Will I never need to use imaginary numbers in the "real" world??
117795
You are taking the expression too literally. "Imaginary" numbers is just that .. an expression to diffeerentiate them from "real" numbers.\n\nIt's hard to understand now, at this point in your life, but you shouldn't worry "if you'll ever use it." That's not the way to go about learning because, as fate would have it, you have no way of knowing what you will or will not need in the future. Imaginry numbers are critical to electrical systems and electrical engineers use them on a daily basis. Also, if you take differential equations, you will use them in order to solve such equations.\n\nTry to learn as much as you can, because you never know when you'll need it. This is something I have learned the hard way. I wish, in some areas in school, I'd paid more attention.
1156569
The problem is not that you can't take the log of a complex number. The problem is that log is multiple valued in the complex plane. In other words, the exponential function is not one-to-one. You can take the 'principle branch' of the logarithm, but then you can lose certain of the rules of logarithms.\n\nThe upshot is that when you take the log of -1, you either get an infinite set of values (.., -pi*i,  pi*i, ...) and another infinite set of possibilities for the log of 1 OR you lose the property that log(a^b)=b*log(a) that you used in your first paradox. This is just a fact of life when working with complex numbers. In the second, BOTH -pi*i and pi*i are logarithms of -1, so the argument fails. It's somewhat like saying that the squares of -1 and 1 are the same, so -1=1.
383124
I believe the book is incorrect as well.\n\nThere are some real-world problems that start and end with numbers that do not belong to the set of Reals.  Consider the Quaternion.  A Quaternion is like a "super imaginary number", it is of the form, a + b*i + c*j + d*k, where i^2 = j^2 = k^2 = -1\n\nThis type of number can be used to represent the position and orientation of an object in 3D.  Since objects do have positions and orientations, it is not unreasonable to represent this with a Quaternion (even though many people don't know what it is).
756332
Hi there!\n\nI've heard the name "hyper real" before, but I don't know much about them.  I had a look on the net though, and came up with this link, which looks quite in depth and reliable: http://mathforum.org/dr.math/faq/analysis_hyperreals.html#construction\n\nAlso try having a look on wikipedia, the link is below.\n\nFrom what I can make out from having a quick read of these webpages is that hyperreal numbers include infinite numbers and infinitesimal numbers (i.e. numbers that are smaller than all positive real numbers and greater than all negative real numbers, and it is possible to have non-zero infinitesimal numbers).  \n\nIt helps to understand a little of real analysis, where you have what's known as an epsilon delta definition of the limit of a function, i.e. the real number R is the limit of f (a function from the real numbers to the real numbers) at a if given epsilon greater than zero there exists a delta greater than zero such that if \n0<|x-a|<delta, then |f(x)-R|<epsilon. (where |x-a| means the modulus of x-a, i.e. the [positive] distance between x and a on the real number line).  Basically, this rigorously defines the idea of "getting as close as you like" to the limit, and it was important in the development of calculus, because it put the limit concept on a rigorous setting, and not just an intuitive one.\n\nHowever, according to Wikipedia, Abraham Robinson gave a rigorous definition of infinitely large and infinitely small numbers, in what's known as non-standard analysis, and this is where the hyperreal numbers come in.\n\nHope this is helpful for you!
681247
Ordinary numbers can be represented as a line, like a ruler.   There's 0, 1 is to the right of it, 2 is further right, 1.5 is halfway between 1 and 2 and so on.   Complex numbers can be represented as points on a flat surface.   There's a real part like an ordinary number, and an imaginary part, which is a multiple of the square root of minus 1, written as i.   So 3-2i is the point you get to when you start off at the origin (like the origin of an x,y graph)You go 3 to the right and 2 down.   3+2i would be 3 to the right and 2 up.   Let's call a complex number z.   Turn z into (z times z) +1.   Call this z.   Do the same thing again, again and again, say 1000 times. z will either get bigger and bigger without limit or it will stay small.   The set of values of z that stay small is the Mandelbrot Set.   You can draw a picture of it.   If your piece of paper is white, and you make a black mark for every point on it corresponding to a complex number in the Mandelbrot Set, you get a black shape looking a bit like a peach with warts on it.   It's infinitely complicated; if you magnify the edge, you see more and more detail.
654434
Yes, there is. The square root of negative 1 is the number i, which is an "Imaginary Number." \n\nDespite the name, imaginary numbers are real quantities that are used in equations to get real answers. However, they are a somewhat confusing concept, mathematically. \n\nOne way to visualize them is to consider that multiplication by an imaginary number rotates a function orthogonally on a number axis. The answers given by performing this procedure once may not seem to be applicable to real situatons, but if you multiply by an imaginary number again, in rotates the function back onto the initial axis, yielding real number results. \n\nI am including links to a number of good sites on imaginary numbers. Although it can be difficult to understand the implications of imaginary numbers beyond the fact that their squares are negative, if you spend some time on these sites, I'm sure you'll come to understand them.
