The Number Decomposing Formula or GOF(Getal Ontledingsformule)

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

$$f(x,y)=\left\lfloor {\frac {|x|\cdot 10^{y}}{10^{(\lfloor \log(|x|\cdot 10^{y})\rfloor +1)-y}}}\right\rfloor \mod 10$$

$y\in {\mathbb {Z} }$

$-1\geq x\geq 1$

The GOF was made by a Dutch mathematician called Kasper van Maasdam. The GOF 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

$$f(x,y)=\left\lfloor {\frac {|x|\cdot 10^{y}}{10^{(\lfloor \log(|x|\cdot 10^{y})\rfloor +1)-y}}}\right\rfloor \mod 10$$

Also written as:

$$f(x,y)=\left\lfloor {\frac {|x|\cdot 10^{y}}{10^{L-y}}}\right\rfloor \mod 10$$

Where:

$$L=\lfloor \log(|x|\cdot 10^{y})\rfloor +1$$

The GOF needs an ${\textstyle x}$ ​​and a ${\textstyle y}$. In this case, the ${\textstyle x}$ is the value to be decomposed. The ${\textstyle y}$ is in this case the variable that indicates which number of ${\textstyle x}$ is the target from left to right.

Example

$$f(143,2)=\left\lfloor {\frac {|143|\cdot 10^{2}}{10^{(\lfloor \log(|143|\cdot 10^{2})\rfloor +1)-2}}}\right\rfloor \mod 10$$

Because:

$$|143|\cdot 10^{2}=14300$$

And:

$$L=(\lfloor \log(14300)\rfloor +1)=5$$

You can write:

$$f(143,2)=\left\lfloor {\frac {14300}{10^{5-2}}}\right\rfloor \mod 10$$

Because:

$${\frac {14300}{10^{3}}}=14.3$$

You can write that:

$$f(143,2)=\lfloor 14.3\rfloor \mod 10$$

Which is equal to:

$$f(143,2)=14\mod 10$$

Then you get:

$$f(143,2)=4$$

Explanation

In the example ${\textstyle x}$ was equal to 143 and ${\textstyle y}$ equal to 2. This meant that the second number from the left side of ${\textstyle x}$ should be the outcome.

1st	2nd	3rd
1	4	3

${\textstyle L}$ became 5 because the ${\textstyle x}$ value was 5 numbers long: 14300
${\textstyle x}$ and ${\textstyle y}$ may also be negative or higher numbers. ${\textstyle x}$ may be negative, because only the absolute value of ${\textstyle x}$ is taken and ${\textstyle y}$ 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

${\textstyle y}$ 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 only works if ${\textstyle x}$ is lower or equal to -1, or if ${\textstyle x}$ is higher than or equal to 1. If ${\textstyle x}$ 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:

$$0.0453\cdot 10^{3}=45.3$$

${\textstyle L}$ should be 5 in this case, but is 2. If ${\textstyle L}$ is two then this is in the denominator:

$$10^{-1}={\frac {1}{10}}$$

Then it will eventually be:

$$\left\lfloor {\frac {45.3}{\frac {1}{10}}}\right\rfloor \mod 10=453\mod 10=3$$

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

$$y\in {\mathbb {Z} }$$

$$-1\geq x\geq 1$$

See also

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• Computer Programming
• Computer science
• Number

References

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