It was a quote from Augustus De Morgan that started it. "Mr.Ballew" he says, "I don't get it," This was a young man who usually gets it, and when he doesn't he has to scratch the itch to know. The quote was on my board, so I was the logical one to ask.The quote (actually a slight modification of the true words of De Morgan) said:
Imagine a person with a gift of ridicule . He might say first that a negative quantity has no logarithm; secondly he might add that a negative quantity has no square root; and finally he could comment that the ratio of the first non-existent to the second is the same as the circumference of a circle is to the diameter.
"So tell me what you DO understand." I asked
He explained that he thought the "negative has no logarithm" means that you couldn't take logarithm of a negative because the domain of logarithms was limited to values greater than zero. The teacher "stayed still, and Br'er Fox, he lay low". "And I guess the second part is that you can't take the square root of a negative number."... and now the teacher raises one eyebrow..."Ok, you can, but you don't get a real number, you get an imaginary number"...
As the teacher unfolds his arms, one long sweep brings a finger to slide beneath the last line... and then ... . "Ok, and the ratio of the circumference of a circle to the diameter is Π " .. at last the teacher speaks...."Good." and he walks to the board to pick up a marker.
"I think you have seen this somewhere, even if you were not sure what it meant." and on the board he writes Euler's beautiful result (which Cotes shoud get more credit for); eπ i = -1
The student admits to have seen it, and the teacher negotiates an agreement to accept it, or at least accept that De Morgan believed it was true. "Now let's take the natural log of both sides, and we get..." and he writes,... Π i = ln(-1) Now we are in the part that the teacher loves... the student is hooked.. its art and entertainment and poetry and math all in one... and the clever student sees the first inkling of the result... Pi is waiting on one side... the log of a negative on the other... and then he sees all.....
" Ahhhh," he says, "just divide by i... "... The teacher yeilds the marker, ...passing the torch to a new generation??? ... and feels a pang of envy.....to be so bright and so young... and the young man writes Π = ln(-1) / i..... and then thinking of the quote, rewrites it as