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I was looking at the different ways the operations +, *, and exponentiation can work on three numbers x, y, and z. I found a weird pattern when the second operation performed is exponentiation. These are the expressions:
(x+y)^z \ x^{(y+z)} \ (x \cdot y)^z \ x^{(y \cdot z)} \ (x^y)^z \ x^{(y^z)}
Notice how I arranged them in a natural way, where the first operation(inside the parantheses) is (+,+,*,*,^,^), and the second operation, exponentiation, is carried out on the (R,L,R,L,R,L) of the parantheses. Now look at the pattern:
(x+y)^z \ \ \ \ \ \ \ \ \ x^{(y+z)} \ \ \ \ \ \ \ \ \ (x \cdot y)^z \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^{(y \cdot z)} \ \ \ \ \ \ \ \ \ (x^y)^z \ \ \ \ \ \ \ \ \ x^{(y^z)}
largest ...<- identical ->... | ...<- indentical ->... largest
written ... written ... | ... value ..... value
formula ... formula
I'm sorry if this doesn't format right, but I'll explain what it means. (x+y)^z has the largest identity expression, in terms of the size of the written formula: the binomial theorem. x^{(y+z)} and (x \cdot y)^z are equal to x^y \cdot x^z and x^z \cdot y^z respectively, so the shape of their written formulas are identical. x^{(y \cdot z)} is equal in value to (x^y)^z. And finally, x^{(y^z)} has the largest value, for x,y,z>>1.
This seems like a very bizarre link between the "man-made" (sort of) written formulas and the "completely natural" values of these expressions. Is there anything to this, or is it just a coincidence?
(x+y)^z \ x^{(y+z)} \ (x \cdot y)^z \ x^{(y \cdot z)} \ (x^y)^z \ x^{(y^z)}
Notice how I arranged them in a natural way, where the first operation(inside the parantheses) is (+,+,*,*,^,^), and the second operation, exponentiation, is carried out on the (R,L,R,L,R,L) of the parantheses. Now look at the pattern:
(x+y)^z \ \ \ \ \ \ \ \ \ x^{(y+z)} \ \ \ \ \ \ \ \ \ (x \cdot y)^z \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^{(y \cdot z)} \ \ \ \ \ \ \ \ \ (x^y)^z \ \ \ \ \ \ \ \ \ x^{(y^z)}
largest ...<- identical ->... | ...<- indentical ->... largest
written ... written ... | ... value ..... value
formula ... formula
I'm sorry if this doesn't format right, but I'll explain what it means. (x+y)^z has the largest identity expression, in terms of the size of the written formula: the binomial theorem. x^{(y+z)} and (x \cdot y)^z are equal to x^y \cdot x^z and x^z \cdot y^z respectively, so the shape of their written formulas are identical. x^{(y \cdot z)} is equal in value to (x^y)^z. And finally, x^{(y^z)} has the largest value, for x,y,z>>1.
This seems like a very bizarre link between the "man-made" (sort of) written formulas and the "completely natural" values of these expressions. Is there anything to this, or is it just a coincidence?
