If you were born in America you undoubtedly were taught that a billion is 10^9, a trillion is 10^12, etc. However in certain parts of the world a billion is 10^12, and a trillion is 10^18. Because of this there is some confusion with these numbers when dealing with international and historical documents. The time and place must be considered to know the correct context in which these terms are used. The system where a billion is 10^9 and a trillion 10^12 is now conventionally referred to as the "short scale", and the system where a billion is 10^12 and a trillion is 10^18 is known as the "long scale".
Why are there two systems for the same numbers? Well when these terms were first coined they were intended to have their long scale meanings. Basically a "billion" was first defined as a "million million", and a trillion as a "million billion". Essentially the latin prefix was meant to specify what power of a million the number represented. The problem is that the gap between a million and a million million is rather large. Perhaps for this reason the alternative short scale came about. Rather than define a billion as a million million, it would be more useful to have a term for a thousand million. A trillion could then be a thousand billion. This usage began with french scientists in the 1700s and then spread across Europe and competed with the long scale. Meanwhile one more refinement was made to long scale. To describe the "inbetween" terms people started using milliard for a thousand million (10^9), and a billiard for a thousand billion (10^15). This provides a way to name every power of a thousand in the long scale.
So what's the dilemma? Now we have two systems which have their adherents and neither side is willing to adopt a universal standard, hence the confusion. There is certainly no confusion in regards to terms such as a milliard. But every -illion technically has a different meaning in each system, with the sole exception of a million, which everyone agrees upon. Russ Rowlett has suggested that we should abandon both systems and adapt a greek based system instead. While such an alternative is logical, people tend to cling to what they know even if it is less than ideal. Certainly this is the case with the short scale. Although I concede it is not very logical, none the less I grew up learning it and so I'm too acustomed to it to "switch over". Although the short scale has been gaining adherents over time we can be sure that those who were raised using the long scale also do not wish to "switch over". So with either side refusing to change how can we resolve the confusion? Do we just have to know the time and place of every single document to avoid confusion? Doesn't that seem like an aweful lot of work to know the meaning of a number?
I have a suggestion to help resolve this dilemma. The key problem is NOT that we have two systems. We can have as many different ways to label numbers as we like. They key is not to have a single term for every number, but to have a single number for every term. This goes back to the identity principle. In mathematics it is fine to use multiple symbols to mean the same thing. For example we can use both "C" and "D" to mean cat. However we can not use the same symbol to mean different things. If "C" means both cat and dog, then every time I use it without specifying which I mean the reader is left to either guess or figure it out through context. So the real problem is the words "billion" and "trillion" are being used to name two different numbers. My suggestion is to create a new term to express a "million million" but that still remains relatively similar to the original term. Since the long scale already uses the alternative -illiard, I came up with a similar alternative for the even powers of a thousand, a "-illiade". This can be pronounced in different ways depending on preference, I suggest "ill-LEE-Ahd", or perhaps "ill-LEE-Aid". The long scale then becomes:
Milliade = 10^6
Milliard = 10^9
Billiade = 10^12
Billiard = 10^15
Trilliade = 10^18
Trilliard = 10^21
etc.
The pronoucation of -illiard and -illiade is significant enough to distinguish them. The idea of using an alternative ending for the even terms in the long scale is certainly one way to resolve the dilemma.
This might seem a bit obtuse, but its no more obtuse than the introduction of the terms Mebi-,Gebi-, and Tebi- for dealing with units of computer storage to avoid confusion with the SI prefixes mega-, giga-, and tera-.
Admittedly, while this move would allow everyone to remain in the scale of their choice and remove any ambiguity in terms, the long scale adherents could certainly ask why they have to be the accomedating ones. Why should they have to change their system? That is certainly a valid question, and one could certainly make the case that the long scale deserves precedence because it came first; the other camp could of coarse site the prevalence of the short scale. Truth is I designed these designations to help myself distinguish the scales. Since the short scale was already entrenched in my mind I naturally felt new terms were needed for the long scale.
Clearly I have a bias. In the interest of fairness however, lets consider the possibility that the short scale should be altered instead to distinguish it from the long scale. Going back to the idea that a billion is a thousand million, we could use the term billiate instead, the "t" standing for thousand. The short scale would then become:
milliate = 10^6
billiate = 10^9
trilliate = 10^12
etc.
