from Hacker News

The scientific “unit” we call the decibel

by Ariarule on 5/22/25, 4:24 AM with 491 comments

by lifthrasiir on 5/22/25, 7:41 AM
The only thing you should know is that any use of bel and thus decibel should ideally have the reference level suffixed (usually in parentheses or subscript), not implied. The absolute sound pressure level is dB(SPL). The human-perceived loudness level is dB(A) and similar. The RMS voltage expressed in power is dB(u) (formerly dB(v), not same as capital dB(V)). And so on. And then each different instance of dB unit is simply distinct, only connected by the fact that it represents some ratio in the logarithmic fashion. Treat any new dB unit you haven't seen as an alien.
by fouronnes3 on 5/22/25, 7:37 AM
When I worked on a radar project, my fellow radar engineers (I'm software) used dB a lot. A lot of them would actually agree with the article, but historical sometimes wins even when you're aware of its shortcomings. Aren't we the same in software anyway? The email protocol, terminal escape sequences, the UX of git command line, etc... Each of those could have an "X is ridiculous" blog post (and I would enjoy every single one).
One upside of dB not touched in the article is that it changes multiplication into addition. So you can do math of gains and attenuations in your head a bit more conveniently. Why this would be useful in the age of computers is confusing, but on some radio projects both gains and losses are actually enormous exponents when expressed linearly, so I sort of see why you would switch to logs (aka decibels). Kinda like how you switch to adding logs instead of multiplying a lot of small floats for numerical computing.
by severusdd on 5/22/25, 8:22 AM
While I thoroughly enjoyed reading this piece of internet-rant, I've to argue that dB is still probably the best we have on this!
In RF engineering, expressing signal levels in dBm or gains in dB means you can add values instead of multiplying, which definitely appeared like a huge convenience for my college assignments! A filter with -3 dB loss and an amplifier with +20 dB gain? Just add. You can also use this short notation to represent a variety of things, such as power, gain, attenuation, SPL, etc.
I guess, engineers don’t use dB because they’re masochists (though many of them surely are). They use it because in the messy world of signals, it works. And because nobody knows anything that might work better!
by svara on 5/22/25, 6:53 AM
A pet peeve I share! An expanded version of this article should become the article on decibels on Wikipedia.
I've read that article many times over my life and for the first couple times came back thinking I was too dim to understand.
Transparently leading it with "Here's something ridiculously overcomplicated that makes no sense whatsoever..." wouldn't fit Wikipedia's serious voice but actually be pedagogically very helpful.
by cb321 on 5/22/25, 11:39 AM
Abbreviation confusability is relative to { in fact one might say measured by ;-) - number? entropy? etc. } the listener/reader's knowledge/exposure, much as sound levels need a reference distance.
I have heard "bare K" refer to a great many different things, not just kilobits (transmission) or kilobytes (storage) or kilograms (drug trade) or kilometers (foot races) and on & on, but pages or items or etc.
The fundamental problem is that some humans like to abbreviate while others get caught and annoyed by the necessary ambiguity of such abbreviation. Sometimes this can be the very same human in different contexts. ;-)
In fact, there even seems to be some effect where "in the know people" do this intentionally - like kids with their slang - as a token of in-group membership. And yes, this membership is at direct odds with broader communication, by definition/construction. To me this article seems to be just complaining about "how people are". So it goes!
This is the primary complaint. The secondary one about voltage and power and the ambiguity of the prefix itself was addressed in another comment (https://news.ycombinator.com/item?id=44059611).
by etskinner on 5/22/25, 2:06 PM
Here's another related one that always bothers me: When you say something's loudness in decibels, you also need to specify a measurement distance.
The author of this article even accidentally makes this omission:
> It’s 94 dB, roughly the loudness of a gas-powered lawnmower
And that distance is very important; the actual sound pressure measured is proportional to distance^2. So for a lawnmower measuring 94dB, let's say we assume that we're measuring at 1m. At 2m away, the sound is actually 91dB.
And don't get me started about the fact that a halving in power is 3dB, that's just wacky. I wish we used base 2.
by ggm on 5/22/25, 7:04 AM
Do a deep dive into audio vu Meters and how they got calibrated. Without being 100% sure, it's basically a totally subjective model, where back in the 1920s the BBC and some US company decided to assert "like us" and two models persist which have been retconned into some BIPM acceptable ground truth but it basically was "test it against the one we made which works"
The hysteresis in the coil-magnet meter response turned out to be a feature, not a bug.
by gregschlom on 5/22/25, 3:48 PM
"The bel is named in the honor of Alexander Bell; this is in the same tradition that prompted us to name the “wat” in honor of James Watt."
