Is the percentage symbol a constant?

1. Isn't the percentage symbol actually just a constant with the value $0.01$? As in $$ 15\% = 15 \times \% = 15 \times 0.01 = 0.15. $$

2. Isn't every unit actually just a constant? But why do we treat them in such a special way then?

Solution 1:

Isn't the percentage symbol actually just a constant with the value $0.01$?

No. If it were, all of the following would be valid constructs:

$$ 30+\%\,50=30.5\\ 90\,\%\,\mathrm{cm}=0.9\,\mathrm{cm}\\ 2-\%=1.99\\ \%^2=0.0001 $$

The percentage symbol is a unit. When converting between units, it's easy to treat them as constants that represent the conversion ratio, and multiply (for example, the $\mathrm{m}$ unit can be thought of as a constant equal to $100\,\mathrm{cm}$, in $2\,\mathrm{m}=2(100\,\mathrm{cm})=200\,\mathrm{cm}$). But that isn't the same as saying they're "just constants", as they represent more than that. A unit is not just a ratio, it's a distance or a weight or an amount of time.

This is less obvious with $\%$ because it's a dimensionless unit, representing something more abstract like "parts of a whole" rather than a physical property like mass or surface area. $1\,\%$ is "one one-hundredth of a thing", measuring an amount of something, anything, often something with its own units. A similarly dimensionless unit is the "degree", where $1^\circ$ is "one three-hundred-sixtieth of the way around". Another one is the "cycle", as in "one $\mathrm{Mhz}$ is one million cycles per second". Things like "wholes", "turns", and "cycles" are more abstract than inches or grams, but when applied they still represent tangible measurements, so they aren't any less powerful when treated as units.

I mean, I guess every unit is actually just a constant, but why do we treat them in such a special way then?

What then would you say the "constant" is that is represented by "inch", or "second", or "ounce"? Would these ideas not have clear numeric values if every unit were simply a constant?

Again, a unit is not just a constant, it represents something more. I don't have exact vocabulary for this, but I would say a unit is an "amount" of a "dimension". The dimension can be time, space, energy, mass, etc. To even begin to treat a unit as a constant, we need to consider it in terms of a different unit in the same dimension. For example, the unit "millisecond" amounts to different constants depending on whether we think about it in terms of a second ($0.001$), hour ($2.77778\times10^{-7}$), microsecond ($1000$), etc. This constant is not intrinsic to the units themselves, as it only arises when comparing to other units.

Solution 2:

Yes, for calculations you can use $\%=\frac{1}{100}$. Of course what is meant by the symbol is an interpretation as "parts of hundred", i.e. as percentage of a given amount.