Fractions With Units

[KWF]

It is common to have the numerator and denominator with units of fractions as seen with rates -- for example 200 miles/2 hours.

Can just the numerator of a fraction have units and not its denominator, and can just the denominator of a fraction have units and not its numerator?

Please provide examples, if possible.

 

[pka]

\(\large\dfrac{2\text{ft}}{4}=\dfrac{1}{2}\text{ft}\)

 

[skeeter]

angular speed example ...

[MATH]\omega = \pi \, s^{-1}[/MATH]

 

[Dr.Peterson]

Similarly, the unit "hertz" is defined as s^-1 (that is, 1/second). In the old days this was expressed as "cycles per second", where "cycles" is just a dimensionless count, like the 4 in pka's example.

 

[HallsofIvy]

If you have a 12 foot length of rope and want to divide it into 4 equal pieces each piece has length \(\displaystyle \frac{12 feet}{4}= 3 feet\).

(Okay, you could argue that it is \(\displaystyle \frac{13 feet}{4 pieces}= 3 \frac{feet}{piece}\) but I don't think most people would do it that way.)

 

[KWF]

So it appears the the numerator will have the unit of measure whenever it is divided by a non-unit-of-measure number. These appear to be common. However, the opposite situation is not so common when the denominator only has the unit of measure and the numerator does not.

 

[topsquark]

So it appears the the numerator will have the unit of measure whenever it is divided by a non-unit-of-measure number. These appear to be common. However, the opposite situation is not so common when the denominator only has the unit of measure and the numerator does not.

There are a few of them in Physics. Hz is one, mho (measures conductivity, and yes that is 1/Ohm) and wave number are two more. Mostly it is the numerator that will have the unit but there are some out there. It's the way that the unit system is set up... For example in the MKS system we start with length in m, mass in kg, and time in s. The other units are defined by some kind of multiplication or division of these terms.

-Dan