This line killed me. I literally laughed out loud.
by leoedin on 5/22/25, 7:58 AM
I worked in RF (radar) for a while and the dB/dBm is an incredibly useful tool there. It makes reasoning about amplifier chains and insertion loss so much more straightforward. It also means you can talk about transmitters and receivers in a comparable unit - in reality the signals are many many orders of magnitude apart.
by cesaref on 5/22/25, 9:54 AM
Generally speaking, the db scale is very useful for many practical situations, and this is overlooked in this critique.
As have been pointed out, it's just a power ratio on a logarithmic scale, but this has many benefits, the main one being that chaining gain/attenuation in a system is just a case of adding the db values together. 'We're loosing 4db in this cable, and the gain through this amp is 6db, so the output is 2db hotter than the input'. Talk to any sound engineer and you'll do this sort of thing successfully without necessarily understanding the science, so that's a massive win.
by smat on 5/22/25, 6:28 AM
Great post, the fusion of scale with unit is a mess.
When using it as a factor, for example when describing attenuation or amplification it is fine and can be used similar to percent. Though the author is right - it would be even more elegant to use scientific notation like 1e-4 in this case.
For using it as a unit it would really help to have a common notation for the reference quantity (e.g. 1mW).
But I guess there is no way to change it now that they are established since decades in the way the author describes.
by kristjank on 5/22/25, 7:30 AM
This seems exceedingly ignorant of the work decibels do in telecommunications, RF and fibre engineering. The voltage vs power relationship is something that exists and is a core memory of beginner blunders in the field, but it boils down to a simple 10 vs 20 division operation. Besides that, the decibel simplifies a lot of multiplying very small and very big numbers to summing of two-digit numbers that you can do in your head, and still preserve a big degree of accuracy.
Whining about it makes me really doubt that the OP has any practical experience about the things they're talking about.
by 49pctber on 5/22/25, 1:59 PM
The idea that helped make decibels click for me is that they're a way to quickly do both dimensional analysis and gain/attenuation calculations at the same time.
"Plain" decibels are simply (power) ratios. These can describe multiplicative changes in power. These are positive for gains (like in a power amplifier) or negative for attenuations (like path loss). They are unitless quantities.
Decibels add. A ten 10 dB gain (x10) followed by a 20 dB loss (x0.01) is -10 dB (x0.1).
"Flavored" decibels are in reference to some power quantity. For example, dBm uses one milliwatt as its reference. So 2 mW / 1 mW = 2 = 10^(3/10) = 3 dBm. These quantities have associated units, but they're still technically dimensionless.
Here's the key insight. You can only have one "flavored" decibel value per computation. Say you have some 3 dBm signal (2 mW). You can add as many regular decibel values as you want, but the unit is still dBm. 3dBm + 4 dB - 7 dB = 0 dBm. In linear units, 2 mW * 2.5 * 0.2 = 1 mW
If you were to do something like 3 dBm + 0 dBm, the linear units would be 2 mW * 1 mW = 2 mW^2, which is probably not what you want.
dBs are confusing. Different fields have slightly different conventions. People talk about any factor of 2 as a 3 dB change, when technically it should only be relative to power-like quantities. It's weird that some of these "units" can be added together, while others can't. The factors of 10 and 20 can be confusing.
But if you consider the units from a dimensional analysis standpoint, decibels are much more sane and intuitive than they appear.
by rebolek on 5/22/25, 6:40 AM
While the author is technically right, I must argue that in the area of sound work, decibels make sense. Zero is base level, -3db is half loudness, +3dB is double. There may be a better way to describe loudness, but decibel is good enough.
by FRidh on 5/22/25, 12:27 PM
The problem with using decibels is that since it is a relative unit you need to know what it is relative to, and that is often not just the unit but also the quantity. Unfortunately, this part is (as expressed) at times omitted or put in the wrong place. And its often also the engineers in the respective fields that keep using this incorrect notation spreading the confusion for those outside the domain.
E.g., acoustic engineers often write db(A) for A-weighted sound pressure levels. Yes, it is often noted this way, but it is incorrect. The correct way is to specify the quantity and that the quantity is A-weighted, `L_{p, A} = 80 dB` for example to express an A-weighted sound pressure level of 80 dB.
Regarding sound pressure and sound power. Sound power is not expressed using A-weighting because it does not make sense. Sound power is a property of the source. A-weighting is a property of the receiver, that is, the human listener.
by teknopaul on 5/22/25, 4:01 PM
"This is nuts: it’s akin to saying that the milli- prefix should have different meanings depending on whether we’re talking about meters or liters."
Were were here recently with "mega": Sometimes mega is squared as in megapixels. Sometime not as in megabytes.
No biggie.