110391
There are many practical applications for imaginary numbers. \n\nFirst, nothing in mathematics really EXISTS. Mathematics is just a way of creating a set of rules, then seeing what can be proven from those rules. Applied mathematics is a procedure where a subset of the material from pure mathematics is used to model something in which we are interested, which usually is something in the real world. \n\nIf the rules for manipulating things for imaginary numbers are useful, then that is a good thing. However, they are useful, because they let one solve a lot of problems easier than if we had to make more cumbersome manipulations without them. \n\nFor example, in any problem involving waves and frequencies, they are very useful, because of Euler's formula:\n\ne^(ix)=cos(x)+i sin(x)\n\nThis lets one manipulate trigonometric quantities easily in a compact form. \n\nAll of electrical engineering makes extensive use of this therefore, especially when power is involved, because AC is so prevalent. \n\nSolving many equations are far easier when imaginary numbers are used, even if the initial problem is all real, and so is the final answer. \n\nIntegrals that are used in all application areas are sometimes far more easily solved using complex integration. \n\nAll of quantum mechanics is deepy connected with imaginary numbers. The Schrodinger equation contains imaginary numbers. The answers are complex functions. Their modulus squared produces the real probability functions we use. \n\nImaginary numbers are used throughout physics. \n\n \nTo paraphrase a famous saying, the quickest way to real results is through the complex plane. \n\n(the real quote by Jacques Hadamard is more accurately translated as "The shortest path between two truths in the real domain passes through the complex domain.")\n\nThe more important question underlying this, is, why is mathematics so unreasonably effective at modelling things we are interested in? (see references).
26912
I give up,what?They don't even make a calculator that will register that many digits and I sure don't want to figure it manually.I'm sure there are a lot of numbers that when multiplied would equal a 400 digit number.400 is only a 3 digit number.
106499
This is the case\ncircumferences no where centres everywhere\n\nJust try,  even you can not imagine NOTHING, because before imagining that, your mind was already there, thereby the NOTHINGNESS is voilated.\n Even the 'ZERO' is not the one we usually think.\nIt is a compromising point of  +(Plus) and - (Minus) infinity !!!!!. Even mathametically all is filled in 'ZERO'. Enven any number, say 100, it is 0-1000, that is to say you have borrowed it from 'ZERO'\nIt enough for now,  there is musch more to say about 'ZERO'
506532
Ahhhhhhh... isn't it great to be 30 going on 10.  Life it too short to be grown up.  Greatest shaped object huh.... okay well lets see.... you have limited my choices here, that is not fair btw. :p ~ How about a circle.  You can not invert it mathematically.  It has no beginning nor ending.  You can do so many wonderful things with it, such as foot ball, balloons, rings and what not.  It is perfect, yet can not be computed to the last digit (PI) and last but not least, because I say so.  LOL  :)
435410
2i or -2i, where i = sqrt(-1)\n\nIf you haven't learned imaginary numbers, then your answer would be that the sqrt(-4) is not a real number.
885741
Do we lose our math skills if we don't use them?\n\nH3LL YES, WE DO! Graw! It's durned frustrating too!\n\n1) Multipications tables? To be quite sadly honest with you there are huge chunks of it that are gone. Just plain ***gone***. I worked in opperational nuclear power as an engineering labrotory techitian for ten years and I can not with any degree of confidence tell you what seven times eight is. No need too... trusty calculator will tell me.\n\n2) No... I have managed too hang on too this. But then again... I use it!\n\n3) ***some of it*** has managed too stick. A few months ago I had too look up some very basic trig info on the net too work out the spacing for some holes I had too drill on an round access cover. That was a touch more than embarassing. "Yeah. Can't get this done today because ... um.... I don't have the right drill bit... yeah... drill bit.... that's the ticket."\n\nSo too reitterate... Yeah. Use it or lose it.
368083
Imaginary and Complex Numbers\n\nDo not let someone argue that all numbers are imaginary...just a side note.
1137735
use  brain\nif not  calculator
517182
I don't think it is possible to actually comprehend i, since it is the square root of -1.  We don't know of any numbers whose square equals a negative number, which is why i is imaginary.  When you square a negative number, the negative signs cancel out, giving a positive value.  In order to get a negative square, you would have to have some sort of crazy negative/positive amalgam or some type of half-positive-half-negative number.\n\n\nimaginary numbers are ones that aren't possible in our normal, continuous mathematics.  They are virtually always represented with i, since i is the basis for imaginary numbers.\n\nI would say that yes, you can have negative somethings, even though you have to imagine them (which is why "imaginary numbers" might be a misnomer).  For example, if you have six apples but you are hungry, you might think of your hunger as -2 apples.
807333
Imaginary numbers, actually complex numbers, are enormously useful in many areas of engineering.  One example is in trigonometry where difficult trigonometric identities become almost trivial to derive.  Another is in analyzing circuits.  Using imaginary numbers allows Ohm's law to be applied to passive (resistors, capacitors and inductors) AC circuits.  Euler's identity allows cubic equations to be solved analytically.  Applying the concept of graphical representation of 2-D vectors provides very powerful tools for the solution of 2-D kinematics problems and difficult geometry problems.\n\nI am a research engineer and I use complex numbers all the time in my work.  It is by far one of the most useful tools that I have to solve difficult problems.
945266
Does 0.9999999999999999999999999....(continued) = 1.0\n\nA great topic in maths which will actually interest people because the above statement is true.