Imaginary numbers (part one)

This concept kind of shoots itself in the foot, so I don’t need to spend much time on it. Using an imaginary number in an equation is like intentionally using a false premise in an argument, which is of course totally inappropriate in scholarship.

In fact, the man who invented Imaginary Numbers was an ophthalmologist named Renes Descartes. He coined the term. And guess what. It was in a criticism of the idea. Orwell would have a field day.

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49 thoughts on “Imaginary numbers (part one)

  1. What is your criticism?

    That you personally don’t like imaginary numbers?

    That you personally cannot solve any math problems that require the use of imaginary numbers?

    More rants of the nature “If I you can’t count it on my fingers, it can’t be real.” Bah.

    If you never have to do any math which is difficult enough to require the use of imaginary numbers that is your own personal loss, and none of my business, but you have honestly descended into pointless anti-intelectual whining at this point.

    I’m sure you fancy yourself intelligent, but I need to inform you that you are delusional.

  2. First of all, who even is this? I will eventually tire of your anonymous trolling.

    I gave arguments of two forms against imaginary numbers.

    1) A structural argument – an analogy to false premises
    and
    2) An historical arguments – an observation of the disreputable origin of the concept, worth looking at for those of us who are serious about the real, not imaginary, world.

    As far as I can tell you have addressed neither. Sure, I can’t count an imaginary number on my finger even in principle, which is a fine argument against imaginary numbers, but it’s not an argument I used.

  3. Wow you may two logically fallacious arguments in barely seven sentences! Hurray!

    “Using an imaginary number in an equation is like intentionally using a false premise in an argument, which is of course totally inappropriate in scholarship.”

    Begging the question: Your statement lends no support to your thesis, it simply says that IF your thesis is true, THEN “Using an imaginary…”. Clearly the question isn’t whether or not a false premise is bad, but whether using and imaginary numbers is comparable to a reasoning from a false premises (which it isn’t). Saying “Imaginary numbers are like false premises.” lends no evidence to this conclusion unless you explain WHY.

    Genetic fallacy: The origin of the concept of imaginary number has no bearing on the current truth or utility of the modern concept. http://en.wikipedia.org/wiki/Genetic_fallacy

    Sheesh.

    Please explain to me what you are trying to accomplish.

    Any one of moderate intelligence and learning will think you are a fool.

    Now, you are an inventive fool, so there is some hope for you, but if you keep posting comments which border on the insane, then you will waste any talent you may have.

    In the mean time solve for theta:

    1) i^(theta)=1;
    2) i^(theta)=i;
    3) i^(theta)=sqrt(2)/2+sqrt(2)/2 * i;
    4) i^(theta)=sqrt(3)/2+1/2 * i;
    5) i^(theta)=1/2+sqrt(3)/2 * i;

    I will post a hint later.

    (sorry about the handle, but you blocked my anonymous posting.)

  4. Wow you may two logically fallacious arguments in barely seven sentences! Hurray!

    “Using an imaginary number in an equation is like intentionally using a false premise in an argument, which is of course totally inappropriate in scholarship.”

    Begging the question: Your statement lends no support to your thesis, it simply says that IF your thesis is true, THEN “Using an imaginary…”. Clearly the question isn’t whether or not a false premise is bad, but whether imaginary numbers are false premises. Saying “Imaginary numbers are like false premises.” lends no evidence to this conclusion unless you explain WHY.

    Genetic fallacy: The origin of the concept of imaginary number has no bearing on the current truth or utility of the modern concept. http://en.wikipedia.org/wiki/Genetic_fallacy

    Sheesh.

    Honestly, what are you trying to accomplish?

    Anyone of moderate intelligence and education will think you are a fool.

    You seem to be an inventive and inquisitive fool, so I would recommend you go to college and utilize your talent in a productive manor, instead of making a mockery of yourself.

    solve for theta:

    1) i^(theta)=1;
    2) i^(theta)=i;
    3) i^(theta)=sqrt(2)/2+sqrt(2)/2 * i;
    4) i^(theta)=sqrt(3)/2+1/2 * i;
    5) i^(theta)=1/2+sqrt(3)/2 * i;

  5. Wow you may two logically fallacious arguments in barely seven sentences! Hurray!

    “Using an imaginary number in an equation is like intentionally using a false premise in an argument, which is of course totally inappropriate in scholarship.”