Db in audio is a relative scale and that makes perfect sense. If you mixer goes + or - 6db that makes sense but can't be measured as power, your mixer might not be plugged in to any speakers so relation to real power is moot in the digital realm.
3 eq bands with -+6db makes sense too. Doesn't need to be precisly specified to be of immediate value, +-12db is clearly something else and users know what.
by vt240 on 5/22/25, 8:00 PM
I see dB scale units used without contextual issues in near uniformity. Unfortunately, I have to agree with the OP, that microphone capsule manufactures seem to be an edge case. I'm not sure where dBV/Pa became the standard. I can understand why given 94dBSPL@1000Hz calibration standards, and the measurement equipment of the time, but I've run into my own fair share of datasheets with lines such as 'Sensitivity -45dB' with no units or other call outs for the standard in use. Thankfully, it seems like most modern datasheets use mV/Pa which seems like a much better unit in my book.
by HPsquared on 5/22/25, 12:44 PM
It's a bit like "percent". A shorthand for dealing with ratios, often with hidden assumptions.
by thinkingtoilet on 5/22/25, 12:33 PM
Not surprisingly, Tom Scott has an interesting video on this:
https://www.youtube.com/watch?v=Is_wu0VRIqQ
by jeremyscanvic on 5/22/25, 7:31 AM
Another surprising place decibels pop up, pretty far from loudness related things, is in image comparison metrics. Peak signal-to-noise ratio is mean squared error normalized by a certain peak intensity and it is generally expressed in dB, i.e. as 10 log10(normalized mse). https://en.m.wikipedia.org/wiki/Peak_signal-to-noise_ratio
Edit: typo
by calmbonsai on 5/22/25, 8:09 AM
If you're specifying decibels in written form you always include the basis or you're simply being incomplete with your units. I don't understand the complaint there.
In casual conversation, the context implies the basis.
Dealing with decibels is also another shorthand to know the domain has a wide enough value gamut such that logarithmic values (where addition is multiplication) makes sense. See also, the Richter scale.
by elfrinjo on 5/22/25, 12:57 PM
I'll summarize: It's annoying that dB is often used as a physical unit without the necessary suffix. It's annoying that the suffix seems quite random for most units It's annoying that the factor for power differs from the factor for voltage, etc. It's slightly annoying that it's usually decibel instead of Bel.
However, we all agree that dBs are really useful.
by aimor on 5/22/25, 2:45 PM
Yes, dB is a mess and if we could do it again we ought to use something more explicit like just annotating the units with log10 such as log10(W). Then it's easy to use other convenient units like log2(W) or if you want to reproduce dBV: 20log10(V). dB is fine shorthand, but it gets used all the time in places where things should be explicit.
by karmakaze on 5/22/25, 2:21 PM
The writing style is antagonistic like a clickbait title. I find it bothersome. I would have much preferred an organized breakdown of all the different ways dB are used and what they mean instead of a random rant. It complains about how unscientific 'the' unit is, knowing that it's not a single unit then adds little to sort it out.
by atoav on 5/22/25, 7:15 AM
But dB is not a unit, it is a multiplier. dB on its own is unitless and if we say "X is reduced by 6 dB" you know that the value of X is half of what it was before (×0.5) if something is amplified by 6 dB it is double of what it was before (×2.0)¹
Note that the unit only starts to play a role when you reference your dB value to some absolute maximum, e.g.:
```
  dBV which is referenced to 1V RMS
  
  dBu which is referenced to 0.775V RMS (1mW into a typical audio system impedance of 600 Ohms)  
  
  dBFS which is referenced to a digital audio maximum level (0dBFS) beyond which your numeric range would clip (meaning all practical values will be negative) 
  
  dBSPL which is refrenced to the Sound Pressure Level that is at the lower edge of hearing (0 dBSPL), this is what people mean when they say the engine of a starting airplane is 120dB loud  
  
```
Now dB is extremely useful in all fields where your values span extremely big ranges, like in audio engineering, where the ratio between high and low values can easily have a ratio of 1:10 Millions. So unless you want people to count zeroes behind the comma, dB is the way to go.
When we think about the connection between analog and digital audio dB is useful because despite you having volts on the one side and bits on the other side a 6dB change on one side translates to a 6dB change on the other, the reference has just changed. If we had no dB we would have to do conversions constantly.
Going from multiplier x to dB: 20×log₁₀(x)
Going from dB to multiplier x: 10^(x/20)
If you use dB to describe the power of a signal that is slightly different (you use 10 instead of 20 as multiplier/divisor)
But you can see, dB is just a way to describe a unreferenced size change in a uniform way or to describe a referenced ratio. And then it would be good to know what that reference is. So if someone says a thing has 40dB you they forgot to tell you the unit.
