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# The Number Decomposing Formula or GOF(Getal Ontledingsformule)

Category:Mathematics The Number Decomposing Formula or GOF[1] is a formula used to decompose a value in a way that is comparable, just as we humans do.

<math display="block">f(x, y) = \left\lfloor \frac {|x| \cdot 10^y } {10^{(\lfloor\log(|x|\cdot10^y)\rfloor+1)-y}}\right\rfloor\mod10[/itex] $y \in {\mathbb {Z}}$

$-1 \geq x \geq 1$

The GOF[1] was made by a Dutch mathematician called Kasper van Maasdam. The GOF[1] gives the possibility, when used in a computer program, to have a computer decompose a value in a way that is comparable, just as we humans do.

## Formula

<math display="block">f(x, y) = \left\lfloor \frac {|x| \cdot 10^y } {10^{(\lfloor\log(|x|\cdot10^y)\rfloor+1)-y}}\right\rfloor\mod10[/itex]

Also written as:

<math display="block">f(x, y) = \left\lfloor \frac {|x| \cdot 10^y } {10^{L-y}}\right\rfloor\mod10[/itex]

Where:

<math display="block">L = \lfloor\log(|x|\cdot10^y)\rfloor+1[/itex]

The GOF[1] needs an <math display="inline">x[/itex] ​​and a <math display="inline">y[/itex]. In this case, the <math display="inline">x[/itex] is the value to be decomposed. The <math display="inline">y[/itex] is in this case the variable that indicates which number of <math display="inline">x[/itex] is the target from left to right.

## Example

<math display="block">f(143, 2) = \left\lfloor \frac {|143| \cdot 10^2 } {10^{(\lfloor\log(|143|\cdot10^2)\rfloor+1)-2}}\right\rfloor\mod10[/itex]

Because:

<math display="block">|143| \cdot 10^2 = 14300[/itex]

And:

<math display="block">L = (\lfloor\log(14300)\rfloor + 1) = 5 [/itex]

You can write:

<math display="block">f(143, 2) = \left\lfloor \frac {14300} {10^{5-2}}\right\rfloor\mod10[/itex]

Because:

<math display="block">\frac {14300} {10^3} = 14.3[/itex]

You can write that:

<math display="block">f(143, 2) = \lfloor 14.3\rfloor\mod10 [/itex]

Which is equal to:

<math display="block">f(143, 2) = 14\mod10[/itex]

Then you get:

<math display="block">f(143, 2) = 4[/itex]

## Explanation

In the example <math display="inline">x[/itex] was equal to 143 and <math display="inline">y[/itex] equal to 2. This meant that the second number from the left side of <math display="inline">x[/itex] should be the outcome.

 1st 2nd 3rd 1 4 3

<math display="inline">L[/itex] became 5 because the <math display="inline">x[/itex] value was 5 numbers long: 14300
<math display="inline">x[/itex] and <math display="inline">y[/itex] may also be negative or higher numbers. <math display="inline">x[/itex] may be negative, because only the absolute value of <math display="inline">x[/itex] is taken and <math display="inline">y[/itex] may be negative or higher, because the number sequence does not end.

 -1th 0th 1st 2nd 3rd 4th 5th 0 0 1 4 3 0 0

<math display="inline">y[/itex] may not be a fraction or irrational number. This is because 1.5th number of a number sequence does not exist.

 1st 1.5th 2nd 2.5th 3rd 3.5th 1 - 4 - 3 -

The GOF[1] only works if <math display="inline">x[/itex] is lower or equal to -1, or if <math display="inline">x[/itex] is higher than or equal to 1. If <math display="inline">x[/itex] has a value such as 0.0453 and you want to know the third number (4) then a wrong value will come out. This is because:

<math display="block">0.0453 \cdot 10^3 = 45.3[/itex]

<math display="inline">L[/itex] should be 5 in this case, but is 2. If <math display="inline">L[/itex] is two then this is in the denominator:

<math display="block">10^{-1} = \frac{1} {10}[/itex]

Then it will eventually be:

<math display="block">\left\lfloor\frac {45.3} {\frac {1} {10}} \right\rfloor\mod10 = 453\mod10 = 3[/itex]

And this is not the correct answer. Therefore, the following conditions are given to the GOF:[1]

<math display="block">y \in {\mathbb {Z}}[/itex]
<math display="block">-1 \geq x \geq 1[/itex]