Precedence of multiplication and division in expressions such as $1/j\omega C$ and $hc/\lambda k T$

Question

One of the first things I learned at math at secondury school is the order of operations:

1. Things between brackets
2. Multiplication and division
3. Addition and substraction

(At that time exponents and roots wheren't introduced.)

Now at university, I've seen formulas like $$Z_C = 1/j\omega C$$ (for the impedance of a capacitor in an electric circuit) which is also written as $$Z_C = \frac{1}{j\omega C}$$

But since both multiplication and division has the same rank on the order of operation, I would interpreted (if I wouldn't know better) the first formula as $$Z_C = 1/j\omega C = \frac{1}{j} \omega C = \frac{\omega C}{j}$$ (which could also be written as $-j\omega C$ since $j$ is the imaginary unit, but that's beside the point).

This is just one example, another would be $$R(\lambda, T) = \frac{2\pi h c^2}{\lambda^5 (e^{hc/\lambda kT} - 1)}$$ for Plank blackbody law, where it is meant that $hc/\lambda kT = \frac{hc}{\lambda k t}$

My question is, does division has higher priority than multiplication in the order of operation? I know my question sounds really basic, but I'm never told the answer.

Answer 1

One thing we always have to remember about order of operations is that it's not an arbitrary set of rules that some random people decided to demand of others. It's a description of what people actually do in practice—habits that writers of math have agreed on over the years, habits that make writing and reading math easier.

People have found that in practice, there's no lack of clarity in expressions like $1/abc$, which I would unambiguously read as $\frac1{abc}$. The reason (in my opinion) is that anybody who intended $1/a \times bc$ would never ever write $1/abc$ when there are so many other easy ways to write it ($bc/a$ being the most straightforward). The fact that we parse $1/abc$ as $\frac1{abc}$ is a consequence of this asymmetry (one form has an easy other way of writing it without using parentheses, the other doesn't) and how that affects people's writing in practice, not a calculated decision by some body of standards.

Learning "order of operations" is a way of trying to help people understand what is done in practice. It's not a law that dictates what must be done—it's an attempt to describe what is done.

Answer 2

Order of operations in a FRACTION always prioritizes numerator and denominator operations FIRST.

Thus, in $\frac{1}{jωC}$ the multiplicative operations of $j*w*C$ will be prioritized. Factoring out $(\frac1j)wC$ will simply lead to an incorrect answer.

Generalizing:

For any number represented in form $\frac xy$, where $x$ and $y$ are placeholders for any number of different mathematical operations:

you must complete all operations on the numerator $x$ and the denominator $y$ FIRST.

Answer 3

No, division and multiplication have equal precedence, but note that Terms are separated by operators and joined by grouping symbols, thus jωC is a single term. More precisely it's a product, which is THE RESULT of a multiplication. "Multiplication" in order of operations refers literally to multiplication signs, not Terms - if a=2 and b=3 then 1/ab=1/6 and 1/axb=3/2 - so there's no "multiplication" in that formula, only division (by a Term/product).

Here's a screenshot from Cajori (1928) saying that bc and (bc) are the same thing (and is the same usage in all modern textbooks, as you've discovered)