| mean of infinity
The geometric mean (g) of a set of n positive data points of a given observable is the nth root of their product. But for the arithmetic mean (a), it is the sum divided by n. The first mean uses the unary operator of root extracting and the binary operator of multiplication. The second mean uses the binary operators of both addition and division.
If all the data points are all unity then the product of n unity is still unity and the nth root of unity is also unity. On the other hand, the sum of n unity is n and n divided by n is simply unity. However, if the order of operations is reversed: extract root first then multiply, divide then add as n tends to infinity then these reversed processes can give different means. For unity data points, the answer is the same for finding the geometric mean but for the arithmetic mean the answer is zero if and only if unity divided by infinity is defined as the value of zero. Furthermore, it can be defined that any number greater than zero but less than infinity where and when divided by infinity would still give zero. Therefore, the arithmetic mean of infinity is zero if and only if division is done before addition as presumed by the principle of superposition for all intensive variables. Moreover, the nth root of any number greater than zero but less than unity as occurs in the theory of probability gives an answer that approaches unity while their product approaches zero. Therefore, only the geometric means can satisfy the law of probability for both dependence and independence events of reality.
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Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
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Linear Mode
