# Why are “degrees” and “bytes” not considered base units

From Wikipedia:

The SI base units and their physical quantities are the metre for measurement of length, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the candela for luminous intensity, and the mole for amount of substance.

As far as I know, a base is a unit which cannot be broken down into units other than those from the above mentioned units. However, degrees (for angles) and 8-bit bytes (for digital data) cannot be expressed using one or more of the base units. So, why are these two units not considered base units?

• The part about degrees has already been asked and answered: Why are angles dimensionless and quantities such as length not? and Are units of angle really dimensionless?. – The Photon Aug 1 '18 at 16:03
• I've removed some brief answers which were posted as comments. – rob Aug 1 '18 at 19:51
• @ThePhoton I can't find the answers that you think address this question. Those are about dimensionality. However this question is specifically about SI base units. In SI, angle (radian and steradian) are derived units, not base units, and this terminology is a matter of convention decided by the CGPM (SI committee) informed by the wider metrology community. I don't think we should expect OP to figure out the answer to this question from the answers I read about dimensionality. – Qsigma Aug 2 '18 at 9:07
• Bytes are broken down to bits, as you have written. This part of the question is self-disproving, as it states a unit "more basic" than byte (btw, there are systems that use "bytes" of size other than 8 bits.) – Agent_L Aug 5 '18 at 16:56
• Yeah you are right. Actually I didn't recall that at the time of writing :) – Wais Kamal Aug 5 '18 at 17:22

The radian (not the degree) is the SI unit of angle, and it's defined in terms of lengths: it is that angle for which the length of a circular arc subtending that angle is equal to the radius of the circle. Since this definition refers to the relative ratio of two lengths, the SI considers it to be a "dimensionless derived unit", rather than a base unit.1

As far as bytes go: Defining a unit amounts to specifying a certain amount of a quantity that we call "one unit". Physical quantities such as mass, length, time, etc., are (effectively) continuous quantities, and so there is no "natural" unit for us to use. We therefore have to make an arbitrary choice about how much of each quantity is equal to one unit.

Digital information, on the other hand, is inherently discrete. All methods of quantifying data simply amount to counting bits; and you don't need to make an arbitrary choice of unit if you can simply count a quantity. There is therefore no need to define a unit for digital information, because there is already a natural unit (the bit).

It's important to note that not every measurable quantity is inherently definable in terms of SI base units. If I count the number of people in my office building right now, and tell you that there are "12 people" in the building right now, then "people" is not expressible in terms of meters, kilograms, and seconds. But I don't need to worry that you're going to come along and use some different unit to count the people in this building, because a natural unit (1 person) exists. It's only when we are measuring a quantity that can take on any real-numbered value (e.g., the mass of all the people in this building) that it becomes important to define a unit; otherwise, you and I have no basis for comparison. Any system of units is essentially a set of these arbitrary choices; "natural" units of quantities that are inherently discrete are unnecessary simply because they're understood to be the obvious choice.

1 It's worth noting that the radian was officially a "supplementary unit" in SI until 1995, when they were reclassified as "dimensionless derived units". A bit of the discussion surrounding this change can be found on p. 210 of the Proceedings of the 20th Conférence Générale des Poids et Mesures (warning: large PDF). Reading between the lines, I suspect that the name "dimensionless derived unit" was something of a compromise between those who thought it should be thought of as a derived unit and those who didn't think it should be thought of as a unit at all; but I wouldn't want to speculate further than that.

• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Aug 28 '18 at 15:22
• Upvoted for the great answer and because I couldn't bear to read $21999$ points. – Les Adieux Sep 16 '20 at 14:56

Another answer (and a linked question) addresses the fact that that the SI derived unit for angles is the radian, which is a ratio of lengths. See e.g.

The bit/byte question is interesting. In information theory, the bit is a unit of entropy. A system which is equally likely to be in one of two states has a thermodynamic entropy of $$S = k_B\ln \Omega = k_B\ln 2 = \rm1\,bit \approx 10^{-23}\,J/K,$$ which must be reduced to zero if you "write" to the bit so that its state is no longer uncertain. That's such a tiny amount of entropy that no one (apart from textbook authors) really thinks about its thermodynamic consequences, which is fine.

