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Imaginary Numbers, Once Rejected, Now Commonplace

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Once again I direct our readers to First Things.  This time Amanda Shaw discusses how imaginary numbers, once rejected as “Impossible, irrational, delusionary, absurd, untrustworthy, fictitious, imaginary,” are now a staple of everyday math.  See http://www.firstthings.com/

Is there an analogy to ID here?  The fact that imaginary numbers were not part of the math “system” did not mean they were not out there waiting to be used by those who were willing to look beyond the blinders of the existing paradigm.  Now, as has been argued at this site before, ID can be fit within the existing scientific paradigm; but even if this were not the case, the point is should we cling to a limiting paradigm that prevents us from seeing greater truths?  Shaw’s last paragraphs are very good:

“Scientific positivists, pencil and paper in hand, peer through shatterproof, UV-protected glasses at a world of animals, vegetables, and minerals. But genuine scientists—true seekers of knowledge—are not afraid to let the sunlight dazzle them, not afraid to seek and imagine what our myopic reason calls absurd.

“Impossible, irrational, delusionary, absurd, untrustworthy, fictitious, imaginary: It is always easier to approach—or rather, ignore—mysteries of math by dismissing them as false or unintelligible. And how much more for mysteries of faith. So is God like an imaginary number, waiting to be discovered and accepted in a renaissance of faith? The simile is ridiculous, on its face. But, in a curious way, the ramblings of scientific history remind those who strive for reason just how vast reality is. The realization is at once unsettling and exhilarating: Truth is far richer than our minds—always confined by the here and now—can prove or even imagine.