¹ this is true for the amplitude of a signal and differs when we talk about the power of a signal, where 3dB is a doubling/halving.
by DonHopkins on 5/22/25, 1:37 PM
Almost as annoying as Euler angles, which force you to choose an arbitrary ordering (XYZ, ZYX, ZXZ, …), while each choice is equally valid yet mutually incompatible, so there is no canonical “true” yaw-pitch-roll.
Not to mention gimbal-lock singularities, wrapping discontinuously at +/-PI, mapping one orientation to many triples, and forcing you to juggle trig identities just to compose two spins.
To pure mathematicians obsessed with elegant unambiguous coordinate-free clarity, Euler’s pick-any-order gimmick is like hammering tacky street signs onto the cosmos: a slapdash, brittle hack that smothers the true geometry while quaternions and rotation matrices sail by in elegant, unambiguous splendor.
by timerol on 5/22/25, 2:29 PM
The one thing that really resonates with me from this article is also the silliest thing: Why decibels instead of bels? Everyone has multiplied in scientific notation before, and making a power ratio of 10^4 = 4 B would be so much more intuitive for conversion to absolute figures. It's a completely silly complaint, given that it's literally just a decimal point instead of a d, but it would make mental math of absolute conversions that much faster, especially for people who don't live and breathe amplifier chains all day.
I want to live in a world where a radio can be specified as a -10.2 Bm sensitivity. -9 is three SI prefixes down from 1 mW, so less than 0.1 pW.
by bob1029 on 5/22/25, 8:20 AM
I think dB is a good way to represent certain user interfaces. I've always preferred to operate my audio equipment using a logarithmic scale. Changing the volume is much more intuitive to me this way than with some linear 0-100 mapping.
by RicoElectrico on 5/22/25, 2:10 PM
This difference in mindset makes me comfortable as an EE that all the laid-off SWEs aren't going to take my job. Granted, demand side of the job market isn't rosy due to macro conditions, but at least supply isn't.
by ziofill on 5/22/25, 6:49 AM
The nice thing about dBs is that they are logarithms: successive applications of gain/attenuation are easily computed by just adding up the dBs. (but yes, I agree that there should be more consistency in their definition)
by cousin_it on 5/22/25, 8:32 AM
Yeah. And don't get me started on the folks who think "6dB/octave" is a reasonable thing to say. Which is, apparently, everyone who works with audio filters except me.
by stdbrouw on 5/22/25, 2:19 PM
One way to illustrate some of the weirdness of dB is to look at the documentation of unit libraries, e.g.:
https://pint.readthedocs.io/en/stable/user/log_units.html
https://painterqubits.github.io/Unitful.jl/v0.7/logarithm/
by tim333 on 5/22/25, 11:37 AM
The author gives the downsides but not the upside of why it is like that.
It's basically so it describes sound levels on an understandable scale with 0db being just audible and 100dB being very loud.
It also corresponds to the energy carried by the sound - 0 dB is 1 pW per square meter so it is kind of a scientific unit. It's probably easier to have a measure that is understandable by the public and let engineers do conversion calculations for signal levels in networks than the other way around.
by lamename on 5/22/25, 12:35 PM
I appreciate the sentiment, but I've found this resource [0] much more direct and comprehensive. It explains all of the nuance regarding dB and related terminology from audio engineering to perception (voltage, power, intensity, volume, loudness, etc.)
The format is a bit circular; just enjoy getting lost in it for half an hour.
https://sengpielaudio.com/TableOfSoundPressureLevels.htm
by hock_ads_ad_hoc on 5/22/25, 7:13 AM
The author seems confused. Decibels aren’t units. They’re a way to express a ratio between a reference value and some other value.
They’re used where they are useful.
by amelius on 5/22/25, 3:26 PM
Speaking of which. Does anyone know why, in Physics, we only use the operators +, * and exponentiation? And why higher operators in the hyperoperation sequence are never used?
https://en.wikipedia.org/wiki/Hyperoperation
by kazinator on 5/22/25, 7:59 AM
The confusing thing about decibels is not the Watts versus Volts thing.
It is the following.
If you mix two identical signals (same shape and amplitude) which are in phase, you double the voltage, and so quadruple the power, which is +6 dB.
But if you mix two unrelated signals which are about the same in amplitude, their power levels merely add, doubling the power: +3 dB.
by bsteinbach on 5/22/25, 4:06 PM
The power vs voltage thing I think comes from the historical connection of dB to sensing waves. Because you can change impedance to match any sensor, but power is conserved, power was just more often more important to discuss than voltage. I still have to look it up every time whether 10x is 10 or 20 dB.
by stevage on 5/22/25, 2:47 PM
Gosh, I didn't know it was that bad - I didn't know it was used for so many different things.
by yuvadam on 5/22/25, 7:45 AM
Decibels aren't actually that ridiculous if you just accept them as plain logarithmic ratios, between your signal and the noise floor or some other signal. (Or between anything else, really.)