    Begging the question: Your statement lends no support to your thesis, it simply says that IF your thesis is true, THEN “Using an imaginary…”. Clearly the question isn’t whether or not a false premise is bad, but whether imaginary numbers are false premises. Saying “Imaginary numbers are like false premises.” lends no evidence to this conclusion unless you explain WHY.

    Genetic fallacy: The origin of the concept of imaginary number has no bearing on the current truth or utility of the modern concept.

    Sheesh.

    solve for theta:

    1) i^(theta)=1;
    2) i^(theta)=i;
    3) i^(theta)=sqrt(2)/2+sqrt(2)/2 * i;
    4) i^(theta)=sqrt(3)/2+1/2 * i;
    5) i^(theta)=1/2+sqrt(3)/2 * i;

  6. You ironically chose “urdumb” for yourself. Almost as Orwellian as Imaginary Numbers. Look:

    sqrt(x*y) = sqrt(x)*sqrt(y)

    Obviously this is correct – almost as obviously as that two and two make four. But Imaginary numbers make nonsense out of commonsense!

  7. So, you can’t solve the math problems then?

    Here’s a hint: Think of the unit circle, and note that I used the letter theta, commonly used to denote an angle.

  8. Also note, we are talking about complex numbers, not multiplying two real numbers. Your response is non-nonsensical.

    b.t.w. does your equation hold when x=-1 and y=1?

  9. You are continually taking back the things that you say! Absurd!

    I agree that my response is non-nonsensical, and take that as a concession on your part. Good day!

  10. Well, I suppose I will post the answers to demonstrate that it is fairly easy.

    1)0
    2)1
    3).5
    4) 1/3
    5) 2/3

    (You can check those on a calculator if you can figure out how.)

    This is true because i^(theta) is defined as cos(theta)+sin(theta) * i;

    You will note that this demonstrates the incredibly useful property of the complex number plane which makes it so indispensable to mathematics.

    Your statement:

    “sqrt(x*y) = sqrt(x)*sqrt(y)

    Obviously this is correct – almost as obviously as that two and two make four. But Imaginary numbers make nonsense out of commonsense!”

    is true, if and only if you use imaginary numbers. Otherwise your statement is false since sqrt(-1 * -1) would be defined, but sqrt(-1) * sqrt(-1) would not.

    The point is, you may not be able to envision what an imaginary number is, but that doesn’t mean they don’t exist. They are an entirely indispensable part of modern mathematics.

    I haven’t taken anything back that I am aware of. The strike out was do to a typo, not exactly an epic intelectual fail.

  11. Dear urdumb,
    You gave 5 equations to solve. I don’t know why, but the first one is unaffected by the presence of the imaginary number (i.e., anything to 0 is 1), and the other 4 use on both sides of your equation. Notedscholar has already addressed this hocus-pocus when he speaks to the falacy of using a false premise in the argument.
    No one debates the fact that you can create some zany, non-empirical, arbitrary, fake, ridiculous – *yet internally consistent* system and teach it to children who don’t think for themselves, calculators, and text-book writers. That’s like saying: I found a new number named “banana” with the property that banana + 2 = 3 but banana*banana=7, and then coming up with some stupid equations where – if one were so hypnotized by this new ineptness – faithfulness to the properties of banana would allow the evaluation of these equations. But banana is still ridiculous, and we all know it.
    Why then, insist on “i”? A “number of the gaps”, if you will, to fill in the wholes on your square-root number line? No one even uses it except as a cheap high school math team trick question (like, say, taxicab geometry, or circles of inversion), for fourier and electrical engineering cheaters who (by their own admission!) lie about reality for a moment because they don’t want to do *real* math, and by Schrödinger who, let’s be honest, we needn’t even bring into this equation – you don’t want to be in that nutcase’s corner…

    If you want to take this question on – and I don’t think you’re up to it – please deal with the real issues discussed.

  12. “No one even uses it except as a cheap high school math team trick question”

    You never took math in college did you?

    Without I there are any number of differential equations that are unsolvable because they have repeated roots.

    The nifty thing is, that it is often that case that you can use i to produce entirely real answers.

