In every single case note that the unique name is either much shorter, or in a few rare instances, equal in length. Inventing 1000 original names is difficult, but what if one was to continue inventing names indefinitely? Eventually one would run into considerable difficulties. One problem is that it is sometimes difficult to make sure a name does not repeat. Also regardless of what scheme I use, I will eventually run out of 2,3,4, and 5 letter words. Eventually the length of the terms would get larger by neccessity. Interestingly this property is not limited to forming "unique names". This is a property inheritent in any attempt to label an infinite number of items. Regardless of the system, eventually the expressions and/or words will become to long for us to reasonably handle. For example , decimal notation eventually has too many digits to be convient to write. And even our counting system eventually breaks down, because of the length of time required to spell out each digit.

So now we have these unique names for the first 1000 counting numbers. How can we continue then? To continue we can use a simple scheme of word combining, just like the one we use to name numbers like "twenty one".

Firstly we can name the first 1000 numbers using the unique names, but to name higher numbers we can switch to an additive approach. Simply add the numbers being called. For example, we can continue after vhamo (1000), with

vhamo nu (1001) , vhamo ud (1002) , vhamo te (1003) , vhamo uc (1004) , vhamo iuq (1005) , etc.

Eventually the 2nd component would approach the size of the 1st...

vhamo ika (1997) , vhamo hoka (1998) , vhamo huka (1999) , ...

rather than continue with "vhamo vhamo", we can say "ud vhamo" (2000).

Then continue with the additive approach...

ud vhamo nu (2001) , ud vhamo ud (2002) , ud vhamo te (2003) , etc.

Once again, once the ending component reaches a thousand, we simply increase the "multiplier" by 1.

ud vhamo hoka (2998) , ud vhamo huka (2999) , te vhamo (3000)

and continue additively. As the multiplier increases it will eventually approach the size of the 2nd component...

ika vhamo (997,000) , hoka vhamo (998,000) , huka vhamo (999,000) ,

Under this scheme we can get as far as huka vhamo huka (999,999). The next logical step would be to use vhamo vhamo (1,000,000) which we understand as a product and not a sum.

How do we generalize these ideas so that it is possible to continue?

Basically we can break everything down into 2 simple rules...

1. If the left of two adjacent components is smaller or equal to the right, then there is a multiplication occuring between the elements.

2. If the right of two adjacent components is smaller than the left, then there is a addition occuring between the elements.

So for example ve vhamo nal = 11 x 1000 + 20 = 11,020 .

This rule also applies when the two components are equal... vhamo vhamo = 1000 x 1000 = 1,000,000

What is interesting is that these two rules are not restricted to decimal like bases. Any double component would be understood as the square of that component...

ud ud = 2 x 2 = 4

te te = 3 x 3 = 9

uc uc = 4 x 4 = 16

etc.

Therefore it is possible to construct a system based on this idea using any base from 2 to 1000. I simply choose 1000 for simplicity and ease of use.

To continue past 1,000,000 we can use the additive rule...

vhamo vhamo nu = 1000 x 1000 + 1 = 1,000,001

vhamo vhamo ud = 1000 x 1000 + 2 = 1,000,002

vhamo vhamo te = 1000 x 1000 + 3 = 1,000,003

...

vhamo vhamo hoka = 1000 x 1000 + 998 = 1,000,998

vhamo vhamo huka = 1000 x 1000 + 999 = 1,000,999

It is here that we run into our first serious problem. How would we name 1,001,000 under this scheme. If we call it vhamo vhamo vhamo, then according to the rules set out above it would be a product of all 3 terms ( ie. 1000^3), not 1,000,000 + 1,000.

The trick now is to insert an erroneous but necessary component. We can label 1,001,000 as...

vhamo vhamo nu vhamo

Under the rules we have...

vhamo vhamo nu vhamo = 1000 x 1000 + 1 x 1000 = 1,001,000

Because of the order of operations, products are essentially seperated by addition. since nu is less than the component to it's left (vhamo) the space between them is additive.

Now this scheme works very well. We can specify millions thousands and units, using this scheme...

vhamo vhamo ud vhamo = 1,002,000... vhamo vhamo huka vhamo = 1,999,000...

ud vhamo vhamo = 2,000,000... ud vhamo vhamo ika vhamo yrp = 2,997,007

Once we reach huka vhamo vhamo huka vhamo huka (999,999,999) we can label 1,000,000,000 as...

vhamo vhamo vhamo.

To continue we can simply create new powers of the base (1000) by repeating vhamo as many times as necessary. In this way it is possible to name any counting number no matter how large, ie. we can continue "indefinitely".

But this is a somewhat deceptive point. Having a scheme that can potentially be extended to infinity does not make it constructable in any practical sense. Eventually the "names" will become unweildly and inconvienent. For example...

vhamo vhamo vhamo vhamo vhamo vhamo vhamo vhamo vhamo vhamo vhamo = 1000^11

Yet this number takes a long time to say and one would also have to listen carefully in order to count the number of vhamo's to determine the magnitude. On the other hand even standard english has a short name for such a number, "decillion". So in the end, certain numbers will still be easier to say in english, then in this "unique naming scheme".