Myriad"
We can think about like this : $\unicode{37}\times\unicode{37}=\unicode{8241}$ : loosely & informally.
Hence , we could say "$10$ Per Myriad ($10\unicode{8241}$) of the over-all Population will require treatment"

What OP is asking :
In general , "Per Myriad == 1 Per Cent of 1 Per Cent"
$\unicode{8241} = \unicode{37} \times \unicode{37}$
We do not use $\unicode{37}^2$ , though we do multiply "Per Cent" values a little carefully.

(C) Between those two , we have $\unicode{8240}$ which is "per Mille" $1/1000$.

(D) Other than that , we have "PCM" == "Parts Per Cent Mille (Milli Per Cent)" & "PPM" == "Parts Per Million" & ETC

Here is a simple way to see the terms , available at the wiki with more details :

More here :

https://en.wikipedia.org/wiki/Basis_point#Permyriad
https://en.wikipedia.org/wiki/Per_cent_mille
https://en.wiktionary.org/wiki/permyriad
https://www.compart.com/en/unicode/U+2030
https://www.compart.com/en/unicode/U+2031
https://www.rapidtables.com/math/number/Per_mille.html
https://percent.info/permille-to-percent/convert-1-per-mille-to-percent.html
https://glossary.ametsoc.org/wiki/Per_mille