3dB is roughly double, 10dB is 10x, but only sounds about twice as loud because our ears are weird.
by kazinator on 5/22/25, 7:51 AM
> This means that if you’re talking about watts, +1 bel is an increase of 10×; but if you’re talking about volts, it’s an increase of √10×. This is nuts: it’s akin to saying that the milli- prefix should have different meanings depending on whether we’re talking about meters or liters.
Well no, because even if you are focusing on a signal measured in volts, the bel continues to be related to power and not voltage. As soon as you mention bels or decibels, you're talking about the power aspect of the signal.
If volume were measured in meters, which were understood to be the length of one edge of a cube whose volume is being given, then one millimeter (1/1000th of distance) would have to be interpreted as one billionth (1/1,000,000,000) of the volume.
When you use voltage to convey the amplitude of a signal, it's like giving an area in meters, where it is understood that 100x more meters is 10,000x the area.
There could exist a logarithmic scale in which +3 units represents a doubling of voltage. We just wouldn't be able to call those units decibels.
by layer8 on 5/22/25, 10:30 AM
Tangential question: Are there any sound meters that can actually measure down to 0 dB? The best I’ve seen specify 20 dB, which to me is still very audible.
(Note that, as the article mentions, 0 dB doesn’t mean “zero sound pressure”, but just “threshold of hearing”.)
by formerly_proven on 5/22/25, 7:08 AM
Decibels make sense and it’s usually only laymen who use only „dB“. I’d be surprised if you’d find „-45 dB“ as the specification for a microphone. Here’s two random examples:
Sensitivity at 1 kHz into 1 kohm: 23 mV/Pa ≙ –32.5 dBV ± 1 dB
Sensitivity: -56 dBV/Pa (1.85 mV)
by taneq on 5/22/25, 8:36 AM
You use a unit that maps well between the thing you want to measure and the thing you want to know about it. If you do this enough with the same until you give it a name. I don’t get the author’s complaint.
by badlibrarian on 5/22/25, 8:17 AM
Horsepower is a unit of measurement. There's hp and bhp and hpE and...
"Other names for the metric horsepower are the Italian cavallo vapore (cv), Dutch paardenkracht (pk), the French cheval-vapeur (ch), the Spanish caballo de vapor and Portuguese cavalo-vapor (cv), the Russian лошадиная сила (л. с.), the Swedish hästkraft (hk), the Finnish hevosvoima (hv), the Estonian hobujõud (hj), the Norwegian and Danish hestekraft (hk), the Hungarian lóerő (LE), the Czech koňská síla and Slovak konská sila (k or ks), the Serbo-Croatian konjska snaga (KS), the Bulgarian конска сила, the Macedonian коњска сила (KC), the Polish koń mechaniczny (KM) (lit. 'mechanical horse'), Slovenian konjska moč (KM), the Ukrainian кінська сила (к. с.), the Romanian cal-putere (CP), and the German Pferdestärke (PS)." [1]
Decibel is not a unit of measurement. Decibels are a relative measurement. It tells you how much louder or powerful something is relative to something else. And frankly far less ridiculous than horsepower, which has a hilarious Wiki article if you read it with a critical mindset.
Deriving some of the constants without Googling is a fun exercise to verify that you're not as smart as you think you are. "Hydraulic horsepower = pressure (pounds per square inch) * flow rate (gallons per minute) / 1714"
[1] https://en.wikipedia.org/wiki/Horsepower
by em3rgent0rdr on 5/22/25, 8:27 AM
> For some reason, the bel — again, what started as a sensible 10× increment — was soon deemed too big to use. I don’t quite know why: in other aspects of life, decimal notation suits us just fine.
Decimal notation can be a tad cumbersome to write and speak. Meanwhile, decibel usage commonly results in nice simple numbers that range between 0 and 100, with the fractional digits often being too insignificant to say out loud. For instance, the dynamic range of 16-bit audio (which is generally all the range that our ears care about) is 96 dB, while volume increments smaller than 1 dB aren't really noticeable, so decibel makes it easy to communicate volume levels without saying "point" or writing a "." or breaking out exponential notation. Even in fields other than audio the common ranges also conveniently will be around 1 dB for being on the verge of significance to around 100 dB or 200 dB for the upper range. (Also the whole power vs root-power caveat is simply something users of dB have to be cognizant of because we need to stick with one or the other to make consistent comparisons, and at the end of the day physical things hapen with power.) So while decibels may seem ridiculous, they actually are quite convenient for dealing with logarithmically-varying numbers in convient range from 1 dB to around 100 dB or so in many engineering fields.
