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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</math>
<math>y \in {\mathbb {Z}}</math>
<math>-1 \geq x \geq 1</math>
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[edit]
<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</math>
Also written as:
<math display="block">f(x, y) = \left\lfloor \frac {|x| \cdot 10^y } {10^{L-y}}\right\rfloor\mod10</math>
Where:
<math display="block">L = \lfloor\log(|x|\cdot10^y)\rfloor+1</math>
The GOF^{[1]} needs an <math display="inline">x</math> and a <math display="inline">y</math>. In this case, the <math display="inline">x</math> is the value to be decomposed. The <math display="inline">y</math> is in this case the variable that indicates which number of <math display="inline">x</math> is the target from left to right.
Example[edit]
<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</math>
Because:
<math display="block">|143| \cdot 10^2 = 14300</math>
And:
<math display="block">L = (\lfloor\log(14300)\rfloor + 1) = 5 </math>
You can write:
<math display="block">f(143, 2) = \left\lfloor \frac {14300} {10^{5-2}}\right\rfloor\mod10</math>
Because:
<math display="block">\frac {14300} {10^3} = 14.3</math>
You can write that:
<math display="block">f(143, 2) = \lfloor 14.3\rfloor\mod10 </math>
Which is equal to:
<math display="block">f(143, 2) = 14\mod10</math>
Then you get:
<math display="block">f(143, 2) = 4</math>
Explanation[edit]
In the example <math display="inline">x</math> was equal to 143 and <math display="inline">y</math> equal to 2. This meant that the second number from the left side of <math display="inline">x</math> should be the outcome.
1_{st} | 2_{nd} | 3_{rd} |
1 | 4 | 3 |
<math display="inline">L</math> became 5 because the <math display="inline">x</math> value was 5 numbers long: 14300
<math display="inline">x</math> and <math display="inline">y</math> may also be negative or higher numbers. <math display="inline">x</math> may be negative, because only the absolute value of <math display="inline">x</math> is taken and <math display="inline">y</math> may be negative or higher, because the number sequence does not end.
-1_{th} | 0_{th} | 1_{st} | 2_{nd} | 3_{rd} | 4_{th} | 5_{th} |
0 | 0 | 1 | 4 | 3 | 0 | 0 |
<math display="inline">y</math> may not be a fraction or irrational number. This is because 1.5th number of a number sequence does not exist.
1_{st} | 1.5_{th} | 2_{nd} | 2.5_{th} | 3_{rd} | 3.5_{th} |
1 | - | 4 | - | 3 | - |
The GOF^{[1]} only works if <math display="inline">x</math> is lower or equal to -1, or if <math display="inline">x</math> is higher than or equal to 1. If <math display="inline">x</math> 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</math>
<math display="inline">L</math> should be 5 in this case, but is 2. If <math display="inline">L</math> is two then this is in the denominator:
<math display="block">10^{-1} = \frac{1} {10}</math>
Then it will eventually be:
<math display="block">\left\lfloor\frac {45.3} {\frac {1} {10}} \right\rfloor\mod10 = 453\mod10 = 3</math>
And this is not the correct answer.
Therefore, the following conditions are given to the GOF:^{[1]}
<math display="block">y \in {\mathbb {Z}}</math>
<math display="block">-1 \geq x \geq 1</math>
See also[edit]
Others articles of the Topic Mathematics : List of positive integers and factors/5, Natural number, Integer, List of positive integers and factors, List of positive integers and factors 4001 to 6000, Mathematics, List of positive integers and factors/3
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References[edit]
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} GOF stands for Getal Ontledingsformule which means Number Decomposing Formula in English
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