Comments
Fun with i, I see. Now, WHY does all of this lead to such interesting real-world connexions? Such as, for example, the Fourier and Laplace transform and the world of frequency and dynamical responses of systems? In short, why is Math -- an issue of mind and abstract logic -- so elegant and magical, with capacity to refer to observed reality? GEM of TKIkairosfocus
September 5, 2007
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So, i ^ i = e ^ (-pi / 2) = 1 / (e ^ pi / 2), a real number.
That's one value, but there are infinitely many. i^i = e^(-pi/2 + 2 pi N) for integer N.guppy
September 5, 2007
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FWIW... Using the formula e ^ (i * pi) + 1 = 0, it can be shown that i ^ i is a real number... x ^ x can be rewritten as: e ^ (x * ln x) so i ^ i can be written as : (1) e ^ (i * ln i) Now, e ^ (i * pi) = -1 = i^2, and i ^ 2 = e ^ (2 * ln i). so e ^ (i * pi) = e ^ (2 * ln i) Equating exponents gives: i * pi = 2 * ln i So ln i = i * pi / 2 Substituting this back in (1), we have: i ^ i = e ^ (i * i * pi / 2) So, i ^ i = e ^ (-pi / 2) = 1 / (e ^ pi / 2), a real number.josephus63
September 4, 2007
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I don't think it is correct to say that the reasoning that leads to Euler's identity is analogous to the reasoning that x^0 = 1.
You are correct. They are only analogous in the sense that multiplying a number by itself zero times and multiplying a number by itself i times seems nonsensical, until one works through the derivations, in which case it all makes perfect sense. This is the beauty of mathematical reasoning.
But actually it is fundamentally no different than 2 + 2 = 4
But isn't it interesting that mathematics describe to a great extent how the universe works? Does this not suggest contrivance, rather than coincidence? BTW, I'm now using a finite element analysis program (LS-DYNA) in my work in aerospace research and development. It models the laws of physics and material properties purely mathematically, and produces amazing results in real-world applications.GilDodgen
September 4, 2007
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Also, to Gil: Your explanation of why x^0 = 1 for all x was good. However, you then write,
Analogous mathematical reasoning (but much more complex) leads to e ^ (i * pi) + 1 = 0.
I don't think it is correct to say that the reasoning that leads to Euler's identity is analogous to the reasoning that x^0 = 1. I think the proofs are far too different to call them analogous to each other. This may be a small point, but I think in math analogous proofs have the same structure, such as the proofs for sin(x + y) and sin(x - y), and I don't think that similarity of structure applies here.Jack Krebs
September 4, 2007
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By the way, in response to the opening post, negative numbers were also rejected for a long time for the same reason that imaginary ones were: in fact, both types of numbers were suggested, argued about, and eventually accepted at approximately the same time after the start of the renaissance. In both cases, the keys to acceptance came from: a) developing rules for manipulating the numbers that worked, b) developing ways of visualizing the numbers on a number line, and c) finding ways to apply the numbers to situations in the real world.Jack Krebs
September 4, 2007
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As a math teacher, let me say that ex-xian is absolutely right when he says that there is no ontological difference between real and imaginary numbers: the choice of the word "imaginary" was an unfortunate and misleading choice, but it has stuck with us. Also, Euler's identity, e^(i * pi) + 1 = 0, is a glorious fact because it is so unobvious and because it ties together constants from all major branches of mathematics. But actually it is fundamentally no different than 2 + 2 = 4: it is a fact about a relationship between numbers that follows from the basic definitions and assumptions that underlie the complex number system. See here for a notesheet I use in calculus that, in paragraphs 3 through 6, outlines a proof.Jack Krebs
September 4, 2007
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Argghhh! HTML is a menace. Try this: http://tinyurl.com/ysybq From the Wiki article:
A reader poll conducted by Mathematical Intelligencer named the [Euler] identity as the most beautiful theorem in mathematics.
GilDodgen
September 4, 2007
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OT: individual genome study by Venter might be interesting post. PLoS http://biology.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journal.pbio.0050254&ct=1 Wired http://blog.wired.com/wiredscience/2007/09/the-genome-is-m.htmlMichaels7
September 4, 2007
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"Didn’t Stephen Hawking use imaginary time to describe the universe. The problem being that imaginary time has no real manifestation. It is just a tool to assist the physics." Yes, he did it to avoid the implications of a universe that has a beginning. I think William Lane Craig takes the position that imaginary time is just that "imaginary" Vividvividblue
September 4, 2007
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By the way, how the Hell do you find anything to the power of i?
An analogous question: How the Hell do you find anything to the power of zero? What does it mean to multiply a number by itself zero times? The answer is always 1. This is the beauty of insights provided by mathematics. Multiply 2^3 times 2^2. That's 2*2*2 * 2*2. Add the exponents (2^5). Now divide 2^3 by 2^2. That's 2*2*2 / 2*2. Two of the 2's cancel, so subtract the exponents. giving the original number. Now divide any number raised to any power by that number raised to the same power, and they all cancel, giving a result of 1 (the exponents are the same, so when you subtract them they equal zero). Thus, any number multiplied by itself zero times is 1. Analogous mathematical reasoning (but much more complex) leads to e ^ (i * pi) + 1 = 0. Euler was one clever dude. http://en.wikipedia.org/wiki/Euler's_identityGilDodgen
September 4, 2007
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bornagain77, I am by no means an expert on relativity and I welcome a comment in response to your question by someone who is. That said, it is my understanding that time goes to zero only relative to the object that is traveling at the speed of light. Relative to other objects, it still moves. Your question is very interesting.BarryA
