2.2.2 - Modern Standard SI

2.2.2

Modern Standard SI

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INTRODUCTION

Today scientists still use the SI prefixes and most people are at least familiar with them. For example, most people would know what a mega , and giga are. This is largely do to familiarity with terms like megabyte and gigabyte. You may have also heard of things like "nanotechnology". However there are plenty of SI prefixes most people don't know. Beyond the gigabyte is the lesser known terabyte, and beyond this the even lesser known petabyte ! How far do these terms extend ? The are only 20 officially recognized SI prefixes that are used. Let's first familiarize ourselves with the standard table of prefixes ...

CANONICAL SI PREFIXES (2009)

SI Prefix Prefix Symbol Multiplier Year added Etymology

yotta- Y 10^24 1991 Latin : "eight"

zetta- Z 10^21 1991 Latin : "seven"

exa- E 10^18 1975 Greek : "six"

peta- P 10^15 1975 Greek : "five"

tera- T 10^12 1960 Greek : "monster"

giga- G 10^9 1960 Greek : "giant"

mega- M 10^6 1960 Greek : "big"

kilo- k 10^3 1795 Greek : "thousand"

hecto- h 10^2 1795 Greek : "hundred"

deca- da 10^1 1795 Greek : "ten"

deci- d 10^-1 1795 Latin : "tenth"

centi- c 10^-2 1795 Latin : "hundredth"

milli- m 10^-3 1795 Latin : "thousandth"

micro- µ [1] 10^-6 1960 Greek : "small"

nano- n 10^-9 1960 Greek : "dwarf"

pico- p 10^-12 1960 Greek : "tiny bit"

femto- f 10^-15 1964 Dano-Norwegian : "fifteen"

atto- a 10^-18 1964 Dano-Norwegian : "eighteen"

zepto- z 10^-21 1991 Latin : "seven"

yocto- y 10^-24 1991 Latin : "eight"

OFFICIAL USAGE

There is also a set of accepted base units which are permitted to be combined with the SI prefixes to form new units. There are seven base units. Each is intended to measure a different kind of physical quantity. Each also has a symbol which can be combined with the SI prefix symbols to form unit symbols. Here is a table of the seven base units...

TABLE OF CANONICAL BASE UNITS (2009) [2]

Base Unit Unit Symbol Measure of

meter m length

gram g mass

second s duration

ampere A electric current

kelvin K thermodynamic temperature

candela cd luminousity

mole mol amount of substance

In any introductory Physics class one of the first things you learn ( often because it is stated explicitly by physics teachers ) is that units of measurement can be treated like variables. For example " 3 m " stands for 3 meters. "m" here stands for the base unit "meter", which is being multiplied by the number. This is exactly the same idea used in algebra when we use things like "3x" meaning 3 times the variable x. In most cases the variable is unknown but is a member of a set of values ( for example we may not know the value of x, but we know it represents a real number, and therefore exhibits all the normal algebraic properties of the real numbers ). However x doesn't neccessarily have to represent a number.

With this basic understanding we can now make use of the SI prefixes to extend the usefulness of the base units. Take meters for example, we can measure the height of people in meters ( usually around 1.5 meters ), or the distances between rooms etc. But longer distances, like a trip to the super market, will require much larger measurements. In these cases distances often range in the thousands of meters. Rather than say something is 1000 meters away we can use the SI prefix "kilo" to form a new unit the "kilometer" which is equal to 1000 meters. This can be symbolized as ...

1 km = 1000 m

The prefix "k" is written to the right of the base unit "m" and acts as a multiplier. In this case k is equivalent to multiplying by 1000. We can then use multiples of this new unit for example " 3 km" means 3 kilometers or 3000 meters.

Other common metric units of length are the "centimeter" symbolized "cm" ( which is 0.01 meters ) , and the "millimeter" symbolized "mm" ( which is 0.001 meters ). We can also form other units , both much bigger and much smaller, using the other SI prefixes. For example we could speak of Mm , Gm , or Tm ( megameters , gigameters , and terameters respectively ). These are almost never used. Instead scientists will usually use measurements like 1000 km , 10^6 km and 10^9 km . None the less, we know from the rules of SI prefixes exactly what Mm, Gm, and Tm stand for. This is one of the appealing aspects of the metric system. In other systems , if we are not familiar with all of the units we may not know that a yard = 3 feet or the a mile = 1760 yards. In the metric system it is only neccessary to understand and be familiar with the SI prefixes and the base unit involved.

You may be wondering ... if SI prefixes can be attached to a unit as a multiplier to form a new unit, what's to prevent that new unit from also being modified ? For example could you have a kkm ( kilo-kilometer) ? Logically this would be ...

1000 x 1000 x meter = 1,000,000 meters

In most cases this would be nonambiguous , although somewhat obtuse since we can simply use Mm for this. The use of combined units could be helpful in the cases such as hkm (hecto-kilometer), which would be 100 kilometers. The use for such units however is limited, and most people will simply use 100 km instead. One could also, in theory , create completely useless terms like mkm ( milli-kilometer). This would be 1/1000 km which is the same a 1 meter.