A byte is a particular number of bits -- usually eight nowadays, but some computers in the past have used a different number. So when you say "I have two bytes of data" what you mean is "these bits of data: I have sixteen of them." The SI does have a unit for expressing collections of many identical objects: it's the mole, which is just like a dozen, only bigger. So I suppose that you could could say that one eight-bit byte is the same as roughly $\rm 13\, yoctomoles$ of bits. I would not recommend this.

• a bit can be a measurement of data or entropy. While those are kinda related, they are not the same. 100bit of data do not have to have 100bit entropy. – Goodbye SE Aug 2 '18 at 7:54
• @kamikaze I think that the difference is not relevant in a discussion about units. Think about drawing shapes. The perimeter of a shape is kinda related to its width, even though they are not the same --- but both are lengths, measured in meters. – rob Aug 2 '18 at 13:09
• @KamiKaze They're really closely related. Suppose we have a random draw $x$ from $P(X)$. The entropy is the expected information that we gain by observing that draw, given that we already know $P(X)$. Or put another way, suppose we know that $X$ follows the distribution $P(X)$, which has $h$ bits of entropy, and we want to be able to uniquely identify any $x \in X$. The optimal encoding will be such that the average length needed to represent a given element is $h$ bits of data. You need at least $h$ bits of data to represent $h$ bits of information. – Ray Aug 2 '18 at 19:00
• @Carl Perhaps. But all of those possibilities have the same units. This question and this answer are about the units. – rob Aug 2 '18 at 20:15
• A mole is like 12, for very large values of 12? – Barmar Aug 3 '18 at 3:12

Units are required to count something which is not obviously countable.

You do not need units to count apples, because you can just do: one apple, two apples, three apples, ... . Just replace "apple" by "bit" and you can count them as easily. A "byte" is just a word we invented to refer to a group of eight bits, as we invented the word "dozen" to refer to twelve objects. Technically, "bits" and "bytes" are as much a unit as "apples" or "cats". I would recommend to consider them as countable objects instead. And of course, countable means that you can talk about fractions too. Half an apple is obviously meaningful, but also half a bit is perfectly fine and useful e.g. in information theory.

However, you cannot count distance/mass/etc. as they are inherently continuous without an obvious subdivision. There is no one distance, two distances, ..., but you need to split distances into finite comparable parts to make it countable. That is what units are for. Back then, this was done with "arbitrary" subdivisions like $1/40,000$ of the circumference of the earths equator ($\approx$ one kilometer). But the modern way is to look for fundamentally given subdivisions, like e.g. the distance light travels in a second, or the mass of an elementary particle.

Angles, while also continues, have a natural subdivision as we can count them in chunks and fractions of "whole turns".

• Angles are just as continuous as distance and mass, yet they are dimensionless. – Jan Hudec Aug 2 '18 at 17:31
• @JanHudec You count them in chunks (and fractions) of "whole turns". They are naturally "discretized", but you could have done different of course. – M. Winter Aug 2 '18 at 17:35
• By the way, fractions of a bit have a perfectly good meaning too (even practically, e.g. in compression algorithms). – Jan Hudec Aug 2 '18 at 18:14
• Modern definitions are equally arbitrary as old definitions; they're just referring to references that are more constant and/or more easily measurable. – gerrit Aug 2 '18 at 20:45
• I don't quite get your reasoning. If counting entropy in bits is "natural", why isn't it natural to count distance in e.g. Planck lengths? – Dmitry Grigoryev Aug 7 '18 at 6:43

Degrees are primarily a historical unit. There are two physically meaningful ways to measure angles: the cycle, and the radian. The cycle is the arc length of a circle subtended divided by the circumference of the circle, and it goes from zero to one. The radian is simply the same arc length divided by the circle's radius instead of its circumference. Physicists and mathematicians have a marked preference for radians because the derivatives of trig functions are substantially simplified in radians, simplifying the ways computers calculate them. These two quantities are, of course, related by a factor of $2\pi$.

The degree is just scaling the cycle up by 360 because it's a number that can be divided by a lot of small integers without producing a fraction: 2, 3, 4, 5, 6, 8, 9, 10, 12, etc. This dates back to a time when decimals had not been invented, and avoiding fractions had a lot of computational advantages.

So, degrees are not a base unit in any sense, neither conceptually nor in terms of overall convenience in a modern setting.