September 4, 2007
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LOL funny Barrybornagain77
September 4, 2007
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I would also like to ask a hypothetical question on "imaginary numbers" In Einsteins special theory of relativity, time as we understand it drops to zero at the speed of light. Zero represents the non-existence of time yet the energy certainly must be real and must exist and as such MUST have a "real" time that it must exist in. That is to say that energy cannot possibly exist in a non-existent time. It is a logical absurdity to define this time of energy to zero, If you can see my point, So my question is how do mathematicians get around this apparent obstacle of logic in math so as to describe the reality of energy more appropriately? Do they "invent" an "imaginary number" to represent the "real" time that energy is logically required exist in?bornagain77
September 4, 2007
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bornagain77, dancing with horses is illegal in most southern states.BarryA
September 4, 2007
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Following the privileged planet principle, This universe seems remarkably designed for our discovery. Without mathematics we definitely could not have progressed to the point technologically to make these discoveries nor could we make heads or tails out of what we have discovered. Thus not only was the universe apparently designed for us to discovery in its full glory but our minds were given the proper mathematical and logical tools to make sense of these astounding discoveries. Indeed, for one to seek to belittle the importance of mathematics in our understanding is to "shoot the horse" that brought you to the dance.bornagain77
September 4, 2007
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Scuba and ex-xian, I'm wondering at your replies. Surely Foljambe is just trying to be funny (not very successfully, I'll admit).BarryA
September 4, 2007
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Charles Foljambe wrote:
Math was originally made as a convenient symbology to describe the universe.
The Scubaredneck replies: This seems to be a rather dubious claim. Mathmatics seems to be much more absolute than say verbal language (which is indeed simply a "convenient symbology to describe the universe"). While two people can disagree about many things related to language, there can be no disagreement on things like the value of pi or other mathmatical descriptions. While there are any number of semantic symbols that could have been chosen to represent the ratio of the circumference of a circle to its diameter, its value is forever unchanged and could be no other. Contra to your claim, mathmatics is not at all ultimately meaningless. Indeed, F=ma (a mathmatical formula) has a specific meaning, one which I employed this past weekend while splitting wood. Indeed, any mathmatical formula or axium has an objective meaning that is generally not open for debate (although the applications of that meaning may vary). Interestingly, you attempt to defeat the notion of i by suggesting that you can't raise anything to the power of i. I would challenge you to raise something to the power of pi or e or any other irrational number (not an approximation but the actual number). The fact that the number itself cannot be exactly defined does not mean that it does not exist. Furthermore (and this is more basic), no has ever suggested that i is a real number. It is, by definition, not a real number but merely a tool used for solving certain problems. The ScubaredneckThe Scubaredneck
September 4, 2007
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So, what you're saying is, it's like some modern art. Ultimately meaningless, but pretty. By the way, how the Hell do you find anything to the power of i? Clearly, I'm no mathemagician, but I'm not a slouching mouth-breather either. http://outrageoracle.blogspot.com/Charles Foljambe
September 4, 2007
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I did not find the exact name for the quote but I found a article that explains the formula in a bit more detail; Here is a quote from the article. Because of the serendipitous elegance of this formula, a mathematics professor at MIT, an atheist, once wrote this formula on the blackboard, saying, "There is no God, but if there were, this formula would be proof of his existence." The whole article is well worth the read. http://www.christianitytoday.com/ct/2006/march/26.44.htmlbornagain77
September 4, 2007
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e ^ (i * pi) + 1 = 0 This equation says that the natural log base raised to the power "the square root of -1 times the ratio of the circumference of a circle to its diameter," plus 1 equals zero. It incorporates all the special constants of mathematics (which would seem to be unrelated in many ways) to produce a completely unintuitive but beautiful relationship and equality. Perhaps someone can track down the exact quote, but (paraphrasing), an atheist mathematician is supposed to have said, "There is no God, but if there were, this would be evidence of it."GilDodgen
September 4, 2007
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Rationality liberates, but rationalism enslaves. Rationality puts reason in its proper place, recognizing that some truths can be above it without contradicting its principles. Rationalism deifies reason and therefore compromises its legitimate function, increasing the possibility that one will fall into error.StephenB
September 4, 2007
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Hah! I say, and balderdash! Imaginary numbers, my eye (i)! Math was originally made as a convenient symbology to describe the universe. Now, arrogant mathematicians and mathematical physicists, thinking that math is an absolute, and indeed, defines the universe rather than describing it, think that because if it didn't, their precious disciplines would be asymmetrical, that the universe must incorporate such nonsense as the square roots of negative numbers, when negative numbers themselves are only symbols of convenience from arbitrary zero points!Charles Foljambe
September 4, 2007
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Didn't Stephen Hawking use imaginary time to describe the universe. The problem being that imaginary time has no real manifestation. It is just a tool to assist the physics.Robo
September 4, 2007
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