by xnx on 5/22/25, 6:53 AM
Weird that the author makes no mention of Sones: https://en.wikipedia.org/wiki/Sone
by dogman1050 on 5/22/25, 10:38 AM
His first equation is wrong. It should be P=V^2/R, not P=V^2R.
by nyanpasu64 on 5/22/25, 9:02 AM
I have a vague report that "root-power" or voltage-like quantities (20 dB per order of magnitude) are signed or vector-like, while power-like quantities are unsigned scalars?
by amai on 5/22/25, 9:27 AM
Fahrenheit on the other hand makes sense...
by undebuggable on 5/22/25, 8:39 AM
If decibels in acoustics confuse you, try learning decibels in optical communication. That's a witchcraft.
by ttoinou on 5/22/25, 11:52 AM
Love this kind of articles but I wished the author suggested solutions.
I go even further than this author : sometimes decibels are computed using logarithms and what we put inside the logarithm has a physical unit. But I can prove mathematically that this is wrong and that whats given to a log function has to be dimensionless. Hence a lot of dB calculus is mathematically wrong and physically meaningless
by thrdbndndn on 5/22/25, 7:20 AM
> this is in the same tradition that prompted us to name the “wat” in honor of James Watt.
The unit is Watt, not Wat.
by waffletower on 5/22/25, 4:05 PM
This is an armchair rant with a very narrow complaint. The decibel unit has objective utility in both analog and digital signal processing and the field of audio engineering. It is far from meaningless in filter design and a useful reference for software and hardware interfaces that employ them.
by kragen on 5/22/25, 12:50 PM
This is an incredibly valuable article for anyone who's trying to make sense of decibels in some context. Michał clearly explains almost all of the gotchas you have to understand.
I realized recently, after years of doing it for signal powers, that dB are a pretty convenient way to do mental logarithmic estimates for things that have nothing to do with power or signals, with only a small amount of memorization. Logarithms are great because they allow you to do multiplication with just addition, and mental addition isn't that hard. For example, if you want to know how many pixels are in a 3840×2160 4K display, well, log₁₀(3840) ≈ 3.58 (35.8dB-pixel) and log₁₀(2160) ≈ 3.33 (33.3dB-pixel), and 3.58 + 3.33 = 6.91 (69.1 dB-square-pixel), and 10⁶·⁹¹ ≈ 8.13 million. The correct number is 8.29 million, so the result is off by about 2%, which is precise enough for many purposes. (To be fair, though, 4000 × 2000 = 8000, which is only off by 3.5%.)
The great difficulty with logarithms is that you need a table of logarithms to use them, and a mental table of logarithms is a lot of rote memorization. You can get pretty decent results linearly interpolating between entries in a table of logarithms, so you can use a lot more logarithms than you know, but you have to know some.
It's pretty commonplace in EE work to make casual use of the fact that a factor of 2× [in power] is about 3 dB, which is a surprisingly good approximation (3.0103dB is a more precise number). This is related to the hacker commonplace that 2¹⁰ = 1024 ≈ 1000 = 10³; 1024× is 30.103dB, while 1000× is precisely 30dB.
To the extent that you're willing to accept this approximation, it allows you to easily derive several other numbers. 4× is 6dB, 8× is 9dB, 16× is 12dB, and therefore 1.6× is 2dB. ½× is -3dB, so 5× is 7dB (10-3). So with just 2× = 3.01dB we already know the base-10 logarithms of 1, 2, 4, 5, and 8, to fairly good precision. That's half of the most basic logarithm table. (The most imprecise of these is 8: 10⁰·⁹ is about 7.94, which is an error of about -0.7% when the right answer was 8.)
If we're willing to add a second magic number to our memorization, 3× ≈ 4.77dB. This allows us to derive 6× ≈ 7.78dB and 9× ≈ 9.54dB. So, with two magic numbers, we have fairly precise logarithms for 1, 2, 3, 4, 5, 6, 8, and 9.
The only multiplier digit we're missing is 7. (Shades of the Pentium's ×3 circuit: http://www.righto.com/2025/03/pentium-multiplier-adder-rever....) So a third magic number to memorize is that 7× ≈ 8.45dB. And now we can mentally approximate products and quotients with mentally interpolated logarithms.
You can do my example above of 3840×2160 as follows. 3.8 is 80% of the way from 3 (4.8dB) to 4 (6.0dB), so it's about 5.8dB. 2.2 is 20% of the way from 2 (3.0dB) to 3 (4.8dB), so about 3.4dB. 35.8dB + 33.4dB = 69.2dB, which is between 8 million (69.0dB) and 9 million (69.5dB), about 40% of the way, so our linear interpolation gives us 8.4 million. This result is high by 1.2%, which is much better than you'd expect from the crudity of the estimation process.