So now either side can opt to alter their nomenclature, just slightly, to distinguish it from the other. But which side should budge? In fairness perhaps the terms billion and trillion should be abandoned all together, each side comprimising by adopting their respective alterations, or some other alternative naming scheme. Most likely people will not like the new terms, no matter what was used. In the computer industry there has certainly been some resistance to using mebi-,gebi-, and tebi-. But when absolute clarity is needed, especially when consumers demand it, they have gradually adopted the new terms. Likewise, we might say that while its perfectly alright for informal purposes or within ones own country to use billion, and trillion is which ever way is standard there, when absolute clarity is needed in the international market, alternatives can be used. That is probably as good a solution as any.
For mathematical purposes, I'd also like to propose the following functions and definitions. For the short scale we can use the function H(n), defined as:
H(n) = 10^(3n+3)
The "H" here stands for "Half Scale". The reason I don't use "S" for short scale is that I use S(n) for the successor function which is important in a foundational theory of arithmetic.
For the long scale we can use the function L(n) defined as:
L(n) = 10^(6n)
For the intermediate terms of the long scale we can use the function iL(n) defined as:
iL(n) = 10^(6n+3)
"iL" stands for "intermediate Long scale". "n" is the latin prefix attached to each of the endings. We can now make the following statements:
n-illiate = H(n) = 10^(3n+3)
n-illiade = L(n) = 10^(6n)
n-illiard = iL(n) = 10^(6n+3)
n-illion = H(n) or L(n) = 10^(3n+3) or 10^(6n)
We can also convert these functions into each other. This allows us to convert from the long to short scale or vice versa. To go from the long scale to the short scale we observe:
L(n) = 10^(6n) = 10^(6n-3+3) = 10^(3(2n)-3+3) = 10^(3(2n-1)+3) = H(2n-1)
iL(n) = 10^(6n+3) = 10^(3(2n)+3) = H(2n)
So to go from any -illiade in the long scale to the corresponding -illiate of the short scale we double the latin prefix and subtract one. For example:
trilliade = 3-illiade = 5-illiate = quintilliate
quadrilliade = 4-illiade = 7-illiate = septilliate
To go from any -illiard of the long scale to the corresponding -illiate of the short scale we double the latin prefix. For example:
milliard = 1-illiard = 2-illiate = billiate
billiard = 2-illiard = 4-illiate = quadrilliate
We can also convert from the short to long scale. To do so however we have to break the short scale into two cases: when the prefix is even, and when its odd. Let's convert the formulas:
Let n be an even number, 2k:
H(n) = 10^(3n+3) = 10^(6k+3) = iL(k) = iL(n/2)
Let n be an odd number, 2k-1:
H(n) = 10^(3n+3) = 10^(3(2k-1)+3) = 10^(6k-3+3) = 10^(6k) = L(k) = L((n+1)/2)
So if n is even we simply divide n by two and add the ending -illiard. If n is odd we add one then divide by two and add the ending -illiade. Some examples:
octilliate = 8-illiate = 4-illiard = quadrilliard
septilliate = 7-illiate = 4-illiade = quadrilliade
So now we can distinguish the systems and convert between them. My E.D.N. (ehanced decimal notation) can also be adapted to suit the long scale.
Notice that the way I defined EDN we can state:
(n) = H(n) = 10^(3n+3)
To adapt EDN to the long scale we can use block braces, " [ ] " and define:
[n] = L(n) = 10^(6n)
These can act as seperators between numbers, essentially acting as commas. For example, to describe the number 789,123,456,789 in EDN we can write:
789(2) 123(1) 456(0) 789
Using the long scale enhanced decimal notation (LEDN), we can write:
789,123 [1] 456,789
by using EDN and LEDN we can define almost any "-illion number" named in any system. For example Bentrizillion is defined as:
10^(6*10^(6*10^(6*10^6,000,000,000)))
We can use LEDN mixed with EDN to simplify this expression:
10^(6*10^(6*10^(6*10^6,000,000,000))) = 10^(6*10^(6*10^(6*[(2)] ))) =
10^(6*10^(6*[[(2)]] )) = 10^(6 * [[[(2)]]] ) = [[[[(2)]]]] = [(2),4]
So...
Bentrizillion = [(2),4]
Looks alot simpler now. To understand what it means start with a billion. Now go to the billionth -illiade, that's (2)-illiade. Now go to the (2)-illiadeth -illiad. That's [(2)]-illiad. Now go to the [(2)]-illiadeth -illiade. That's [[(2)]]-illiade. Now go to the [[(2)]]-lliadeth -illiad. That's a Bentrizillion. Now should that be a Bentrizilliade or a Bentrizilliard? ... :)
-- Sbiis Saibian
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