In Ohm's Law, why do we not take the constant of proportionality as $\dfrac{1}{R}$?

Question

This is the standard derivation for Ohm's Law :

$$I \propto V$$ $$V \propto I$$ $$\therefore V = k.I$$

Where $k$ is the constant of proportionality.

We define this $k$ as resistance of $R$.

Why can't we derive it like this: $$I \propto V$$ $$\therefore I = k_2V$$

Where $k_2$ is the constant of proportionality

So, in this case : $$k_2=\dfrac{1}{R}$$

Why was resistance not defined as how $k_2$ is defined and why was it defined as it is?

I actually think that this might be the reason :

Most of the time, current(($I$) is less than the voltage($V$)

Let's say that resistance was defined as $k_2$, in that case, $$k_2=\dfrac{I}{V}$$ if $I$ is less than $V$ (like it generally is), then the value of $k_2$ will come out to be a fraction which is not too convenient. So, if we define the constant of proportionality as $R$, then $$R=\dfrac{V}{I}$$ and since $I$ is less than $V$ in most cases, we would get $R > 0$ unlike the case with $k_2$ in which we would get $0 <k_2 < 1$

Is this reasonable?

Answer 1

What you are talking about is conductance. You can use that instead if you want to.

I actually think that this might be the reason : Most of the time, current(I) is less than the voltage(V)

This isn't correct. You can't compare values that are physically different. Currents can't be more or less than voltages.

...then the value of $k_2$ will come out to be a fraction which is not too convenient

I'm not sure where this is coming from. Fractions are totally fine. Beginners in algebra might hate fractions, but they aren't inconvenient in their actual application. Plus, many numbers that are fractions have reciprocals that are also fractions. e.g. $3/2$ and $2/3$.

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Is this reasonable?

To summarize, it's totally fine to use $1/R$ which is the conductance. However, your arguments as to why one would want to do this either aren't valid, or are nonissues.