For a more difficult problem, what's the diameter of a round cable with 1.5 square centimeters of cross-sectional area? That's 150mm², half of 300mm², so 24.77 dB-square-millimeters minus 3.01, 21.76dB. A = πr². Divide by π by subtracting 5dB (okay, I guess that's a fourth magic number: log₁₀(π) ≈ 4.97dB) and you're at 16.76dB. Take the square root to get the radius by dividing that by 2: 8.38dB-millimeters. That's less than 7× ≈ 8.45dB by only 0.07dB, so 7-millimeter radius is a pretty decent approximation, 14mm diameter. The precise answer is closer to 13.82mm.
For approximating small corrections like that, it can be useful to keep in mind that ln(10) ≈ 2.303 (a fifth magic number to memorize), so every 1% of a dB (10¹·⁰⁰¹) is a change of about 0.23%. So that leftover 0.07dB meant that 7mm was high by a couple percent.
More crudely: 150mm² is 22dB, ÷π is 17dB, √ is 8½ (pace Fellini), 7×.
It's pretty common in engineering and scientific calculations like this to have a lot of factors to multiply and divide, increasing the number of additions and subtractions relative to the number of logarithmic conversions; this is why slide rules were so popular. Maybe you derived the 1.5cm² number from copper's conductivity and a resistance bound, or from the yield strength of a steel and a load, say. 3840×2160 pixels × 4 bytes/pixel / (10.8 gigabytes/second), as I was calculating last night in https://news.ycombinator.com/item?id=44056923? That's just 35.8 dB + 33.3dB + 6dB - 100.3dB = -25.2dB-seconds, which is 3.0 milliseconds to memcpy that 4K framebuffer. (I didn't do that mentally, though.) Even 36 + 33 + 6 - 100 = -25, so π ms, is a fine approximation if what you want to know is mostly whether it's more or less than 16.7 ms.
So here's a full list of the seven magic numbers to memorize for these purposes:
```
  2× ≈ 3.01dB (∴ 4×, 8×, 5×)
  3× ≈ 4.77dB (∴ 6×, 9×, 1.5×)
  7× ≈ 8.45dB
  π× ≈ 4.97dB
  ln(10) ≈ 2.303 (∴ 0.01dB ≈ 0.23%, etc.)
  1.259× ≈ 1dB (+1dB ≈ +25.9%)
  (1 - .206)× ≈ -1dB (-1dB ≈ -20.6%)
```
I haven't been applying this approach long; I'll try to report on results later.

by numpad0 on 5/22/25, 11:27 AM

I just can't understand why "[specialized domain] uses these stupid wrong units, it should be [unit that no educated smart SMEs use]" types don't do their researches to understand why those are used at all. This type of weird non-SI units appear when means of measurement and unit is related to one another and has downstream dependencies.

Just look at aviation. An airplane's:

  - speed is measured in knots, or minute of angle of latitude per hour, which is measured by ratio of static and dynamic pressure as a proxy.
  - vertical speed or rate of climb is measured in feet per minute, which is a leaky pressure gauge, probably all designed in inches.
  - altitude is measured in feet, through pressure, which scale is corrected by local barometric pressures advertised on radio, with the fallback default of 29.92 inHg. When they say "1000ft" vertical separation, it's more like 1 inHg or 30 hPa of separation.
  - engine power is often measured in "N1 RPM %" in jet engines, which obviously has nothing to do with anything. It's an rev/minutes figure of a windmilling shaft in an engine. Sometimes it's EPR or Engine Pressure Ratio or pressure ratio between intake and exhaust. They could install a force sensor on the engine mount but they don't.
  - tire pressure is psi or pound per square inch, screw tightening torques MAY be N-m, ft-lbs, or in-lbs, even within a same machine.

Sure, you can design a battery charging circuits in Joules, fly an airplane with a GPS speedometer, analog audio-radio circuitry in millivolts. Absolutely no one does. I think that cognitive dissonance should trigger curiosity circuitry, not rant mode.

I mean, just type in "use of decibels[dB] considered harmful" at the box at chatgpt.com. It'll generate basically this article with an armchair version of the top comment here as the conclusion.

by halayli on 5/22/25, 10:43 AM
this is one of the best videos to understand what a dB is and when it is used, and how to interpret them. what's unique about this unit is it relies a lot on the context.
https://www.youtube.com/watch?v=1mulRI-EZ80
by 867-5309 on 5/22/25, 10:37 AM
still doesn't answer what 0.3 Sone is in dBA
by varjag on 5/22/25, 8:45 AM
Decibels are not anywhere near as ridiculous as Marketing Kilobytes introduced in the 1990s. Pot, meet kettle.
by adrian_b on 5/22/25, 12:11 PM
Decibels are not ridiculous, but very frequently the notations for quantities expressed in decibels are ridiculous.