    My point was simply to show that notedschoolar was ignorant of basic high school mathematics.

    I am unaware of any real issues that have been discussed.

  13. “Without I there are any number of differential equations that are unsolvable because they have repeated roots.”

    In this single sentence, I rest my case. When you say that there are “any number” of such differential equations, you are of course not including i in that “any number.” You mean “any number – like 4 or 9 or a billion or any number at all, i of course not included, for it is not ‘any number’ and not in play here.” If the defender of i will use such loose language as “any number” and rule out i (and, as it were, rule out irrational numbers, which we must tackle another day…), I don’t think this lunacy need continue.

    But to beat the dead horse, as they say, I would re-note that “it is often that case that you can use i to produce entirely real answers” is the very case I list above, that i is used to, “lie about reality for a moment because they don’t want to do *real* math” – see especially Euler’s formula.

    Be that all as it may, urdumb, I don’t think you’ll drop the issue and so here is my challenge:
    You state that there are “any number” of differential equations unsolvable without i. Frankly, that’s a big show-offy, as you could say there are any number of “square-root-of-negative-numbers unsolvable without i”, and the “thin air and empty shadows” from which you compose this argument would stand for what it’s worth.
    So please provide one such such differential equation (or square root), that requires i, with any affixed tangible representation. If you can find me a spring which oscillates imaginarily, or a predator-prey mapping with imaginary populations, or a wave equation which yields an imaginary wave, or a speaker kicking some imaginary driven resonance feedback – any empirical context at all for which we need an imaginary number, then I shall throw my writings into the fire and you’ll have the day.
    Until then, I’ll find any number of *real* things to do with my life, rather than waste it in imaginations.

  14. “””
    In this single sentence, I rest my case. When you say that there are “any number” of such differential equations, you are of course not including i in that “any number.” You mean “any number – like 4 or 9 or a billion or any number at all, i of course not included, for it is not ‘any number’ and not in play here.” If the defender of i will use such loose language as “any number” and rule out i (and, as it were, rule out irrational numbers, which we must tackle another day…), I don’t think this lunacy need continue.
    “””

    That phrase also rules out non-natural numbers, such as 3.5. After all, it is absurd to consider half of a differential equation. Yet fractions exist, as is plain as day to all intelligent people. But since half a differential equation is just as meaningless as sqrt(-2) diff-equations, fractional numbers must not exist, either! So your argument is absurd; there is a reason unnatural numbers

    As for your demanding of a “tangible representation” of complex/imaginary numbers, I direct you to the Wikipedia article on complex analysis. Electromagnetic fields are rather real.

  15. PC,
    Sorry for the delayed responses – I had considered this a settled issue.

    First, 3.5, one-half, fractions, etc. are all within the set of so-called “rational” numbers, the simple derivation of which is by all accepted. You are correct that there cannot be 3.5 such differential equations in mind, but there can indeed be 7 or 2, and you see where I’m going.
    In fact, to press the analogy, NotedScholar and I could together produce 7 such differential equations which urdumb doesn’t know how to solve. It would be reasonable to attribute 3.5 to each of us.

    On complex analysis, I’d recommend that you learn a bit about either of the subject materials at hand (and not just make arbitrary google searches) before suggesting the empirical existence of anything “imaginary.”
    I am opposed to the actual existence of imaginary numbers – I believe in the existence of half-wit scientists who have used such absurd self-contradictory creations as these to attempt to better express their limited understanding of any particular field.
    Imagine that you’re driving and have gotten lost, so you call me for directions. You tell me you’re heading South on I-55, headed for such-and-such place. I tell you to continue 70i miles in that direction, by which I mean that you should turn and go not-at-all in that direction. I have used a false mode of representation in order to lie to you.
    This is the deceit of using imaginary numbers for fields or differential equations. The fact that you and I may’ve previously agreed on this notation does not excuse the arbitrary nature of what has just occurred.

    As above, no em field will every produce an “imaginary” result.

  16. anwei40:

    “I believe in the existence of half-wit scientists who have used such absurd self-contradictory creations as these to attempt to better express their limited understanding of any particular field.”

    Where is the contradiction?

    Oh, wait! Earlier you admitted that it is consistent…

    “No one debates the fact that you can create some zany, non-empirical, arbitrary, fake, ridiculous – *yet internally consistent* system and teach it to children”

    They sure are consistent! That’s right, free from contradiction as you say so yourself!

