Originally posted by sonhouseI'm not actually sure why i^i=e^(-pi/2) was considered paradoxical by Peirce. Perhaps because we see here an imaginary exponentiated by an imaginary, and the result is a real exponentiated by a real. "Paradoxical" is being used here to mean "unexpected" or "strange," I think, as it often is in everyday language.
the paradox being i squared we define as being minus one? i being the square root of minus one.
The post that was quoted here has been removedActually, no, nobody thought that was a paradox. What seems paradoxical is the concept that an object can traverse the infinite sequence 1/2 + 1/4 + 1/8 + ... and yet later proceed to further locations. The paradox is that it appears that the object has got to infinity and beyond.
Originally posted by SoothfastIt's possible to represent i as a two by two matrix, so in that representation the formula i^i has a matrix raised to the power of a matrix, which probably is paradoxical.
I'm not actually sure why i^i=e^(-pi/2) was considered paradoxical by Peirce. Perhaps because we see here an imaginary exponentiated by an imaginary, and the result is a real exponentiated by a real. "Paradoxical" is being used here to mean "unexpected" or "strange," I think, as it often is in everyday language.
Originally posted by DeepThoughtYes, though it's much more common to formally regard complex numbers as ordered pairs: a+ib = (a,b). A plane is the natural habitat of the complex numbers, as opposed to presenting them as some subspace of a 4-dimensional space. Then the real numbers are simply identified with ordered pairs having second coordinate equal to zero: (a,0) = a. I'm sure in Peirce's time this was being done. I'm not sure whether two-by-two matrix representations were being entertained back then. It's possible. Looking around, it seems that Arthur Cayley may have conceived of the idea in 1858.
It's possible to represent i as a two by two matrix, so in that representation the formula i^i has a matrix raised to the power of a matrix, which probably is paradoxical.
Originally posted by twhiteheadThe problem I have with 1/2 + 1/4 + 1/8 + ... = 1 is that it appears to me that on the left hand side we have infinity whereas on the right hand side that same infinity is actually given a number, namely 1. And as far as I know it is considered wrong to treat infinity as a number. I also remember reading some thread(s) in the spirituality forum of this website where that was stated (maybe even by you), but it was some time ago and don’t remember which thread it was.
Actually, no, nobody thought that was a paradox. What seems paradoxical is the concept that an object can traverse the infinite sequence 1/2 + 1/4 + 1/8 + ... and yet later proceed to further locations. The paradox is that it appears that the object has got to infinity and beyond.
My answer to what is the sum of 1/2 + 1/4 + 1/8 + ... would be infinitely close to 1.
Originally posted by Great King RatYou say "infinitely close to 1[/i]" and not exact 1, right?
The problem I have with 1/2 + 1/4 + 1/8 + ... = 1 is that it appears to me that on the left hand side we have infinity whereas on the right hand side that same infinity is actually given a number, namely 1. And as far as I know it is considered wrong to treat infinity as a number. I also remember reading some thread(s) in the spirituality forum of thi ...[text shortened]...
My answer to what is the sum of 1/2 + 1/4 + 1/8 + ... would be infinitely close to 1.
As this is mathematics, then this would be provable. Please give us the proof, along with the definitions needed for the proof to be stringent.
Originally posted by humySince you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?
Slightly confusing question:
Tim is twice the age of what Sue was when Sue was 8 years younger than the age Tim is now.
How old is Tim?
( just an attempt to lighten-up this atmosphere )
-Gustave Flaubert
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Originally posted by Great King RatHere's some arithmetic sleight of hand you see in "survey of mathematics" texts: what is the difference between 1 and the nonterminating decimal 0.999999…?
The problem I have with 1/2 + 1/4 + 1/8 + ... = 1 is that it appears to me that on the left hand side we have infinity whereas on the right hand side that same infinity is actually given a number, namely 1. And as far as I know it is considered wrong to treat infinity as a number. I also remember reading some thread(s) in the spirituality forum of thi ...[text shortened]...
My answer to what is the sum of 1/2 + 1/4 + 1/8 + ... would be infinitely close to 1.
Let N = 0.999999…
Then 10N = 9.999999…
Now, 9N = 10N - N = 9.999999… - 0.999999… = 9.000000…
Hence 9N = 9.
So N = 1.
Therefore 0.999999… = 1.
Tah-dah!
(The argument is somewhat informal, but it is sound.)