Lastly , what does it mean if there is no base unit ? For example "M" by itself would just be a "Mega", the multiplier of a million, but a million of what ?!

As it turns out official policy in regards to the use of compound SI prefixes is that it is not allowed. Furthermore SI prefixes must be coupled with a base unit. The base unit should always come last (furthest right). This avoids certain ambiguities.

Given that we have 7 base units, and 20 allowed prefixes to attach to them we have potentially 140 different units to use ! This system works well, and we can usually interpret them properly by context, although units like "dcd" may look confusing at first ( this would be a deci-candela ).

The situation becomes alittle more difficult when we consider complex physical quantities that are formed by combining the base units together. For example "speed" is measured as "distance over time". So for example "m/s" stands for meters per second. Now consider what does "km/s" mean ? The k could be best understand as multiplying the numerator. In otherwords we can think of this as traveling a "1000 meters" every second. Can we simply apply the "k" to the entire unit "m/s". In theory, but this actually makes little difference because we would still end up having to interpret it the same way.

Things don't always work out this nicely however. Recall that a measure of area is understood as the product of lengths. So for example "m^2" means "meter squared" or a "square meter". What then should mm^2 be ?

Is it a "square millimeter " or is it a " milli- square meter ". In otherwords is it (mm)^2 or m(m^2).

In physics class this detail is usually glossed over. Basically we assume that the SI prefix always applies first. It's as if the base unit and the multiplier are acting as a whole unit. So a cm^2 is a square centimeter. Algebraically this is unusual. For example an expression like 3x^2 would never be understood as (3x)^2 = 9(x^2). Instead the "higher operation" of exponentiation gets carried out before the multiplication ( we will go into more depth into matters of resolution much later ).

COMPUTER MEMORY

The SI prefixes are also commonly used in computer terminology for both processing speed and memory. Although SI prefixes are only specified for certain units, conceptually you can apply them to just about anything.

The smallest unit of memory in a computer is called a bit. It can be thought of as a simple switch which stores only one piece of information, that of being either off ( symbolized as 0 ) or on ( symbolized as 1 ). In and of itself this information is not very useful. However if you use millions and billions of bits you can use them to encode virtually any kind of information imaginable. Interestingly computers usually don't operate on a bit by bit basis, but rather work with packets of bits. A "byte" is equal to 8 bits. An example of a byte would be something like "00101101". Because each bit within a byte can be either a 0 or 1 there are 2^8 possible configurations for every byte, that's 256 possible configurations. Each byte can then be used to encode a computer command. "b" is the unit symbol for "bit", and "B" is the unit symbol for "byte".

You may notice that the memory in most simple text files is measured in kB. This is the kilobyte. We can understand it as roughly a 1000 bytes. In actuality computers measure a kilobyte as 1024 bytes ( 2^10 = 1024 ), but this detail is glossed over in the use of SI units here.

Image files can often measure in the MB, or megabytes. A megabyte is 1024 kilobytes. Computer gaming software can measure in the GB, or gigabytes. A gigabyte is 1024 megabytes. You get the picture.

With this understanding we can then define things like TB (terabytes) , PB (petabytes) , and even EB (exabytes). Although TB is coming into use, PB is much more obscure and only relevent to today's super computers. As far as I know, EB has not been used in practice yet. We can of coarse finnish this off by using all of the large scale SI prefixes with ZB , and finally YB.

Note that because the multiplier is 1024 instead of 1000 we are taking some liberty in the interpretation of the SI units. It is a common mistake to assume that a Megabyte is exactly a million bytes. We could of coarse simply take the position that the SI prefixes are multiples of 1000 in all applications except in regards to computers where they are multiples of 1024. This seems to be the unspoken but accepted standard. Other people however think this needs correction. It has been proposed that a set of special SI prefixes be used to specify multiples of 1024 instead of 1000 for exclusive use in computer terminology.

To correct these apparent ambiguities a special set of "binary prefixes" were proposed in 1996 as a joint effort between the IEEE , ISO , and the IEC [3] . Here is a table of these special prefixes ...

Binary Prefix Prefix Symbol Power of 2 Decimal Value

kibi- Ki 2^10 1024

mebi- Mi 2^20 1,048,576

gibi- Gi 2^30 1,073,741,824

tebi- Ti 2^40 1,099,511,627,776

pebi- Pi 2^50 1,125,899,906,842,624

exbi- Ei 2^60 1,152,921,504,606,846,976

zebi- Zi 2^70 1,180,591,620,717,411,303,424

yobi- Yi 2^80 1,208,925,819,614,629,174,706,176

Acceptance of these prefixes have been slow. One major problem is that the computer industry itself seems to have little interest in modifying an entrenched standard of labeling. Oddly it is this very industries inconsistency which has lead for the need of greater clarity. Memory sizes are usually recorded using the "binary values", meaning when you buy a TB hard drive your purchasing not 1 trillion, but 1,099,511,627,776 bytes. Who could complain though ? Your getting more than you expect. The problem is that processing speeds uses the SI prefixes in the standard way, so a processing speed of 1 billion hertz would be GHz ( Hz is the symbol for hertz ). Believe it or not this ambiguity has lead to real lawsuits against companies, because the defendants have argued this is a violation of fair reporting [3] . On the one hand this may be seen as incredible, even ridiculous. On the other this may eventually force the industry to adopt the unconventional prefixes proposed by the IEC. Then instead of buying a TB hard drive, you will see it labeled as a TiB hard drive.

