The decimal counter is born from the necessity of finding a system of decimal metric units of wide spectrum, but as I soon saw, we can also use this as exponential notation in several expressions and mathematical operations.

Principle and foundations of the decimal counter "k":

"Any decimal metric unit, quantity, set or decimal level can be exposed in simplified or compressed way by means of a dual or bi-parametric expression formed by a base ( a ) or numeric extract , and a decimal counter "k" ak "

Procedure:

" To express a quantity with decimal counter "k", we count and fix the quantity of decimals that we want to compress, and as substitution, we put this number of decimals (+/-) next (behind) to the "k" letter (k3, k6, k-9, etc.) "

This system takes the advantage of the ordered situation of decimals in quantities to avoid the use of powers of ten (10^n) by mean of counting the number of decimals simply.

The problem arose me in August of 2010 when studying the energy of waves.
This way a tsunami is a wave of enormous dimensions and potential energy, while an electromagnetic wave is of minimum energy power.

But how we can measure and relate the energy of both waves with the same energy unit, when for example the joule is insignificant for the tsunami and too big for the electromagnetic wave.
And the solution would be: Applying a method of units of wide spectrum that embraced from the infinitely small things to the infinitely big things.
And this method would be the one of getting a form of indefinite multiples of exponential decimal units, to know, the method of decimal counter k.

Let us see: With the unit of longitude, the meter, we make decimal units that are multiples of the same one, such as decametre, hectometre, kilometre, etc.
But this method soon stops to have simple and clear expressions when it arrives to certain values such as: 10⁸ metres; 10¹¹ metres; 10¹⁷ metres, etc.

The method of decimal counter applied on decimal metric units kn m (metres), i.e. k5 metres

This method consists on applying a decimal counter to the symbol of the chosen unit k m (ka: meters) which values and names to the new resulting unit.

General use of the decimal counter

In practice, the decimal counter "k" (ak) means the number of decimals that we should apply to the base a.

As we can see, the decimal counter k allows us any quantity of integer and decimal numbers in the base a, as for instance:

Although it doesn't correspond to me the definition of the pronunciation of this notation method, I would propose the following one:
To express firstly the base (a), fallowed of the expression k (ka:) and finally the value of the decimal counter k.
In the case of having chosen the meter, it would be: base a - ka: - value of K.
For example:

As we see, in these cases the definitions "ka: thirty two metres", "ka: twenty Joules", "ka: minus thirty four Joules", etc. they serve as name of the chosen unit, just as if they were decametres, kilometres; decimetres, millimetres, picometres, etc., but with a limitless application ambit .

4.000.000.000.000.000.000.000 = 4k21 => "four ka: twenty-one"
9.000.000.000.000 = 9k12 => "nine ka: twelve".
0,000.000.000.000.000.7 = 7k-16 => "seven ka: minus sixteen"

k7 m => "ka: seven metres" => Longitude unit equivalent to 10,000.000 metres.
k6 J => "ka: six joules" => Energy unit equivalent to 1,000,000 joules.
k-20 J => "ka: minus twenty joules" => Energy unit equivalent to 0,000,000,000,000,000,000,01 joules.
k12 g => "ka: twelve grams" => Weight unit equivalent to 1.000.000.000.000 grams.
k42 Kg => "ka: forty two Kgs." => Weight unit equivalent to 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000 Kgs.
k6 years => "ka: six years" ." => Time unit equivalent to 1,000,000 years

In the dimensional or physical parameters (e.g. meter, square meter, cubic meter) the decimal counter follows a logical rule considering the structural reality of the units firstly, to which we can apply later its corresponding multiples.

This way, we firstly look at the consistency of the unit (for example the square meter), and later on we apply its decimal multiples.

* The flexibility of the decimal counter allows us also the use of any other classic unit as for example:

In the mathematical operations, the decimal counter simplifies a lot the expressions and therefore I believe that it could also be very useful sometimes.
* But let remember an important characteristic:

-- The notation with decimal counter is simply a compressed number (ak)
-- While the scientific notation is a set of operations (a x 10^n)

(12 x 10^12) x ( 6 x 10^8) x (5 x 10^7) = 36 x 10^28
12k12 x 6k8 x 5k7 = 36k28

( 12 x 10^12) : ( 8 x 10^6 ) = 2 x 10^4.
12k12 : 6k8 = 2k4

* In multiplication/division the base or numeric extracts are multiplied/divided,
And the decimal counters are summed/subtracted.

Additions and subtractions

To add and subtract quantities with decimal counter we can make it by means of the equalization to the same counter k.

For example:

7k17 + 8k16 +12k15

Then we equal them, as for example to k15

..7k17 = 700k15
..8k16 = ..80k15
12k15 = ..12k15
------------------------
...............792k15

Powers and roots

Powers with decimal counter akⁿ ).

