Fun With Math-Superpowerition: Inventing a Super Power Operation -- Decimal 0 < a < 1 ✱ Decimal 0 < b

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Decimal 0 < a < 1 ✱ Decimal 0 < b < 1

Now let's let a be a decimal number between 0 and 1 and b be a decimal value between 0 and 1 using steps of 0.25.

Let's try out some variables for a using the formula a ^ (a ^ (b - 1)).

0.00 ✱ 0.00 = 0.00 ^ (0.00 ^ (0.00 - 1)) = 0.00 ^ (0.00 ^ (-1.00)) = 0.00 ^ 0.00 = undefined

0.25 ✱ 0.00 = 0.25 ^ (0.25 ^ (0.00 - 1)) = 0.25 ^ (0.25 ^ (-1.00)) = 0.25 ^ 4 ≈ 0.0039062

0.50 ✱ 0.00 = 0.50 ^ (0.50 ^ (0.00 - 1)) = 0.50 ^ (0.50 ^ (-1.00)) = 0.50 ^ 2 = 0.25

0.75 ✱ 0.00 = 0.75 ^ (0.75 ^ (0.00 - 1)) = 0.75 ^ (0.75 ^ (-1.00)) ≈ 0.75 ^ 1.3333333 ≈ 0.6814202

1.00 ✱ 0.00 = 1.00 ^ (1.00 ^ (0.00 - 1)) = 1.00 ^ (1.00 ^ (-1.00)) = 1.00 ^ 1.00 = 1.00

0.00 ✱ 0.25 = 0.00 ^ (0.00 ^ (0.25 - 1)) = 0.00 ^ (0.00 ^ (-0.75)) = 0.00 ^ 0.00 = undefined

0.25 ✱ 0.25 = 0.25 ^ (0.25 ^ (0.25 - 1)) = 0.25 ^ (0.25 ^ (-0.75)) ≈ 0.25 ^ 2.8284271 ≈ 0.0198206

0.50 ✱ 0.25 = 0.50 ^ (0.50 ^ (0.25 - 1)) = 0.50 ^ (0.50 ^ (-0.75)) ≈ 0.50 ^ 1.6817928 ≈ 0.3116950

0.75 ✱ 0.25 = 0.75 ^ (0.75 ^ (0.25 - 1)) = 0.75 ^ (0.75 ^ (-0.75)) ≈ 0.75 ^ 1.2408064 ≈ 0.6998020

1.00 ✱ 0.25 = 1.00 ^ (1.00 ^ (0.25 - 1)) = 1.00 ^ (1.00 ^ (-0.75)) = 1.00 ^ 1.00 = 1.00

0.00 ✱ 0.50 = 0.00 ^ (0.00 ^ (0.50 - 1)) = 0.00 ^ (0.00 ^ (-0.50)) = 0.00 ^ 0.00 = undefined

0.25 ✱ 0.50 = 0.25 ^ (0.25 ^ (0.50 - 1)) = 0.25 ^ (0.25 ^ (-0.50)) = 0.25 ^ 2 = 0.0625

0.50 ✱ 0.50 = 0.50 ^ (0.50 ^ (0.50 - 1)) = 0.50 ^ (0.50 ^ (-0.50)) ≈ 0.50 ^ 1.4142135 ≈ 0.3752142

0.75 ✱ 0.50 = 0.75 ^ (0.75 ^ (0.50 - 1)) = 0.75 ^ (0.75 ^ (-0.50)) ≈ 0.75 ^ 1.1547005 ≈ 0.7173534

1.00 ✱ 0.50 = 1.00 ^ (1.00 ^ (0.50 - 1)) = 1.00 ^ (1.00 ^ (-0.50)) = 1.00 ^ 1.00 = 1.00

0.00 ✱ 0.75 = 0.00 ^ (0.00 ^ (0.75 - 1)) = 0.00 ^ (0.00 ^ (-0.25)) = 0.00 ^ 0.00 = undefined

0.25 ✱ 0.75 = 0.25 ^ (0.25 ^ (0.75 - 1)) = 0.25 ^ (0.25 ^ (-0.25)) ≈ 0.25 ^ 1.4142135 ≈ 0.1407857

0.50 ✱ 0.75 = 0.50 ^ (0.50 ^ (0.75 - 1)) = 0.50 ^ (0.50 ^ (-0.25)) ≈ 0.50 ^ 1.1892071 ≈ 0.4385438

0.75 ✱ 0.75 = 0.75 ^ (0.75 ^ (0.75 - 1)) = 0.75 ^ (0.75 ^ (-0.25)) ≈ 0.75 ^ 1.0745699 ≈ 0.7340820

1.00 ✱ 0.75 = 1.00 ^ (1.00 ^ (0.75 - 1)) = 1.00 ^ (1.00 ^ (-0.25)) = 1.00 ^ 1.00 = 1.00

0.00 ✱ 1.00 = 0.00 ^ (0.00 ^ (1.00 - 1)) = 0.00 ^ (0.00 ^ ( 0.00)) = 0.00 ^ undefined = undefined

0.25 ✱ 1.00 = 0.25 ^ (0.25 ^ (1.00 - 1)) = 0.25 ^ (0.25 ^ ( 0.00)) = 0.25 ^ 1.00 = 0.25

0.50 ✱ 1.00 = 0.50 ^ (0.50 ^ (1.00 - 1)) = 0.50 ^ (0.50 ^ ( 0.00)) = 0.50 ^ 1.00 = 0.50

0.75 ✱ 1.00 = 0.75 ^ (0.75 ^ (1.00 - 1)) = 0.75 ^ (0.75 ^ ( 0.00)) = 0.75 ^ 1.00 = 0.75

1.00 ✱ 1.00 = 1.00 ^ (1.00 ^ (1.00 - 1)) = 1.00 ^ (1.00 ^ ( 0.00)) = 1.00 ^ 1.00 = 1.00

Once again, 0.00 ✱ [any number] = undefined