[nofriend]

The hertz is formally defined as 1/s, except this leaves open the question of 1 what each second. I've seen it argued that since the numerator is unitless, and radians are also unitless, that the hertz as defined refers to one radian per second, and that it should have instead been defined as rev/s. While this argument might be specious, it suggests to us that even if our numerator is unitless, we should still be clear about what kind of thing we are describing rates of. So say "requests per second" if that is what you are talking about, and things will be clearer for everyone.

[tshaddox]

Why are you asking “what happened each second” for the SI unit hertz, but not asking “what happened for a second” for the SI unit second?

The hertz is dimensionally identified as 1/T, and the second is dimensionally identified as T. These are both equally clear, at least dimensionally.

And if anything, the hertz is actually more specified in SI than the second, because the hertz is specifically reserved for describing periodic events, while the second can be used for many things beyond the amount of time between consecutive periodic events.

[adrian_b]

The hertz is not dimensionally identified as 1/T, despite the fact that this has been voted in 1995 by the clueless delegates to the CGPM.

The hertz is a name for cycle per second and the physical quantity that is measured in hertz is plane angle per time a.k.a. phase angle per time, not reciprocal time.

[tshaddox]

According to your proposal, phase angle must not be dimensionless, right?

[adrian_b]

Phase angle is not dimensionless.

I have explained in another comment why this is so.

This misconception about angles being dimensionless quantities is unfortunately taught in many school textbooks and it cripples the thinking about physics of many people.

The foundation of physics is the theory of the measurement of physical quantities. However, when teaching physics frequently this is taught either badly or not at all, instead of following classical examples, like that of James Clerk Maxwell, who started his Treatise on Electricity and Magnetism with an exposition of the theory of the measurement of physical quantities that was complete and up-to-date for that time.

The plane angle a.k.a. phase angle not only is not a dimensionless quantity, but its unit of measurement plays a crucial role in determining the other base units of any modern system of units of measurement.

The reason why the unit of plane angle is so important is that plane angle is the only fundamental continuous quantity for which it is possible to approximate measurement by counting, because the numeric value of a plane angle measured in cycles can be decomposed in a sum of a discrete quantity, the integer part of the numeric value, with a continuous quantity, the fractional part of the numeric value. The value of the discrete quantity that is the integer part of the numeric value of an angle measured in cycles can be obtained by counting.

This unique property is the cause that for the highest possible precision in measurements all continuous quantities are converted by various methods into phase angles, before the analog-to-digital conversion.

The fact that plane angle has a natural unit, which is the cycle (or its integer multiples or submultiples) is exploited in defining almost all units of other continuous quantities.

For instance, the unit of length is said to be the wavelength of a certain wave. Because wavelength is a physical quantity equal to the ratio between length and plane angle, the previous definition stated correctly says that the unit of length is equal to the unit of plane angle multiplied with a certain wavelength (i.e. 1 meter = 1 cycle multiplied by a wavelength measured in meter per cycle). Similarly for many other continuous quantities, whose units are also derived in one form or another from the unit of plane angle, because only it can be defined intrinsically, instead of being derived from other units.

[smilekzs]

> that the hertz as defined refers to one radian per second, and that it should have instead been defined as rev/s

This is precisely what leads to the "forgot to multiply 2pi or 1/(2pi)" problem in the EE / signal processing domain where you FFT and s-/fourier-transform back and forth. Heck, I can never remember which convention any particular library / package decides to adopt (esp. mathematica vs. matlab which IIRC caused tons of confusion and headache during my undergrad).

[nvader]

So something that is entirely revolving once a second is spinning at 2 * pi Hertz?

Hertz just thinking about it.

[Sesse__]

https://en.wikipedia.org/wiki/Angular_frequency

It's very convenient when you're working with periodic signals (say, when discussing the FFT); you don't have to drag a 2pi factor around everywhere.

[rusk]

> 1 what

Cycles. I suppose you might say it’s a derivative.

How many times per second the measurement returns to the original value.

In engineering school we used to tie this directly to radians using the Euler notation pow(e, j * 2 * pi * f) where 2pif is your angular frequency expressed in radians per second!

