Greg Hewlett passed away on January 17th after nearly eight years of battling colon cancer. While we grieve his loss, we are comforted to know that he is with his Lord.
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The Struggle Has Ended
Tuesday, September 4, 2007
Imaginary things
Of all of the theorems, formulas, transforms, and properties I have attempted to understand in math and science classes, one continues to fill me with awe: Euler's Identity. Every couple of months in my work, I sit back and sigh in wonderment at the ring of truth associated with some aspect of it. A colleague and I will be at the white board and some facet of it will emerge, and one of us will say, "Isn't it amazing?" For me, it delights more than E=mc2, Pyhtagoras' theorem, or many of the other gems that have emerged in the history of the discovery of truth.
In its simplest expression, the identity states that e to the power of pi times i equals -1. It magically combines the three most mysterious constants conceived, or rather discovered, by mankind:
- e: the constant, the derivative of which, when raised to x, is 1. e is not representable by decimal numbers, but is close to 2.72.
- pi: the constant equivalent to the ratio of the circumfrance to the diameter of a circle. pi is also not representable by decimal numbers, but is close to 3.14.
- i: the imaginary square root of -1. i is not only not representable in decimal numbers, it is so beyond imagination it seems plain silly.
The identity is, at the same time, beautifully elegant and laughingly non-sensical. It seems surely to have been fabricated in the wishful thinking of a naive, wanna-be mathematician.
This odd identity regularly bears fruit in engineering. In my particular branch of engineering, it allows us to get a grip on images, audio, and video in the frequency domain. While we experience life in the space-time domain, the frequency domain allows us to "see" things from a perspective that enables things impossible when looking through space-time. It allows us, for example, to compress images into small fragments that can fit on camera cards. It allows cellphones to talk to one another through the air. It allows us to cram hundreds of albums into iPods. It allows us to simultaneously put hundred's of TV channels on a single thin wire. It allows an MRI to see tumors without cutting. It is what made the Speak-n-Spell speak. And it allows us to make DLP video the best picture in the world (ok, so I'm biased).
When I speak of how it inspires awe (and even joy?), many of my nerd friends know what I'm talking about. If you have experienced this, or are curious enough to want to, then I think you will appreciate Amanda Shaw's latest essay in First Things blog.
In its simplest expression, the identity states that e to the power of pi times i equals -1. It magically combines the three most mysterious constants conceived, or rather discovered, by mankind:
- e: the constant, the derivative of which, when raised to x, is 1. e is not representable by decimal numbers, but is close to 2.72.
- pi: the constant equivalent to the ratio of the circumfrance to the diameter of a circle. pi is also not representable by decimal numbers, but is close to 3.14.
- i: the imaginary square root of -1. i is not only not representable in decimal numbers, it is so beyond imagination it seems plain silly.
The identity is, at the same time, beautifully elegant and laughingly non-sensical. It seems surely to have been fabricated in the wishful thinking of a naive, wanna-be mathematician.
This odd identity regularly bears fruit in engineering. In my particular branch of engineering, it allows us to get a grip on images, audio, and video in the frequency domain. While we experience life in the space-time domain, the frequency domain allows us to "see" things from a perspective that enables things impossible when looking through space-time. It allows us, for example, to compress images into small fragments that can fit on camera cards. It allows cellphones to talk to one another through the air. It allows us to cram hundreds of albums into iPods. It allows us to simultaneously put hundred's of TV channels on a single thin wire. It allows an MRI to see tumors without cutting. It is what made the Speak-n-Spell speak. And it allows us to make DLP video the best picture in the world (ok, so I'm biased).
When I speak of how it inspires awe (and even joy?), many of my nerd friends know what I'm talking about. If you have experienced this, or are curious enough to want to, then I think you will appreciate Amanda Shaw's latest essay in First Things blog.
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5 comments:
Your churchy friends are oddly silent on these technical subjects. :) Just thought I'd chime in with my shared awe of that equation.
That's fascinating. Thx for the link to the first things essay -- so appropriate: I don't understand Euler's identity or i but am in awe of how even in math God uses "foolish" things -- his ways are higher than our ways. And then he allows us (er, you) to use such mysteries for cool technology -- how gracious of him, how humbling for us.
a churchy friend,
Kristi
Yup, Max. Good to hear from you, by the way.
Don't forget that the closely related Laplace & Fourier transforms make the solution of all linear differential equations into plain ol' linear equations. So mechanics and other physical domains should also tip their hat to Euler.
Now for true nerdiness: AND if e^(i*pi) = -1 then [e^pi]^i = -1, so the ith root of -1 equals e^pi. Ponder that for a while.
Glen said: Now for true nerdiness: AND if e^(i*pi) = -1 then [e^pi]^i = -1, so the ith root of -1 equals e^pi. Ponder that for a while.
I get (-1)^i = 1/(e^pi) instead of e^pi.
(-1)^-i = (e^pi)
No less ponderable since an imaginary power gives a real result. I tell my students to think of e^(phi x i) as notation for a unit phasor with angle phi and not to dig any deeper or it will worm its way into their brains like a maggot.
another churchy friend
Don't know if you'll ever track back to this, Greg, but I am glad I discovered your Blog. Now I know what is going on and need to wait for you to contact me.
Check your facebook page for a message sent from a stranger, who happen to be my inlaws, for my contact info
Anyway, why comment here? I always loved that equation too. Was even talking about it the other day
So even though we are separated by miles and years and a lot more, reading this makes me think about playing Backgammon w/ you in our dorm room and how much we do have in common
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