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Comment by kazinator | Hacker News Reader

"When scientists measure any quantity they must specify the units being used."

No they don't. For instance, a simple aspect ratio measures how something is wide relative to how it is long. It has no units.

This here desk I have here is about 1.7 times as long as it is wide. No inches or centimeters required.

There isn't anything amazing or mystical about a unitless quantity.

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You've got the right intuition here. The unitless part is key for several reasons. It means that the number itself is meaningful. In your example, the ratio of 1.7 long/wide would be the same no matter what units used to make the measurement. (Obviously, the units DO have to actually cancel cm/inches doesn't work). The number for the speed of light (299 792 458 m/s) is a defined property, so that number doesn't mean anything deeply. However, 1/137... itself is directly meaningful. This is why people who work in fundamental constant research use dimensionless ratios.

Now for a piece that's more interesting. The fine-structure constant (alpha) is the coupling constant that sets the strength of electromagnetism. This means that its value is the thing that matters in the equation: e^2/\hbar c. Each of the other values is a derived quantity. Further, to speak a bit loosely, only changes in alpha matter -- in the sense that if the speed of light (c) changes, but the other constants (e and \hbar) change in a way that keeps alpha the same, then you wouldn't be able to tell with an experiment that anything has changed.

Contrast this situation w/ a change in alpha -- a table-top experiment would be able to detect the change (given that it's large enough, and we have methods of measuring changes on year-time scales that are a few parts in ~10^-18 (Rosenband, 2008)), as it would mean that physics has changed in a fundamental way.

http://phys.columbia.edu/~millis/1900/readings/Science-2008-...

Uh, pardon the stupid question, but isn't it safer to consider the fine structure constant to be the IR fixed point of \alpha (i.e. \alpha(\mu = m_e)^-1 = 137...)?

\alpha(\mu = M_Z) is about 128. But m_0(photon), h, and e are the same at both energy scales (we can determine the numerical values experimentally). To the best of our ability to measure, none of them (including, if we ignore the main posting, \alpha_em) varies anywhere in a background fixed by the isometry group of SR (where we find c as the sole free parameter, corresponding to the speed of a particle where m_0(particle) = 0).

So I'm having trouble understanding your assertion that "Alpha is the fundamental physical constant: c, e and \hbar are the derived quantities" (in your other comment, but also reflected in your second paragraph above).

I think we both totally agree that varying \alpha_em at different points in space or time leads to a mess.

> This here desk I have here is about 1.7 times as long as it is wide. No inches or centimeters required.

But you are using units. Your unit is the width of your desk. If it were a different width but the same length, your measurement would be different.

So tell me, how large is my desk?

Note that the aspect ratio lets us say things like: "these two desks have the same aspect ratio".

The "desk widths" interpretation of aspect ratio precludes this useful sentence from having semantics.

The aspect ratio 1.7 has a life of its own, independently of that desk.

Let's say the width of your desk is denoted by "w". The length is then 1.7w. The aspect ratio is dimensionless because it is = 1.7w/1w. w/w = dimensionless. But you still used a unit, w, to make the measurement in the first place.

He could have measured the angle of the desk's diagonal makes with the longer side, that would give him tan(w/l), no inches or centimetres required.

You do need radians, but that's not a unit in the same way a 'meter' is, you don't need any kind of reference to know how much, say, 1 pi radians is. For most intents and purposes angles are unitless.

That is inescapable for any quantity whatsoever that we extract from nature. In dimensionless quantities, the units just canceled out. Amperes divided by amperes, or whatever.

See root comment. I do not have to specify the units that I used when I quote the aspect ratio, or the method that I used.

So, if he measured the width and height again, using different units, would he get a different answer?

If you measure something which has units using different units, then you get a different answer. If you measure something which doesn't, like aspect ratio, then you don't.

The match between this rhetorical question and your username is awesome.

I think a lot of the confusion in this discussion is coming from the specific example. The fine structure constant, an aspect ratio, or something like Reynold's number for quantifying fluid flow turbulence -- these are all dimensionless numbers that scientists use, sure and there's probably nothing special about them.

I think the confusion is in actually measuring these things. Scientists use dimensionless numbers all the time, but you have to directly measure quantities with units to indirectly get the dimensionless quantity.

Isn't that still a "unit"? (Your desk is 1.7 "desk widths" long?)

Scientific units are:

1. Standardised. 2. Defined by a numerical relationship to observable physical constants, like c, by definition.

Desk widths aren't a scientific unit. You can only measure the ratio of width to height by reference to a standard unit like the metre, which in turn is based on a constant observable quantity - the distance travelled by light in one second.

You can pick your derived units using any relationship to c - like the distance travelled in 3.2 seconds. But that's still a derived unit, not a fundamental observable unit.

And ratios are dimensionless because they stay the same whatever derived unit system you choose, as long as it's consistent.

To make things more complicated: the size of your desk isn't fixed because the universe is expanding. So the units you are using are just a convention to relate the size of an object to the size of other objects.

Expressing the size of the desk in a coordinate system specifically chosen such that the size doesn't change is much more than a convention. Conventions are arbitrary, like whether to drive on the left or right side of a two-way road.

Also, I tend to move around while my desk stays still, so from my desks point of view I am not aging as fast.

So long as the fine structure constant remains the same, your desk (and ruler) won't change size. Rather they will heat slightly as the expanding universe stretches the chemical bonds and those same bonds spring back to average rest length.

/nitpick

The effects of air eroding the desk are much more significant than the expansion of the universe. And both are dwarfed by the fact that the texture of the surface keeps the exact length and width from being well-defined.