The decibel is an arbitrary unit for the quantity named "logarithmic ratio".
Logarithmic ratio, plane angle and solid angle are 3 quantities for which arbitrary units must be chosen by a mathematical convention and these 3 units are base units, i.e. units that cannot be derived from other units. For a complete system of base units for the physical quantities, there are other 3 base units for dynamic quantities that must be chosen arbitrarily by choosing some physical object characterized by those quantities, i.e. a physical standard (originally the 3 dynamic quantities were length, time and mass, but in the present SI the reality is that mass has been replaced by electric voltage, despite the fact that the text of the SI specification hides this fact, for the purpose of backward compatibility), and there also are other 2 base units for discrete quantities (amount of substance and electric charge) which must be established by convention.
Like for the plane angle one may choose various arbitrary units, e.g. right angles, cycles, degrees, centesimal degrees, radians, or any other plane angles, for the logarithmic ratio one may choose various arbitrary units, e.g. octave, neper, bel, decibel.
So if we choose decibel all is OK. Decibels have the advantage that for those used to them it is very easy to convert in mind between a logarithmic ratio expressed in decibels and the corresponding linear ratio, so it is very easy to make very approximate computations in mind, but good enough for many engineering debugging tasks, in order to replace multiplications, divisions and exponentiations with additions, subtractions and rare simple multiplications, for a quick estimate of what should be seen in a measurement in a lab or in the field.
The problem is that whenever a logarithmic ratio is specified in decibels, it must be accompanied by 2 quantities, what kind of physical quantities have been divided and which is the reference value. Humans are lazy, so they usually do not bother to write these things, assuming that the reader will guess them from the context, but frequently the context is lost and guessing becomes difficult or impossible.
An additional complication is that one never uses logarithmic ratios for electric voltages or currents, but only for powers. When it is said that a logarithmic ratio refers to a voltage or a current, what is meant is that the logarithmic ratio refers to the power that would be generated by that voltage or current into an 1 ohm resistor. A similar problem exists for sound pressures, because logarithmic ratios are used only for sound intensities, so where sound pressure is mentioned, actually the corresponding sound intensity is meant.
This complication has appeared because voltages, currents and sound pressures are what are actually measured, but powers and sound intensities are frequently needed and using logarithmic ratios with different values for related quantities, while omitting frequently to mention the reference value, would have caused even more confusion than the current practice.
by lambdaone on 5/22/25, 11:40 AM
It's a bit like hating numbers, and saying "What do you mean by 'three'; what is it - three volts, three amps, three metres? Clearly 'three' is meaningless, and we should stop using it and all the other numbers besides."
decibels are simply a dimensionless ratio, used as a multiplier for some known value of some known quantity.
In every context where decibels are used, either the unit they qualify is explicitly specified, or the unit is implicity known from the context. For instance, in the case of loudness of noise to human ears in air, the unit can be taken to be dBA (in all but rare cases which will be specified) measured with an appropriate A-weighted sensor, relative to the standard reference power level.
And similar (but different) principles apply to every other thing measured in dB; either theres an implicit convention, or the 0 dB point and measurement basis are specified.
People who assume that everyone is an idiot but themselves are rarely correct.
I look forward to the author discovering about (for example) the measurement of light, or colorimetry, and the many and various subtleties involved. The apparent excessive complexity is necessary, not invented to create confusion.
by nabla9 on 5/22/25, 9:03 AM
Decibel is ridiculously overloaded concept.
It can be used to express and calculate relative change in power, amplitude ratios, and absolute change. All of these are different units and should always use different notation, but sometimes it's skipped.
by pajko on 5/22/25, 9:41 AM
Loudness is not measured in dB, but in phon (https://en.wikipedia.org/wiki/Phon) or sone (https://en.wikipedia.org/wiki/Sone).
by timewizard on 5/22/25, 6:58 AM
> That said, I don’t know how to pronounce “3e5”
"Three to the exponent of five." Or "Three Exponent Five." Or "Three Exp Five."
> Seeing this, some madman decided that 1 bel should always describe a 10× increase in power, even if it’s applied to another base unit. This means that if you’re talking about watts, +1 bel is an increase of 10×; but if you’re talking about volts, it’s an increase of √10×
This is power vs. amplitude. This is the specific reason the dB is so useful in these systems.
> the value is meaningless unless we know the base unit and the reference point
No you just need to know if you have a power or a root-power quantity. Which should generally be obvious.
https://en.wikipedia.org/wiki/Power,_root-power,_and_field_q...