Similarly for bytes. A byte is just 8 bits. Why 8? Probably because it's the smallest power of two that can encode an entire ASCII character (7 bit code). Computer scientists have a thing for bits, and it makes it possible to easily detect many cases where a file is not ASCII text without making text files unnecessarily large. I believe that a long time ago a lot of machines had different word/character lengths, but the 8 bit byte became a de-facto standard.

All of that said, the byte is, fundamentally, a unit of information, and therefore of entropy. As far as units go, especially in physics, we have to deal with systems where the number of degrees of freedom are countable only in principle, not in practice. It is situations like that where you need units like the mole, where you know it's an integer but have no way to actually count it. That's why we derive our unit for entropy as a Joule per Kelvin.

In the context of information entropy, on the other hand, everything is actually countable. There, a more natural unit for the machines is, of course, the bit, but that's a question of technological convenience, not anything fundamental. We could also use the trit for ternary, the oct for octal, the hex for hexadecimal, the digit for decimal, etc. Notice how those correspond to different numbering systems, where we characterize them by the number of symbols in the system. In that thinking, treating the 8-bit byte as a unit is the same as using a base-256 counting system. There's no fundamental feature of reality that makes that number system more special than any other.

Point being, both bytes and degrees are not actual units. They're more akin to the percent, or the SI prefixes (e.g. kilo, centi, etc), but they're not a power of 10, and so not "metric". It could also be argued that a byte is more closely related to the decibel or the "magnitude" in astronomy, given the presence of logarithms in the definition of those and in entropy, but those aren't base units, either.

• Degrees are hardly just historical; it is meaningful to measure using degrees. Their value comes in practical applications such as navigation and building, where the multiple divisors, 2, 3, 4, 5, 6, 8, etc. make it convenient for quick estimation and communication. – xxyzzy Aug 8 '18 at 7:20

There's a misapprehension in your question: a byte or octet can be expressed in terms of the bit. An octet is represented by two hexadecimal (more correctly, senidenary) numbers, each of which represents a nybble, or 4 bits. The byte is simple a special common word of a fixed length. The word octet isn't used so much in English, but in French it's the preferred term over byte.

Now: why is the bit or binary digit not a base unit? Perhaps because there are ternary digit, denary digit, senidenary digit, which all say the same thing, the unitless number 1.

The key is what are you measuring? If you say "1 bit", what is that? Until you express it in terms of storage, memory, or registers, you're not even using a unit. The failure in the SI is the lack of a unit for these things; so we still express storage size in powers of ten, memory size in powers of 2 (or 8, or 16, or 1024), and register size in terms of whatever the fabricator thinks of this year.

• "The byte is simple a special common word of a fixed length". Not quite. The word size is the size of the numbers passed to the CPU instructions. The byte is the minimum addressable size, which these days is usually, but not necessarily, 8 bits. An octet is always exactly 8 bits by definition. Both byte and word length are hardware dependent; they may be equal, but usually aren't. A bit is a unit of information, and is independent of hardware (and is definitely unaffected by whether you're storing it in memory or register, and whether you're using base 2 or base 10 to count). – Ray Aug 2 '18 at 19:13

A quantity is dimensional if you can rescale it and all the relations remain the same. It is dimensionless if the numeric value has direct meaning in the equations.

Distance is dimensional. Whether you use meters, feet or astronomical units, the relations with them stay the same, except related units, e.g. velocity, scale along with it. But angle is dimensionless. The value in radians is a ratio of lengths and if you use degrees instead, a conversion factor pops up in the relations. And bit is also dimensionless, being the 1 of information entropy, defined in terms of counts and probabilities.

Now the dimensional quantities are still related to each other. Since velocity is distance per time, if you scale the distance unit, the velocity unit scales with it.

The base units are a set that can be scaled independently of each other (in your problem domain!). Note that the choice is somewhat arbitrary. For example electric current was selected as base dimension, but electric charge would arguably make more sense. The other units are derived.

The problem domain is actually important. It turns out plenty of constants are actually just conversion factors due to choice of scaling. For example once special relativity gets involved, time becomes just another spatial dimension, distances can be measured in seconds, and velocity becomes a dimensionless ratio.

In fact, all the dimensional constants ones are and the natural units, especially in the planck variant, leave you with no dimensions at all and only the three dimensionless constants $\pi$, $\alpha$ and $\alpha_G$.