Indeed, the author conveys the wrong message when implying that only pure numbers have no units. Measurable quantities, as you point out, can also have no units.

Another point that throws me off is the Vulcan scientist part. They would need to be using the same numerical base in order to get that number. There might be some base-less representations of numbers, but they are usually restricted to representing integer numbers. The constant could still be represented by a fraction, but there would be no math-neutral way of saying "hey, this here is a fraction/division".

The same number in a different base is still the same number..

Ratios and Quantities are completely different things

Not according to any set of definitions that I have ever heard of. Can you provide an explanation of the difference as you understand it?

To give one example, my piece of US Letter paper is 93.5 in^2 in area. One side is 8.5 in long. If I divide these, I find that 93.5 in^2 / 8.5 in = 11 in. Is that eleven inches a ratio or a quantity?

It's a quantity. However, the aspect ratio of US Letter paper, 8.5/11 is a ratio. If you used a copier to reduce a US Letter page by 50%, it would still have the same aspect ratio.

> This here desk I have here is about 1.7 times as long as it is wide. No inches or centimeters required.

How did you measure your desk in order to compute its length/width ratio?

I fail to see where you're trying to get. c, e and h all have units and are measurable.

The quote is "When scientists measure any quantity they must specify the units being used."

I am not convinced they determined the ratio of his table's length and width without first measuring those quantities in some unit.

A measurement doesn't have to be direct to be a measurement. In other words, the aspect ratio of a desk is still obtained by measurement and thus is a measured quantity, subject to error and so on.

Of course, both the quantity and the associated error come from previous measurements.

i'm just wondering if there's an example of a direct measurement of a ratio, rather than one computed from the measurement of dimensional quantities.

Well, you've got to measure some quantity. How about an angle? Measure any angle in a right triangle and you know something about the ratio of the sides.

At first i thought "aha! yes, how obvious.", but then i thought about how one might measure an angle. i would use a protractor. but how is a protractor calibrated? surely the angles determine a relationship between commensurable quantities, and a protractor is calibrated by performing measurements on lengths held at angles and measuring the observed relationship. can one measure an angle directly without performing an experiment on commensurate lengths? a protractor only works because a measurement of an angle can be transported and remain the same (to an approximation). that's a nontrivial fact that i took for granted, but that's really what a protractor does: it is the stored result of a bunch of experiments to determine angles. those experiments must have been done using observations of dimensional quantities.

i would like an example of an experiment that directly measures a dimensionless quantity without inferring it from direct measurements of dimensional quantities. it's just a curiosity.

However, we can use angles in such a way that we never have any estimate that correlates with the size of the rectangle whose aspect ratio we are measuring.

Suppose some rectangle is visible at a distance. We can use parallax angles from our vantage point to estimate its aspect ratio.

We have no idea how long or wide that rectangle is because we don't know how far away it is.

(Indeed, we don't even know whether it is a rectangle, or whether it is some quadrilateral that appears as a rectangle under perspective.)

I think you're saying something a little too strong. I think such a measurement does provide some information about the dimensions of the quadrilateral being measured. I think it would be possible with a bounded number of observations to measure the lengths purely by measuring the angles. You would have to shift your perspective each time, but it shouldn't take too many. So, each such measurement must contribute a non-zero amount of information about the dimensions.

Either that, or i'm misinterpreting your first sentence.

Imagine a balance with an arm that can be somehow folded in half again and again (think a snap bracelet. Those are rigid in one direction and stiff bent in the other). Fold the beam in half and mark the center. Use this point as the fulcrum to balance two objects. Pick the side that comes up. Divide this side in half and use this new half way point as the new fulcrum. starting with 0. if the beam reads on the heavy side, write down a 1. If reads on the light side, write a 0. Eventually you'll have 0.101000101110101 or some such. This is the ratio of the weights of the two objects in base 2. You can binary search the ratio of the weights.

Here's another. Imagine you want the ratio of two clock sources. Put them both through frequency dividers, with one divider output put through a phase shifter. compare the resultant divided frequencies by xoring the signals together. The ratio of the frequencies is given by the ratio of the dividers when you can find a phase shift such that the xor output doesn't change. The time you wait to see if the xor output doesn't change represents the certainty you have in your ratio. This is called the time frequency uncertainty relation, also known as the Heisenberg uncertainty principle.

Okay we have mass and time nailed down. I think the subdivision method should work for length as well. Let's try voltage.

Consider two voltage sources that you want to measure the ratio of. Place one source on one arm of a resistor, and the other through a pulse width modulator, (ideal) low pass filter with a cutoff frequency much lower than the pulse width modulator frequency, and then attach this to the other lead of the resistor. Turn on the voltage sources and adjust the pulse width modulator until the resistor doesn't heat up. You could place it in a cup of water and monitor the temperature. Remember we're free to push up the current in this thought experiment

Resistance should be easy as well. I think the wikipedia page for a Wheatstone bridge explains this best.

If this sort of thing interests you, I'd encourage you to look at the quantum metrology triangle. Frequency, resistance, and current are all currently related to a josephson voltage standard, the quantum hall resistance, and the current generated by a single electron transistor. This allows us to standardize all these back to a frequency. Length can be related to frequency by wavelength of light. Everything has been related back to frequency. Basically the only thing not referred back to something absolute is mass, which I think is still referred back to a mass standard, which is a physical thing stored in a vault.

Okay well more than you ever wanted to know.

All your posts are great! These are all really illuminating experiments.

I'm likewise not convinced that the fine structure constant and other constants of its ilk were obtained without taking measurements.

The units involved in the aspect ratio disappear because they are divided out. After that, we don't have to (in fact, must not) cite any units when giving the aspect ratio.

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