@siriusmartM to Daily Maths Challenges • 6 months ago[2024/05/03] Infinite exponentialimagemessage-square6arrow-up156arrow-down11file-text
arrow-up155arrow-down1image[2024/05/03] Infinite exponential@siriusmartM to Daily Maths Challenges • 6 months agomessage-square6file-text
minus-square@[email protected]linkfedilink9•6 months agoNow try xxx^… = 4. And then tell me why it’s not a logical crisis that you get exactly the same answer when you solve xxx^… = 2 instead.
minus-square@[email protected]linkfedilink7•edit-26 months agoThe reason is that xx… doesn’t converge to anything greater than e. Here’s what happens with sqrt(2) as x with a 12 large power tower (WolframAlpha doesn’t like power towers with roots): https://www.wolframalpha.com/input?i2d=true&i=Nest+\(91)Power[sqrt\(40)2\(41)%2C%23]%26\(44)+1\(44)+12\(93) Here’s what happens with e(1/e), the max value for x with a 500,000 large power tower: https://www.wolframalpha.com/input?i2d=true&i=Nest+\(91)Power[\(40)1.44466786\(41)%2C%23]%26\(44)+1\(44)+500000\(93) (Note: I was unable to enter that value so I approximated it. The exact result should be e) Here’s what happens with the value above +0.00001 and a mere 1023 large power tower: https://www.wolframalpha.com/input?i2d=true&i=Nest+\(91)Power[\(40)1.44467787\(41)%2C%23]%26\(44)+1\(44)+1023\(93) You can read the reasons here, I couldn’t follow for very much but maybe you can get something out of it: https://mathworld.wolfram.com/PowerTower.html
minus-square@[email protected]linkfedilink1•6 months agoWhat is weird is that xxxx… does have multiple results as for e.g. x=√2 it can yield 2, 4, and possibly more. Maybe it is due to the concept of infinity that we cannot really grasp.
![[2024/05/03] Infinite exponential](https://lemmy.world/pictrs/image/2c780fd5-54b5-4943-aa4a-5a932a7c4674.png){kind=link}
Now try xxx^… = 4. And then tell me why it’s not a logical crisis that you get exactly the same answer when you solve xxx^… = 2 instead.
The reason is that xx… doesn’t converge to anything greater than e.
Here’s what happens with sqrt(2) as x with a 12 large power tower (WolframAlpha doesn’t like power towers with roots):
https://www.wolframalpha.com/input?i2d=true&i=Nest+\(91)Power[sqrt\(40)2\(41)%2C%23]%26\(44)+1\(44)+12\(93)
Here’s what happens with e(1/e), the max value for x with a 500,000 large power tower:
https://www.wolframalpha.com/input?i2d=true&i=Nest+\(91)Power[\(40)1.44466786\(41)%2C%23]%26\(44)+1\(44)+500000\(93)
(Note: I was unable to enter that value so I approximated it. The exact result should be e)
Here’s what happens with the value above +0.00001 and a mere 1023 large power tower:
https://www.wolframalpha.com/input?i2d=true&i=Nest+\(91)Power[\(40)1.44467787\(41)%2C%23]%26\(44)+1\(44)+1023\(93)
You can read the reasons here, I couldn’t follow for very much but maybe you can get something out of it:
https://mathworld.wolfram.com/PowerTower.html
What is weird is that xxxx… does have multiple results as for e.g. x=√2 it can yield 2, 4, and possibly more. Maybe it is due to the concept of infinity that we cannot really grasp.