On the other hand, there are cases where you can distinguish, say, parallel distance and perpendicular distance and then suddenly angle becomes perpendicular distance over parallel distance and is dimensional. If in what you do you don't mix the two, making them distinct units significantly improves utility of dimensional analysis as a verification.

The SI base units were simply chosen to be practical for classical physics and everyday engineering and is somewhat arbitrary (especially the candella, unit of luminous intensity, is not really a basic unit; it is just energy weighted-averaged over light spectrum using specific weighing function).

Unit standardization has gone through several stages. First, there were ill-defined units, such as "the length of a forearm". Then reference objects were established: for instance, there might be an Official Foot Rod, a rod that was officially one foot long, and all measurements were done by comparing an object to the Official Foot Rod, or to rulers that were marked with reference to the Official Food Rod. The metric system started out using the reference object system: there was a physical object that was defined to be one kilogram, a rod that was defined to be one meter, etc. Later, scientists moved towards units defined by physical properties of the universe: for instance, the second is defined in terms of the emission spectrum of the cesium atom. So, now, if you want to know how long a second is, you need to measure the emission spectrum of a cesium atom.

Degrees and bytes need neither a reference object nor measurement. There is no need for an international organization to establish a standard "degree" or "byte", any more than words in general need a standard definition. A degree is simply 1/360th of a circle, and a byte is a unit that denotes logarithm base 256.

• Just a minor comment on the original SI metre - although it was referenced by a unit rod, it was defined as 1/10,000,000 of the distance from the north geographic pole to the equator through the meridian of Paris. Though this is no longer exactly true, it does give a very easy way to "remember" the size of the Earth... – Grimxn Aug 2 '18 at 16:05
• Both degree and byte are standard definitions just like metre and second. The natural unit of angle is radian (circumference over radius) and natural unit of information is bit. – Jan Hudec Aug 2 '18 at 18:20
• @Grimxn That was the original source of the length, but once they created the bar based on that distance, the bar was the official length of a meter. – Acccumulation Aug 7 '18 at 17:29
• @Acccumulation - absolutely, but the bar came second (unlike the “King’s foot”). That’s why I used “defined” rather than “standardised as”... :) – Grimxn Aug 7 '18 at 21:08

Why are “degrees” and “bytes” not considered base units ?

Pretty much for the same reason for which percentages and digits are not considered base units either. After all, a degree represents the $360^{th}$ part, just as a percent signifies the hundredth part. Likewise, a byte denotes a group of eight bits, the latter being short for binary digits. In other words, they are abstract mathematical concepts, devoid of any physicality.

• Bytes are what I take when I'm eating pi. – Hot Licks Aug 4 '18 at 1:05

Interestingly, the discussion is regarding Base Units, rather than dimensions.

SI takes great convoluted care to carefully weave its way through the minefield of conventions and misunderstandings. It was also started as a convention from the time before modern computers were common place, so many things we may expect to be done with the support of a computer were done manually and that required their own techniques.

In particular, dimensional analysis was done independently of the numerical calculations, and relativity was not even a consideration.

The metre is a base base unit of Length, but we live in a 3d world, so Length can't be both a single 'dimension' (equivalent to a base unit) and a 3d space.

For the byte, you should look to the base unit of the Neper. The Neper takes a power of 'e', rather than a power of '2' (bits). This leads to the other power of 'e', the imaginary Angle base unit (rabbit hole opens here).

When using a modern Computer Algebra System that is capable of carrying the Base Units (dimensions) through the calculations, you see a potential error for Length dimensions when we divide two values of Length that are in different dimensions, and claim that the result has no dimensions, yet for any other pair of dimensioned values the dimension indicators would be retained.

It is for those cases that an indication of the previously supplementary unit of Angle should be retained. That is, it is, or should be, about error detection and correction.

I have had colleagues that think one can take the tangent of 10 metres [tan(10m)], simply by separating the units from the calculation to produce "tan(10) * m", which if I understand the SI rules (if taken pedantically) is what should be done.

In summary, the SI system is a slowly developed ponderous set of rules that does not take even small steps without great and careful consideration. Until folk start noticing the errors they make (see Panko, Errors in spreadsheets), then little will happen unless one of the big CAS systems (MathCAD, Maple, Mathematica,..) take the plunge and extend their dimensional analysis systems to show the way, then little will change.

• The neper isn't an SI base unit, just a unit.... – Philip Oakley Aug 2 '18 at 21:56