The decimal counter "k" [HOTLIST]
Compressed numbers with "k"
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The decimal counter "k" [HOTLIST]
System of decimal units and levels, as well as decimal notation for mathe operations
** Note: We will use here the text form mainly ( ak ) due to it is of most general application

Preamble

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.

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Principle and foundations of the decimal counter "k": [HOTLIST]

"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 "

The base a contains the extract of the numeric value.
The decimal counter k expresses the number of deduced or compressed decimals from the initial expression.

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.

Example for text way

2,300,000,000-----> 2.3k9 --- Where k9 is the mumber of compressed decimals
0.000000035 ------> 35k-9

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Development

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.

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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.

For example, the meter in longitudinal units (k m)

----- divisors -------
.etc.
Femtometre = k-15 meters----read as --------k, minus fifteen meters.
Angstrom = k-10 metres------ read as ------ k, minus ten metres.
Nanometre = k-9 meters ----- read as ------- k, minus nine meters.
Millimetre = k-3 meters ----- read as -------- k, minus three meters.
Centimeter = k-2 meters ----- read as -------k, minus two meters.
Decimetre = k-1 meters ------- read as ------k, minus one meter
------- multiples---
Decametre = k1 meters -------- read as -------k, one meters
Hectometre = k2 meters ------- read as --------k, two meters.
Kilometre = k3 meters --------- read as -------k, three meters
A light-year = k16 metres (approx.)----------------- read as -------------- k sixteen metres.
Unit equivalent to our galaxy diametre = k21 metres----------read as -------k, twenty one metres.
Known Universe as unit = k27 metres ------------------read as------ -----------k, twenty seven metres.
.etc.

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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.

For example:

12,85k12 = 12850000000000.
12,85k4 = 128500.
12,85k-4 = 0.001285.
12,85k-12 = 0.00000000001285.

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

12.85k12 = 12850000000000.
1.285k13 = 12850000000000.
0.1285k14 = 12850000000000.
1285k10 = 12850000000000.

Nomenclature

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:

k12 m. = ka twelve metres = unit of 1,000,000,000,000 metres. ( Currently Terametre)
k15 m. = ka fifteen metres = unit of 1,000,000,000,000,000 metres (Currently Petametre)
K13 m. = ka thirteen metres = Unit of 10,000,000,000,000 metres (no name currently)
k16 m. = ka sixteen metres = Unit of 10,000,000,000.000,000 metres (no name currently)
k8 m. = ka eight metres = Unit of 100,000,000 metres (no name currentlyl)
k17 m. = ka seventeen metres = Unit of 100,000,000,000,000,000 metres (no name currentlyl)

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 .

Examples of nomenclature:
In quantities

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"

In decimal metric units.

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

Dimensional or physical parameters

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.

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

k5m = Unit of 100,000 metres
k5m^2 = Unit of 100,000 square metres
k5m^3 = Unit of 100,000 cubic metres.
k5 hm^3 = unit of 100,000 cubic hectometer.
K3 degrees C = unit of 1000 C degrees
K6 Tn. = unit of 1,000,000 tons. etc.

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

The kilogram ---- k3 kg = Unit of 1,000 kg.
The kilometre ---- k3 m = Unit of 1,000 km.
The square kilometre ---- K6 m^2 = Unit of 1,000,000 square kilometres.);
The cubic hectometre ---- k4 hm^3 = Unit of 10,000 cubic hectometres ), etc.

In mathematical operations:

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)

Let us do some comparisons with the current method (scientific notation) and with the decimal counter.

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

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( 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.

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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

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2k3 + 625

Equal them to k3

2k3 = 2.......... k3
625 = 0,625.... k3
----------------------------
.........2,625k3 = 2625

Powers and roots

Powers with decimal counter akn ).

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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.

Examples:

3k2^3 = 27k6
4k2^4 = 256k8

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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:

k2 Ships = Fleet of 100 ships.
k3 Birds = Goup or flock formed by 1,000 birds.
k4 Trees = Group of trees or forest formed by 10,000 trees.
k11 Stars = Group of stars (or galaxy) formed by 100,000,000,000 stars.

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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.

Use of the decimal counter in other bases

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

Hexadecimal

2Ak2 * 2Bk3 = (2A*2B)k5 = 70Ek5 = 70E00000

2Ak-2 * 2Bk-3 = (2A*2B)k-5 = 70Ek-5 = 0.0070E

Octal

12k3 * 7k2 = 106K5 = 10600000

12k-3 * 7k-2 = 106K-5 = 0.00106

Binary

11k2 * 11k3 = 1001k5 = 100100000

11k-2 * 11k-3 = 1001k-5 = 0.01001

Floating point

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.

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* 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".

Discussion

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.

Level Characteristic.

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.

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And to finish let us remember this difference:

* 1k4 is a compressed number, equal to 1 followed by 4 zeros = 10,000. (1k4 = compressed number)

* Whereas 1 x 10^4 are two operations from which we should extract its result, which also is 10,000. (1 x 10^4 = group of operations)

In computation we can simply putting the letter E, but it remains being the something "one multiplied by ten to the power E "

Thank you. Ferman. 2010-8-26

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