[adrian_b]

The modern SI contains several serious mistakes, and the fact that they are the result of votes demonstrates that democracy is inapplicable in sciences like mathematics and physics, because the majority of the people are incompetent enough to vote things equivalent with saying that "2 + 2 = 5", but regardless if such things are voted by a majority of humans, they remain false.

The hertz is not "1/s" and everyone who has been brainwashed by school education to believe this misses some of the most important concepts on which physics is based.

Originally, "hertz" has been defined as a name for "cycle/s", not for "1/s", and that is the correct definition, which has always been the one used in practice, regardless of what is written in the SI brochure.

"cycle" is a unit for plane angle a.k.a. phase angle, so "hertz" is the SI unit for the physical quantity "angle/time", not for the quantity "reciprocal time". SI is an inconsistent system of units, because it uses 2 units for plane angles: cycles and radians. In practice, the right choice is to always measure angle in cycles, not in radians, because using cycles ensures both a higher accuracy in computations and faster computations (because the argument range reductions in trigonometric functions become exact and fast) and because all high-precision sensors and actuators must use cycles not radians, so when radians are used in computations that requires additional conversions.

The current wrong definition of the hertz is a consequence of an outrageous and shameful resolution voted in 1995: "Resolution 8 of the 20th CGPM", where it was voted that the units of plane angle and solid angle are not base units a.k.a. fundamental units, so these quantities should not be used in dimensional formulae.

This resolution is just an example of human stupidity. You cannot establish by vote whether a unit of measurement is fundamental or derived. Any unit of measurement that cannot be derived from other units is a base unit a.k.a. fundamental unit.

There are 3 fundamental units that are missing from SI (the units of logarithms, plane angles and solid angles), despite the fact that their use is extremely frequent in practice, and one of them, the unit of plane angle, is actually the most important fundamental unit, because for the highest precision in measurements all other continuous physical quantities are eventually converted into phase angles before the analog-to-digital conversion (because when phase angles are measured in cycles, the measurement can be reduced to counting, i.e. to cycle counting).

In theory, one could choose as fundamental only one of the 3 units of logarithms, plane angles and solid angles, and derive the others from it. The unit of solid angle can be derived from the unit of plane angle by setting to "1" the proportionality factor in Albert Girard’s theorem (by using this method, the steradian is derived from the radian, the hemisphere is derived from the cycle and the solid angle degree is derived from the sexagesimal degree, where a full sphere has 720 solid angle degrees; during the first 2 centuries after the beginning of the 17th cetury, when it was defined how to measure solid angles, the solid angle degree was the main unit of measurement). The unit of logarithms can be derived from the unit of plane angle by setting to "1" the ratio between the proportionality factors that appear in the formulae for the derivatives of exponential and trigonometric functions.

Choosing only 1 of these 3 units as base unit and deriving the other 2 from it results in the 3 units neper, radian and steradian. However, neper and radian are very inconvenient units, so in practice it is far better to choose independently the units of logarithms, plane angles and solid angles.

Regardless whether one chooses independent units for logarithms and solid angles, or they are derived from the unit of plane angle, it is impossible to derive the unit of plane angle from the currently official units of SI.

Some publications claim that the plane angle unit can be derived from the unit of length, supposedly because the measure of an angle is the ratio between the length of a corresponding arc and the radius of the circle.

This pseudo-argument demonstrates a complete lack of understanding of how physical quantities are measured. There are 2 variants of this pseudo-argument, which contradict each other and both are equally false. First, saying that this shows that the unit of angle is derived from the unit of length is obviously false. The reason is that when you divide two lengths, the unit of length is divided by itself, so the result of the division is independent of the unit of length, therefore this formula cannot derive anything from the unit of length.

The second formulation of the pseudo-argument is that this formula shows that the plane angle is an "adimensional quantity" a.k.a. "dimensionless quantity". This claim is equally ridiculous. As understood very well more than a century and a half ago (e.g. the theory of measurements was explained more clearly by James Clerk Maxwell than by most modern handbooks of physics), the measurement of any physical quantity is expressed by the product of 2 factors, a unit of measurement and a dimensionless numeric value.

