Now, let's use the operation for decimals, fractions, irrational numbers, constants, pi and more.
On this page, let's make the a in "a ✱ b" an integer > 1 and the b a decimal number > 1.
Let's try out some variables for a using the formula a ^ (a ^ (b - 1)). Let's let b go from 1 to 2 in 0.25 steps.
Let's let a = 2.
2 ✱ 1.00 = 2 ^ (2 ^ (1.00 - 1)) = 2 ^ (2 ^ (0.00)) = 2 ^ 1 = 2 |
2 ✱ 1.25 = 2 ^ (2 ^ (1.25 - 1)) = 2 ^ (2 ^ (0.25)) ≈ 2 ^ 1.1892071 ≈ 2.2802738 |
2 ✱ 1.50 = 2 ^ (2 ^ (1.50 - 1)) = 2 ^ (2 ^ (0.50)) ≈ 2 ^ 1.4142135 ≈ 2.6651440 |
2 ✱ 1.75 = 2 ^ (2 ^ (1.75 - 1)) = 2 ^ (2 ^ (0.75)) ≈ 2 ^ 1.6817928 ≈ 3.2082616 |
2 ✱ 2.00 = 2 ^ (2 ^ (2.00 - 1)) = 2 ^ (2 ^ (1.00)) = 2 ^ 2 = 4 |
It seems that when b is a decimal number, the answers are irrational numbers in many cases. So, let's use this formula instead a√(a(ab)) or (a ^ (a ^ b)) ^ (1 / a)
Let's let a = 4.
2 ✱ 1.00 = (2 ^ (2 ^ 1.00)) ^ (1 / 2) = (2 ^ (2.0000000)) ^ (1/2) = (4.0000000) ^ (1/2) = 2 |
2 ✱ 1.25 = (2 ^ (2 ^ 1.25)) ^ (1 / 2) = (2 ^ (2.3784142)) ^ (1/2) ≈ (5.1996488) ^ (1/2) ≈ 2.2802738 |
2 ✱ 1.50 = (2 ^ (2 ^ 1.50)) ^ (1 / 2) = (2 ^ (2.8284271)) ^ (1/2) ≈ (7.1029921) ^ (1/2) ≈ 2.6651440 |
2 ✱ 1.75 = (2 ^ (2 ^ 1.75)) ^ (1 / 2) = (2 ^ (3.3635856)) ^ (1/2) ≈ (10.292957) ^ (1/2) ≈ 3.2082616 |
2 ✱ 2.00 = (2 ^ (2 ^ 2.00)) ^ (1 / 2) = (2 ^ (4.0000000)) ^ (1/2) = (16.0000000) ^ (1/2) = 4 |
The answers using the two formulas are the same.
Let's let a = 4 and b = go from 1 to 2 in 0.25 steps, using the formula a ^ (a ^ (b - 1)).
4 ✱ 1.00 = 4 ^ (4 ^ (1.00 - 1)) = 4 ^ (4 ^ (0.00)) = 4 ^ 1 = 4 |
4 ✱ 1.25 = 4 ^ (4 ^ (1.25 - 1)) = 4 ^ (4 ^ (0.25)) ≈ 4 ^ 1.4142135 ≈ 7.1029926 |
4 ✱ 1.50 = 4 ^ (4 ^ (1.50 - 1)) = 4 ^ (4 ^ (0.50)) = 4 ^ 2 = 16 |
4 ✱ 1.75 = 4 ^ (4 ^ (1.75 - 1)) = 4 ^ (4 ^ (0.75)) ≈ 4 ^ 2.8284270 ≈ 50.4525051 |
4 ✱ 2.00 = 4 ^ (4 ^ (2.00 - 1)) = 4 ^ (4 ^ (1.00)) = 4 ^ 4 = 256 |
You may have noticed that an exception was found for the expression 4 ✱ 1.50 in which the answer is a rational number of 16.
Let's let a = 2 and b = go from 2 to 3 in 0.25 steps, using the formula a ^ (a ^ (b - 1)).
2 ✱ 2.00 = 2 ^ (2 ^ (2.00 - 1)) = 2 ^ (2 ^ (1.00)) = 2 ^ 2 = 4 |
2 ✱ 1.25 = 2 ^ (2 ^ (2.25 - 1)) = 2 ^ (2 ^ (1.25)) ≈ 2 ^ 2.3784142 ≈ 5.1996488 |
2 ✱ 1.50 = 2 ^ (2 ^ (2.50 - 1)) = 2 ^ (2 ^ (1.50)) ≈ 2 ^ 2.8284271 ≈ 7.1029931 |
2 ✱ 1.75 = 2 ^ (2 ^ (2.75 - 1)) = 2 ^ (2 ^ (1.75)) ≈ 2 ^ 3.3645856 ≈ 10.3000940 |
2 ✱ 2.00 = 2 ^ (2 ^ (3.00 - 1)) = 2 ^ (2 ^ (2.00)) = 2 ^ 4 = 16 |
We discover that 4 ✱ 1.00 = 2 ✱ 2.00, and 4 ✱ 1.50 = 2 ✱ 2.00, but due to approximation, 4 ✱ 1.25 is very close to equalling 2 ✱ 1.50, but since I'm using seven decimal places, and due to the nature of irrational numbers, they might indeed be equal.
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