Personally I don't mind the nomenclature as far as the prefix symbols go, though the names do sound alittle silly. Were used to hearing "terabyte", but who wants to sound pretentious by calling it a "tebibyte". Then again, the SI prefixes are quite unusual in themselves. Where as usually scientists are the champions and rationality and simplicity, the SI prefixes are riddled with oddities and seem down right arbitrary at times.

This debate is only likely to get more poignant as computer technology increases. This is because as computer power increases the discrepancy between powers of 1000 and powers of 1024 becomes greater and greater. For example ( here I will adopt the IEC's proposal for greater clarity )...

1 KiB = 1.024 KB

while

1 KB ~ 0.9766 KiB

The discrepancy is only about 2 or 3 percent either way. But now consider the next level ...

1 MiB ~ 1.049 MB

while

1 MB ~ 0.9537 MiB

As you can see the discrepancy has now increased to about 4 to 5 percent. Now we can jump ahead all the way to ...

1 YiB ~ 1.209 YB

while

1 YB ~ 0.8272 YiB

Here there is at least a 20 percent discrepancy ! Now let's say someone bought a 1 YB hard drive expecting a 1 YiB hard drive. Let's say that the computer company decided that in this case the labeling of YB meant literally 10^24 bytes. The costumer would be missing a fifth of their expected hard drive space ! That sounds like good reason to be upset. Would such a dispute lead to the future adoption of the IEC binary prefixes ? Who knows.

While were on the subject of YB it should be mentioned that there are limits to Moore's Law [4] . Basically Moore's Law states that computer capacity doubles every 2 years. If projected indefinitely of coarse computers can reach any processing speed and memory storage. However this does not take into account that all circuitry made up to this point involves constructing physical components. Eventually the integrated circuit will become so small that it's components will be built on the atomic scale. At this point the miniaturization process would hit a solid brick wall. This would be the limits of "conventional computer technology". Another problem that occurs even before we hit this wall is that as computer components get smaller and smaller they build greater and greater amounts of heat. Heat eventually destroys a computer, so it has always been necessary to develop cooling systems to counter this. As the integrated circuits get smaller and smaller however, it becomes increasingly difficult to keep up with the "heating problem". All of this puts definite limits on conventional computing.

One can then make reasonable estimates to the limits of conventional computing. A simple idea is to take the volume of a typical computer and fill it solid with atoms, each representing a bit. This usually leads to something in the YiB level. If this is true than there will be no need for SI prefixes above Yotta and Yobi for conventional computers.

So is that it ? The end of technological progress ? Well that depends on the feasibility of non-conventional computer technology. One popular idea is to construct a "Quantum Computer". The idea is to exploit the nature of the quantum realm itself to produce a computer which can be in several states at once. In other words it would have massive parallel processing abilities. If such a computer could be constructed it would quickly outstrip any kind of conventional computer that could be constructed, and then the sky is the limit. From what I've heard, the computing power of quantum computers wouldn't grow exponentially, but HYPER-exponentially! It has been said that such computers would be so powerful that they could in fact model the earth's entire weather system leading to incredible and unprecedented predicting power. These exciting and frightening possibilities may seem the stuff of science fiction, but there are scientists now trying to actually construct quantum computers. So far these "computers" are only a few molecules large, and can only perform arithmetic tasks. Still , if such computers are possible, we may one day see today's computers eclipsed by quantum computing. Whether larger functioning quantum computers can actually be constructed is still I think anyone's guess.

In any case, SI prefixes seem to be here to stay. Their popularity among scientists and ordinary people is surprising. For once everyone can agree on something, however arbitrary it may seem. But why does the system end with Yotta and Yocto ? Will they ever be extended ?

In the next article we explore the attempts that have been made over the years and until the present to extend and/or modify the canonical system at the time. The key to understand is that ideas, what we can call "memes", can be generated by anyone. Whether such memes are widely accepted is another story. From this point on we will be discussing "non-canonical" prefixes which have been proposed by all sorts of individuals from professionals to amateurs to pure pranksters.

NEXT>> 2.2.3 - Non-Canonical SI Prefixes

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Source Material & Footnotes :

[1] This letter is not a lower case u , but is actually the lower case greek letter mu. In html this non-ascii character can be encoded as &#181;

[2] http://en.wikipedia.org/wiki/SI : This table was constructed based on the same table appearing on this wikipedia page.

[3] http://en.wikipedia.org/wiki/Binary_prefix : Wikipedia article which goes into depth into the history of "computer prefixes", including a mention of the lastest proposals. The article also mentions a few lawsuits involving the ambiguities within the computer industry with regards to labeling.

[4] http://en.wikipedia.org/wiki/Moore's_law : Wikipedia article discussing Moore's Law.