# What's the bloody deal?

1. May 30, 2015

### Xavier692912

Why do physicists continually use the term "per" when they are multiplying units? Division is the appropriate operation when the term "per" is encountered. For example, Joules per kilogram per Kelvin. Physicists denote this as J/kg*K. Why do so many physics textbooks write "per" when kilograms are being multiplied with Kelvins? There are other examples, such as N*m^2 / kg^2. Here N*m^2 is recited as "newton-meters squared", not "newtons per meter squared". So, shouldn't the units in the initial example also be read as "kilogram-Kelvins"?

2. May 30, 2015

### moriheru

1) Live with it
2) Joules per kilogram per Kelvin with per being the division, one would end up with J*K/kg.

3. May 30, 2015

### Staff: Mentor

This is the way that division by a fraction works. If you divide by a fraction, it is the same as multiplying by the reciprocal of the fraction.
$$\frac{a}{\frac b c} = a \cdot \frac c b = \frac {ac}{b}$$
So $$\frac{\text{Joules}}{\frac {\text{Kg}} {\text{deg K}}} = \frac{\text{Joules }\text{ deg K}}{\text{Kg}}$$

4. May 30, 2015

### symbolipoint

Not even fully reading the rest of the posting;
"per" is used both in the human languages and in the mathematical numeric form as a language. It is used because it is necessary for communication and understanding.

5. May 31, 2015

### certainly

Ummmm...... I don't see anything wrong with what the OP said.
You're quite right, whoever said "Joule per kilogram per Kelvin" is saying it wrong.

6. May 31, 2015

### MrAnchovy

There are some important lessons here.
1. Nobody likes a smartass.
2. Conventions are useful, but they are essentialy arbitrary - when something can be expressed in two different ways the convention says which is to be used, not which is "correct".
3. Conventions are useful. but they are often counter-intuitive in certain contexts - for instance when read aloud "joules per kilogram per kelvin" unambiguously implies to many listeners $\frac{\frac{J}{kg}}{K} = \frac J{kg K}$ which is correct, whereas "joules per kilogram kelvin" implies $\frac J{kg} K$, requiring the arbitrary insertion of brackets or a hyphen to imply "joules per (kilogram-kelvin)".
4. Conventions are useful, but they are not immutable.
5. Use of a convention shows only that you know the convention, it says nothing about your understanding of the underlying concept. Heat capacity is measured as "joules per kelvin", and so intuitively specific heat capacity is "(joules per kelvin) per kilogram" - it is only by (current) convention that we restate this as "joules per (kilogram-kelvin)". Similarly the rate of change of a quantity is measured in units "per second", and so intuitively the rate of change of velocity is "(metres per second) per second", and in understanding this you understand acceleration. Knowing that the conventional unit of acceleration is "metres per (second squared)" does not help - indeed the misunderstanding that this convention causes leads many people to drop Physics at an elementary level.
6. Nobody likes a smartass.

7. May 31, 2015

### HallsofIvy

Staff Emeritus
I have never heard or seen a physicist say "per" for multiplication. If you have seen that in a text or paper, please cite and quote it.

8. May 31, 2015

### certainly

lol.
I think what's happening here, is a misinterpretation. Even I'm confused now.
Some people (including me) are thinking Joule per kilogram per Kelvin is $\frac{J}{\frac{kg}{K}}$ while others are thinking it's $\frac{\frac{J}{kg}}{K}$ (which would represent what it's supposed to, I guess.)
Anyway, what exactly is this per thing ?
[googling both Joule per kilogram-Kelvin and Joule per kilogram per Kelvin give the same results, which one is correct ?]

9. May 31, 2015

### Staff: Mentor

We also say "meters per second per second" for acceleration. By convention, such constructions implicitly group the first two units together as if there were parentheses: "(meters per second) per second" or "(joules per kilogram) per kelvin", which agrees with certainly's second formulation.

If it makes you feel better, you're free to say something like "left-parenthesis joules per kilogram right-parenthesis per kelvin", but don't be surprised if some people look at you a bit oddly.

Or insert a little pause when speaking it: "meters per second, per second."

All the programming languages that I've used (Fortran, Pascal, C, C++, Perl) handle division this way: a/b/c is evaluated the same way as (a/b)/c. Computer-science people call this "left-associativity."

Last edited: May 31, 2015
10. May 31, 2015

### MrAnchovy

Neither is more correct than the other, that is the whole point - they just follow different conventions.