I always thought it was cool, and figured there would be a use for it divulged later.
It’s later, but I still don’t know what it’s for.
Maybe because it doesn’t exist?
It exists, and it seems like you could calculate magnetic fields with it, but I’ve never seen it in action, not even a real-life story problem.
Sorry, there is no such thing as a square root of a negative number. Reverse the logic: what number squared would result in a negative number? The answer is clear and simple: none.
Well, I use it all the time, but I’ve got one of those technical jobs.
Aside from complex impedances and Optical Transfer Functions and the Cauchy-Riemann theorem and things lik that, the main purpose of the square root of minus one (or “i”) is to dazzle or befuddle people by proclaiming that “e” raised to the “i times pi” power is minu one (-1).
“Tell Zeno I’m willing to meet him halfway”
All those years of school and university in vain? There IS a square root of -1??
All my imaginary friends use it for their imaginary math.
Actually, it’s extremely useful in only certain fields (like knowing Sanskrit fluently). I mean, that’s not the field it’s useful in, that’s an example of interesting but not broadly applicable knowledge.
I use it almost every day (j, not Sanskrit). Oh, and I call it j rather than i (which is what most others call it) because I’m using it relative to electricity, and we use i for current (and j means current density too, but nobody minds that).
Anyway, as for it’s use – it makes a lot of problems much simpler to deal with. The best way to explain it is in regards to the complex plane and vectors. So there’s some vector in the plane that represents some quantity like current; the complex plane has reals on one axis and complex numbers on the other. What we observe makes a lot of sense if you take the vector and swing it around in a circle in the complex plane. Then you realize that we only see the real part of it ( whatever the vector’s projection on the real axis is). Imagine (since you’re thinking about j, I know you can imagine) looking sideways at a thin rod moving in a circle, but you have no depth perception and the rod’s apparent thickness doesn’t change. You’ll see it move side to side only, but in this sinusoidal pattern. Doesn’t that sound confusing, though? It’s much easier to look at the circle pattern.
There’s a lot more history and probably other reasons behind this. J is useful in solving differential equations as well, and for analysis of stability of control systems.
panama jack
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I used to collect sticks when I was a young boy.
Sometimes after reading posts like panamajack’s, I feel like collecting sticks again.
It was something that I was good at, something that I understood and no matter how completely I explained it to an intelligent person they would respond with a blank stare.
pj,
::blank stare::
Huh?
panamajack wrote
I like that one.
Or should I say “i like that one”?
For a long time, square roots of negative numbers were dismissed as being “nonexistent”, just as, for example, the square root of 2 was considered nonexistent.
The Greeks (I believe) knew the quadratic formula, allowing them to find the roots of any 2nd degree polynomial. Of course, sometimes this formula would result in square roots of negative numbers–these were rejected (since such a thing doesn’t even exist), and the polynomial was considered to have no roots.
When people were discovering the cubic formula (roots of a 3rd degree polynomial) things started getting interesting. Sometimes the formula would give intermediate results involving these “fictitious (imaginary) numbers”, but the end result would be the proper roots of the cubic. People started thinking that if, along the way to the solution, the cubic formula involves “fictitious” numbers, and STILL gets the correct solution, there just may be something to these “fictitious” numbers after all.
I believe it was Cardano or Tartaglia (I can’t remember for sure, and I don’t have a reference handy) who began to accept these numbers as “existing”, but he never got all the details worked out. It wasn’t for some time later that mathematicians such as Euler were able to develop the imaginary/complex numbers more fully.
It was the cubic formula that got it all started. I think it’s somewhat unfortunate that many people take the words “real” and “imaginary” numbers at face value, and don’t understand that the usual connotations of these words don’t apply when being used to describe numbers. “Imaginary” numbers are no more “imaginary” than real numbers, the word came into usage because of the history in the development of these numbers–some were reluctant to accept them. Same thing with “rational/irrational”–again, the words have a good/bad connotation. This may apply also to positive/negative numbers, but I’m not sure of that etymology.
Reminds me of a joke I told to a co-worker one time, who happend to have majored in math.
I picked on her a lot, and one time after some practical joke or wisecrack, she told me one time that “I was giving her a complex.”
I responded “Your complex is imaginary.”
See?
A complex number contains a real component and an imaginary component.
If the real component is a zero, then you’re left with just an imaginary.
Get it?
Thus the “complex” number becomes “imaginary” because it has “zero real.”
Went plumb over her head.
Yup. It makes for a few very useful mathematical relations.
For example, (e^ix + e^-ix)/2 = cos x.
Cool, huh?
– Sylence
Outside of the classroom, I have used i only once. A few years ago i was the number I chose for my softball jersey. Excepting my team mates, no one has ever figured out for what the “i” meant.
The year before that my jersey number was infinity (we used a sideways eight).
I tried to get pi this year. I thought that might be easy with all the iron-on decals for the college greeks. The shirt vendor could not fulfill my order. Dammit! Bastards!
Next year I’m thinking of trying for omega.
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Define real life.
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If we can say that academic competitions (similar to It’s Academic, but college-version) are real life, then yes, I have used (-1)^(1/2) in real life. It’s also an incredibly cool thing to bring up at parties. And it’s a really cool way to piss off a math major at a party when he’s drunk (you give him what looks like a simple series . . . 1, -1, i, -i . . . he’s so piss-drunk he can’t remember).
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If we can’t say academic competitions are real life, you can still use it for party tricks. And of course there’s always the thought of replacing the E in E=MC^2 with e, thus totally messing with drunk people’s minds . . .<eg>
Patrick
Rational/irrational isn’t a good/bad connotation. You’re using the wrong definition of irrational.
A rational (RATIO-nal) number is any number that is the RATIO of 2 integers. An irrational (ir-RATIO-nal) number is any number that cannot. It doesn’t mean that they don’t make any sense or that they don’t properly think out their decisions. Just that they are not ratios of integers.
sigh forgot to use preview. Forgot to use grammar, too.
a rational number is any number that can be expressed as the ratio of 2 integers. An irrational number is any number that cannot.
If you’re looking for something non-mathematical, read George Bernard Shaw’s “The Adventures of the Black Girl in Search of God.” He has a section on “the square root of Myna’s sex (minus x)”. Someone later told him it was supposed to be “the square root of minus one”, but Myna’s Sex made a better pun.
If you’re looking fo something historicalmathematical try “A history of i”, which is obviously made in imitation of Petr Beckman’s “A History of pi.” It’s not as good, because the authors aren’t as cranky as Beckmann, but it is accurate.
>> Have you ever used the square root of -1 in real life?
Real life? What’s that? I wish I had a real life.
I was Pledgemaster of Pledgeline square root of negative one. These were friends from other schools who we pledged our frat. It was most excellent 'cause we could haze those bastards.
One of my pledges became a brother after he married a brother (co-ed frat). We figured he had suffered enough.
V.