# Order of precedence for parallel operator

It looks like the parallel operator is surprisingly common, and there are precedence rules defined by *some* sources, but not all.

MIT in their courseware suggest using it, and set exam questions on it. They provide no information regarding precedence in their lecture notes. They say the following, with no description of precedence.

This mathematical relationship comes up often enough that it actually has a name: the “parallel operator“,denoted by the symbol \$\parallel\$. When we say \$x‖y\$, it means \$\frac{xy}{x+y}\$

The university of Utah use it to analyse circuits in their course notes, and provide precedence information as follows:

In the absence of parentheses, the parallel operators act before the plus signs, just as multiplies act before the plus signs in a calculator expression. In the parlance of computer science, "the parallel operator binds more tightly than the plus operator."

Finally, this book also defines the operator, and provides precedence rules, in the same way when referring to the equation $$I = \frac{V_{in}}{R_1+R_2 \parallel R_3},$$

the parallel operator has precedence over the addition: \$R_2 \parallel R_3\$ is computed first and then added to \$R_1\$

Note that neither of the references that define precedence of the \$\parallel\$ operator with respect to addition do so with respect to multiplication.

As I have rarely come across this operator, I would make no assumptions regarding precedence, and use brackets to clear up any ambiguity. E.g.: $$I = \frac{V_{in}}{R_1+(R_2 \parallel R_3)}$$

There is no question of operator precedence here. \$R2\$ **cannot** be in series with \$kR1\$ and also in parallel with \$R3\$. There is some information missing from the equation and the only way to clarify the problem is to look at the schematic.

To start: \$R_1||R_2 = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} = \frac{R_1 R_2}{R_1 + R_2}\$

Then: \$(kR_1)||R_2 = \frac{1}{\frac{1}{k R_1} + \frac{1}{R_2}} = \frac{1}{\frac{kR_1 + R_2}{kR_1R_2}} = \frac{kR_1R_2}{kR_1 + R_2}\$

Which is **not** equal to: \$k(R_1||R_2) = k\frac{R_1R_2}{R_1 + R_2}\$

So order definitely matters, but as mentioned in the comments \$||\$ is not really an operator per se. I'd recommend sticking with parentheses to indicate order of operations and avoid ambiguity.

Likewise with addition:

\$R_1 || (R_2 + R_3) = \frac{R_1(R_2 + R_3)}{R_1 + R_2 + R_3} \neq (R_1 || R_2) + R_3 = \frac{R_1R_2}{R_1 + R_2} + R_3\$