The solution or result of a power with decimal counter is an expression with decimal counter and with power exponent, such that:
The decimal exponent is the product of the decimal counter by the exponent; and the new base would be the power of the initial base.

Roots with decimal counter.

For the resolution of roots with decimal counter we will follow the next phases:
1.- In the first place we make to be multiple the decimal counter regarding to the root number.
2.- Subsequently we solve by means of two steps, which will give us as solution the resulting decimal expression:
A----We divide the decimal counter by the root and we put it as resulting decimal counter.
B----We solve the root of the initial base to obtain us the resulting base.

Extensive variety of concepts.

We can also use the decimal counter in variety of symbols and concepts, as for instance in the mathematical set:

Summarizing, in one hand the decimal counter helps us to express big (or extremely small) numeric quantities in simple or easy way.
And on other hand, it allows us to use a limitless set of units of any type.

In the practice, if we put the decimal counter on a number, this will take the exponential value that the decimal counter has; and if we apply it to a symbol, this will become another unit with the level that the decimal counter has.

The decimal counter can be used with any other base, apart from the decimal one (10)
For example,

The decimal counter has a direct relation with the method of floating points since, by itself, the decimal notation (k) marks us the displacement toward the right ( + ) or toward the left ( - ) that we must give to the point that separates the integer part from the decimal one in the base ( a ) to get the initial extended number.

* To understand us better with the application of the decimal counter, to the initial number we will call it "extended number" and to the number with the decimal application will call it "compressed number".

The following explanation and discussion is for those who like to know the antecedents and reasons of this proposal.

At the beginning of August of 2010 I was studying and revising the different metric units and their corresponding multiples when I notice that for any unit type, for example the meter, we establish a letter or symbol to designate it and a value-pattern to value it.

Now then, as we need multiples and divisors of any unit-pattern and we use in mathematics the decimal systems, because we conceive the multiples and divisors following this decimal system.
This way, the multiples of the meter would have:
10 decametre, 100 hectometre, 1000 kilometre, 10.000 miriametre, etc.
And the divisors:
1/10 decimetre, 1/100 centimetre, 1/1.000 millimetre, etc.

Then, to designate these multiples we use a relative prefix to their first written letters:
Dm--decametre; hm--hectometre; km--kilometre, etc.
Dcm--decimetre; cm--centimetre; mm--millimetre etc.

And it is here where the first problem arose me, since the applicable letters are scarce and contrarily the numbers, and therefore the multiples, are infinite.

But also in the previous or "classic" form of representation and expression of multiples and divisors of metric units, another problem or complication exists: we should know the numeric value that we have given to each letter.
Therefore we must translate the letters with which we designate to the metric units in numeric values to take conscience of real value of the unit that we are valuing.
And due to these units and their representative letters are diverse, because to remember their real values can be complicated and can induce us to errors.

k--kilo, M--mega, G--giga, T--tera, P--peta, E--exa, Z--zetta, Y--yotta.,

So the logical question arises immediately:
What reason exists to use letters that produce us so much confusion?
Then, we could forget the letters and to put numeric decimal values directly as prefix of the chosen metric unit.

k__kilo,---M__mega,--- G__giga,--- T__tera, ---P__peta,--- E__exa, ---Z__zetta,--- Y__yotta.,
k3----------- k6----------- k9----------- k12--------- k15--------- k18--------- k21--------- k24--------- k27--------- k32---------............... Kn

This way the number of multiples and divisors it is limitless and there is not confusion possible of value since each number indicates us the value of the unit.
So, k9 m "ka:nine meters" means an unit with value in meters equal to 1 followed by 9 zeros.

As we have seen before, this form of mathematical expression is also very useful and simple to make mathematical operations.

With the decimal counter "k" we can take clear conscience of the level concept through the fourth dimension of space.
For example:
If we choose our level, (k meters), then we can go measuring a road meter after meter, or simply making footing as in the inferior drawing.
Rising of level to k5 meters, we can go measuring or jumping from city to city.
With the level k6 meters we can go displacing us or jumping from country to country.
If now we choose the level k11 meters, we can go jumping from planet to planet.
If we consider, adopt and situate mentally on the level of the unit k22 (meters) then we could think and see us measuring k22 after k22, or simply jumping and running among galaxies.
If contrarily, we adopt the level of the k-11 (meters), then we are measuring, moving and jumping among electrons.
But if we go to low levels as the k-19 (meters), we can see ourselves measuring or making footing on the surface of an electron.
Say, with the decimal counter we can situate (mathematically) on any level of space in an easy and simple way.

And to finish let us remember this difference:

Thank you. Ferman. 2010-8-26