    If you had taken complex analysis, you would probably know that the complex numbers form a field (just like the real numbers).

    “I found a new number named “banana” with the property that banana + 2 = 3 but banana*banana=7″

    This is obviously not consistent. Taken together, this would imply sqrt(7) = 1, which is a contradiction.

    “As above, no em field will every produce an “imaginary” result.”
    That’s fine. Nobody expects the result to be imaginary either. Have you ever heard of contour integration? It is a technique that allows you to do improper integrals (using a really nice switch to evaluating in terms of the complex plane), but giving a real answer. I hope you can understand that introducing complex variables into a problem does not necessarily imply that the result ought to be imaginary. Gamelin’s complex analysis (http://www.amazon.com/Complex-Analysis-Theodore-W-Gamelin/dp/0387950699) is a pretty good introduction which I would recommend that you read before making any more posts here. Good luck!

  17. For anyone interested in a real world application (and if you study electrical engineering, this is critically important understanding).
    “Reactive power does not transfer energy, so it is represented as the imaginary axis of the vector diagram. Real power moves energy, so it is the real axis.”
    - http://en.wikipedia.org/wiki/Real_power#Real.2C_reactive.2C_and_apparent_power
    So, you can still deny the math developed to explain and illustrate imaginary numbers and claim that it is only consistent within the context of of math problems, but with that math you can accurately predict the real and reactive power that CAN BE MEASURED in electrical systems. Without this math we would not be able efficiently deliver power for most applications which I am sure you use it for on a daily basis.

    Hopefully that fact will reassure anyone who stumbled onto this page and started to doubt the validity of the math established around imaginary numbers.

  18. Dear Nah,

    I’m glad you get the point. This post isn’t particularly complex, since it mainly points out that mathematicians have (in Orwellian fashion) taken a derogatory term and made it a virtue. In the sociology of science this is called scientism.

    Cheers,
    NS

  19. Dear Nobody,

    It’s fitting that your name is “nobody,” because the whole point of the discussion you link to is that there is “no” energy or “no” power actually represented in the calculations. This is a completely different issue from imaginary numbers, which are what (allegedly) happens when you try to figure out the square root of a negative number. Of course the whole concept is absurd, as any first grader can tell you.

    Cheers,
    NS

  20. “Most men in departments of maths are chauvinists, although fortunately this is changing in the last ten years or so.”

    “Nice try. Instead of proving that 0.999…=1, you have instead proven the age-old thesis that women are deceivers.”

    So who exactly is being deceptive?

  21. Hi Renfield. So, I’m afraid you’ve misread my remark. It was reductio ad hominem, or “bringing the person into absurdity.” It’s shocking to me that advocates of these views don’t realize that their positions lead directly to chauvinism, racism, and hatred.

    Anyway, thanks for stopping by, and I hope I’ve cleared things up.

    Best,
    NS

  22. Why then is it that you address to the male respondents by only their name, but pad the responses to female posters with “Sweet Aria” and “Gentle Molly”? I strongly suspect that you hold chauvinistic views. The term “simple woman” has also escaped your fingers.

  23. Renfield,

    I cannot guarantee that all of the comments under “notedscholar” here have been my own. But aside from parodying the male bias intrinsic in western science, I have not meant to show chauvinism. But I honor your concern.

    Best,
    NS

  24. Notedscholar,

    Thank you for taking my concerns seriously. A defensive or flippant response would have caused further doubt in my mind, but we have avoided that slope.

    On to the math:

    It seems to me that your general beef with some of these concepts is related to an impulse to ground mathematics in the realm of observable evidence.

    Consider the following narrative:

    Euclid postulated that parallel lines never intersect, but was unable to demonstrate this by way of formal proof. It seems intuitively obvious to anybody with human vision that it’s true, and rather than get hung up on this point, he went on to describe other geometric properties, many of which were built upon this postulate.

    Attempting to prove it by way of proof-by-contradiction, other mathematicians conducted the following thought experiment: assume instead that parallel lines DO intersect, and base further theorems and properties on this assumption. Once a contradiction arises, it proves that the postulate must have been correct. Instead, they arrived at internally consistent rules (ie, non-Euclidian geometry) that seemed to have no relationship with the physical world we inhabit, but fail to mathematically prove the parallel postulate.