The ratio between the length of an arc and the radius is not the complete value of a plane angle. It is just the dimensionless numeric value of an angle, in the special case when the angle is measured in radians. To obtain the complete angle value you must multiply that ratio by 1 radian. The "radius" in that formula is not truly the radius, but it is the length of the arc corresponding to the unit angle. When angles are measured in cycles, the length of a unit angle is the perimeter, so the dimensionless numeric value of an angle is equal to the ratio between the length of the arc and the perimeter.

Applying the pseudo-argument that plane angle is dimensionless to length "proves" that length is dimensionless, because the length of any object is obtained by dividing its length by the length of an 1 m ruler. Similarly for any other physical quantity.

The fact that the unit of plane angle is fundamental is demonstrated beyond any reasonable doubt by the fact that you can choose any angle unit you want, and many such units are really used in practice, e.g. cycle, radian, sexagesimal degree, centesimal degree, right angle, etc., and for each choice of a unit of plane angle you obtain a different system of units for the physical quantities. Between the numeric values measured in the different system of units there are conversion formulae that are constructed exactly in the same way as when changing for example the unit of length from meter to inch.

In order to be able to generate automatically the conversion formulae when the unit of plane angle is changed, you need to have the plane angle in the dimensional formulae of the physical quantities. The fact that most physics books usually omit the plane angle from dimensional formulae is a source of mistakes, especially when comparing some modern numeric values with values from old publications, where non-SI systems of units were used, which sometimes were based on implicit assumptions that were different from the implicit assumptions used by SI.

In SI, implicit assumptions are made everywhere. In most cases it is assumed that angles are measured in radians, and when other angle units are used the displayed formulae become wrong, despite the fact that it is claimed that they are written in a form that does not depend on which units are used. Nevertheless, wherever "hertz" is involved, it is implicitly assumed that plane angles are measured in cycles, not in radians.

The term "frequency" has 2 meanings in physics. Originally, "frequency" was the name for the ratio between the number of some random events that occur during some time interval and the duration of that time interval.

At earlier authors, before the last years of the 19th century, "frequency" was used only with this meaning. It was never used for periodic phenomena. Periodic phenomena were described using period and wave-length.

Some time around 1890, "frequency" began to be used with a second meaning, as the ratio between the number of cycles of a periodic phenomena and the duration of a time interval. After that time it has become more common to describe periodic phenomena using frequency and wave-number (i.e. number of waves), instead of using period and wave-length.

The two kinds of frequencies are distinct physical quantities, with distinct units of measurement. Frequency with the second meaning is the ratio between plane angle and time. Possible units are radian per second and cycle per second, where the latter is named hertz. Frequency with the first meaning is the ratio between a number of events (which is a discrete quantity, not a continuous quantity, like for the other kind of frequency) and time. SI has the name becquerel for this unit of measurement.

The author of the parent article is perfectly right that the becquerel is the appropriate unit of measurement for the frequency a.k.a. rate of any random events.

When you see "frequency" in a physics text, you must always pay attention to recognize which of its 2 meanings is used. Similarly, "phase" has 2 meanings, which must be distinguished. Originally, "phase" was the fractional part of a rotation angle measured in cycles. This meaning was more useful, but later "phase" began to be used for the integral of frequency with the second meaning, i.e. for the total rotation angle, not only for its fractional part.

[pants2]

Is a "count" not a valid unit?

[orangesilk]

"count" is an incomplete type, the same as "array" is an incomplete type. "count of fruit" would be a complete type, compare to "Array of int64".

SI leaves this underspecified, which causes confusion with dimensionless units.

[nlitened]

> "count" is an incomplete type, the same as "array" is an incomplete type. "count of fruit" would be a complete type, compare to "Array of int64".

Javascript has arrays of unspecified types, and count of items of such arrays. I don’t think your analogy holds

[pants2]

Perhaps the SI system needs generics :)

[hasley]

Yeah, that is really something that bothers me from time to time. For instance, in communications we count symbols per second and people still use the unit Hz. But, as it was said before, Hz is revelations per second.

There is a problem in general: Units do not refer to what they count. For instance, when you have dashed road markings and you want to quantify the length of the dashes per road length you get as unit "meters per meters". And then people go and say this is unitless since the units "cancel". But for me, they do not: It stays "dash meter per road meter".