    Fast forward to the 20th century, in which the behavior of photons cannot be explained by Euclid’s geometry, but specifically obey the non-Euclidian rules. (Indeed, anticipating a straw man, it should be noted that Heizenberg’s principle is taken into account in these studies).

    The point of which is that, even if mathematical thought experiments seem to have no relationship with our shared reality, who is to say that it’s not simply the limitations of our powers of observation that lead to this impression? That discoveries will be made that eventually demonstrate the relationship between the number theory and physics?

    Unrelatedly, consider that there’s nothing fundamentally wrong with mathematicians simply agreeing on certain conventions in order to create a standardized method by which they can explore numerical properties. The same thing occurs with language and allows people to communicate with one another. It has been theorized that words have no actual meaning and do not truly represent the “things” they attempt to represent, but by assuming that they do, I can say that an object is “green” or “Republican” or “underwater” and most people will be able to understand my meaning based on these conventions. Negative numbers, infinity, and imaginary numbers have the same property: whether or not they correspond to the true Platonic form is irrelevant; the point is that they give mathematicians a common ground by which to have a conversation.

    What is your investment in shutting down this conversation? The usual suspects would be religious fundamentalism or something of that nature, but that doesn’t seem to be the case on this blog.

  25. Having read Descartes’s writings on the subject, I noticed that he though of them as a nessecary part of solving certain equations

  26. NS: Technically you should be bringing up Cardano, the 14th-century mathematician who first used imaginary numbers. ;) He also said they were an “absurdity,” but could still be worked with like regular numbers.

  27. …imaginary numbers are simply concepts like other numbers.

    It doesn’t matter what YOU think of them, imaginary numbers are fucking bitching! I mean, without imaginary numbers, how would be able to use polar values like r-cis-theta (e^i-theta) which may not be provable if your theorem is right, but IT WORKS! I mean, just about everything that has a measured oscillation can make use of phase-angle stuff from r-cis-theta, like the AC currents that give power to your house, the clock oscillators in every computer, the wireless or wired signals these things take, and so on.

    These things are so touchy and precise that our math with imaginary numbers would lead to major failure if it didn’t work.

    But it does.

    You’re reading this on a computer so QED. Quite Effectively Demonstrated!

  28. It has been pointed out that “imaginary number” are no less real than any others. That is to say, numbers such as three, five, and 5962 have no physical form. You can show me that many items, or a symbol for that number, but you can’t show that number itself;
    BTW, ALL HAIL ANGEL PLUME!

  29. Alfonso,

    I appreciate the energy and enthusiasm with which you approach these difficult, pressing issues. That being said, I am disappointed very much with your ideas. Certainly imaginary numbers work – so do nuclear bombs. Certainly our math wouldn’t work without them – just as the housing bubble wouldn’t have burst without bad mortgages in the first place. This is no argument for the latter items.

    See you tomorrow,
    NS

  30. Nice try Angel Plums. Just because you use a different name doesn’t mean I don’t know that you are praising yourself.

    Um…. as for physical form….. um definitely imaginary numbers don’t have the same kind of existence as real numbers. Find a single scholar who thinks they do answer you can’t because they don’t exist (like imaginary numbers).

    Not best,
    NS

  31. NS.
    One the one hand, I thank you for complimenting me, but on the other hand I will swear by the concept/deity of your choice that I am not Angel Plume.
    Also, it is impossible to show me or anyone three. You can show me three of something, or a symbol for three, but not three.

  32. Also note that just because they don’t have a physical form doesn’t mean I can’t work with numbers. For example, 2+7=9, just as (3+4i)*(3-4i)=25. You Daft Ninny.

  33. The only way anyone works with numbers is through thoughts and symbols just like I did.

  34. Why are you even discussing with this guy?

    BTW, “sqrt(x*y) = sqrt(x)*sqrt(y)” is not correct. I rest my case

  35. Btw, I Wanst accusing you of ad hominem, I was just stating that you are Mr. Ad Hominem.

    And a question can very well be a fallacy. I see you have yet to understand fallacies. Don’t worry, it will grow on you as you get older ;)

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