What are some of the practical applications of imaginary numbers?
371301
1. Solving AC circuits that have resistors, inductors, and capacitors in them.\n2. Signal processing\n3. Solving differential equations.\n\nA smart guy by the name of Euler showed that e^(i*x) = cos(x) + i*sin(x), thus allowing the conversion of a complex number in rectangular form (a + b*i) into polar form (r*e^(i*x)). It also allows us to write sinusoids in terms of complex exponentials, as in:\ncos(x) = (1/2)(e^(i*x)-e^(-i*x))\nIt turns out, perhaps unexpectedly, that complex exponentials are oftentimes easier to deal with then sinusoids. It also turns out, according to a guy by the name of Fourier, that most "well-behaved" functions can be written in terms of sinusoids (and thus can be written in terms of complex exponentials as well). This paved the way for a lot of neat applications. Google "complex exponential" (or any of the terms or names I dropped here) if you want more details.
322093
Hello.  First you have to determine if you are going to find 'real' answers.  use b^2 - 4*a*c, where b=-3, a=1, c=5 (from ax^2 + bx + c).  \n\nNow, b^2 - 4*a*c = 3^2 - (4*1*5) = 9-20=-11, hence, you will have imaginary roots in your answer.\n\nNote that sqrt(-1) = i, then,\n\nx^2 - 3x + 5 = 0\n(x - [3/2])^2 + 5 - (3/2)^2 = 0 ;(Completing the roots)\n(x - [3/2])^2 = -11/4\n(x - [3/2])^2 = (i^2) * 11/4 ; (note that i^2 = -1)\n\nx - 3/2 = +/- i * 0.5 * sqrt(11)\nx = (3/2) +/- i * 0.5 * sqrt(11)\n\n------------\nAlways remember to check if b^2 - 4*a*c > 0 or < 0 then you will know if there is imaginary i in the answer..
29898
Well it depends on what you want to define as a "number". Since you have already included imaginary and complex numbers, I might suggest you look at Hamilton's quaternions, which are noncommutative, and the nonassociative Cayley numbers, as well as Cantor's transfinite numbers. \n\nOf course, if we follow Leopold Kronecker, we might ignore everything except the integers:\n\n"God created the integers," wrote mathematician Leopold Kronecker, "All the rest is the work of Man."
120558
Another answer suggested you could write your own\ncomplex functions, and while strictly possible, puts\na lot more burden on you than you might want.  There's\nvery little reason to reinvent the wheel.\n\nThe good news is that complex numbers are supported\nby both the current C ("C99) and C++ ANSI/ISO standards.\nSo if you have a recent compiler, you will just be\nable to use the complex type.\n\nIn the C++ standard, see chapter 26 (specifically 26.2).\nThe header is <complex>, and all of the expected operators\nare provided.\n\nIn the C standard, see section 7.3.  The header is\n<complex.h>, and it uses the new built-in type "complex"\nto provide all of the functions that used to only\nbe provided for float and double.
455504
I will explain a method that is the most easy on this planet, and a unique one.\n\nI will illustrate with the help of a simple example.\n\nEg. 4-2.\n\nDraw no. line as follows:\n\n<-6--5--4--3--2--1--0--1--2--3--4--5--6--7-->\n\nNumbers left to zero are negative integers\nand to the right are the positive ones.\n\nLocate 4 on this line. Now move 2 steps behind 4 as the operation to be carried out is subtraction.\n\nThis will give answer as 2. \n\nSimilarly for addition move the steps ahead.
1085635
First of all, you (along with some other people here, seem to be confusing imaginary and complex numbers).  Complex numbers make a big set.  Imaginary numbers make a smaller set inside the complex numbers and real numbers make a smaller set inside the complex numbers.  These two sets, however, do not make the entire set of complex numbers.  In other words,\n1.All real numbers are complex numbers.\n2.All imaginary numbers are complex numbers.\n3.There are complex numbers that are not real number and they are not imaginary numbers either.\n\nDraw a big circle and draw two smaller circles inside the big circle.\n\nSo from now on, everybody, be careful when you use the words imaginary and complex.  They are not the same things.\n\nNow for some clarifications:\n1.A complex number can always be written as a+bi where i is the imaginary unit (square root of -1).\n2.In any complex number a+bi, a is called the real part.\n3.And b is called the imaginary part.\n\nSo the number 5 for example, is a real number AND it is also complex because 5 can be written as 5=5+0i.\n\nThe number 2i for example, is an imaginary number AND it is also a complex number because 2i=0+2i.\n\nSo now the light bulb goes off and everyone realizes that a real number is just a complex number with the imaginary part being zero and an imaginary number is a complex number with the real part being zero.\n\nThe number 1+2i is the last example.  It is a number that is not real not imaginary.\n\nHere is an interesting thing.  On the real numbers, we have this notion of ordering, meaning if I give you any two real numbers, you can always put <, >, or = between them, meaning you can always compare them.  The real numbers are ordered, in fact, they are "well-ordered".  What about the complex numbers that?  How in the world do you compare 1+i to 1-i?  Is it bigger than, less than, or even equal that goes between them?  Well, the answer is that they are both equal to each other.  But dig in, and see if you can find out why.\n\n----------------------------------------\nNow for the question you asked,  pretty much all the engineering fields make extensive use of complex numbers but electrical engineers live, eat, drink, and breathe them.\n\nBut no worries, they were not invented only for them.  In math, if you go far enough, it turns out that it actually is a lot easier to work with complex numbers than with real numbers only.\n\nHere is an example you should be familiar with, when we want to find roots of polynomials.\nIf we consider a polynomial, as a polynomial with real numbers, then we know for a fact that you can't always find all of the zeros.  You might not find any roots at all. x^2+1=0 has all real coefficients but no roots which is a problem.\nNow, if we consider the same polynomial as a polynomial with complex numbers, we are ALWAYS guaranteed ALL the roots.  Because remember, all real numbers are complex numbers as well. So x^2+1=0 is a polynomial with complex coefficients.
977033
I happen to care quite a bit about imaginary numbers as a person who studies electrical engineering.  In electrical engineering, we use the imaginary number (denoted i or j) to help represent sinusoidal waveforms, work with linear systems in the frequency domain, solve sinusoidally-excited RLC circuits etc.  It shows up quite a bit in my chosen area of study.  Even something as mundane as the voltage sine wave at a common household outlet can be represented by a phasor 120*e^(i*theta) where theta is the phase shift of the sinusoid.  Also, I hear physicists use imaginary numbers in some applications.\n\nAnyway, one of the interesting properties of the imaginary number is that when you square it you have -1 (since i = sqrt(-1)).\n\nSo:\ni^300 = (i^2)(i^2)(i^2)...(i^2)\ni^300 = (-1)(-1)(-1)...(-1)\n\nThe above set of (i^2) factors will repeat 150 times (the exponents add).  Since (-1)^150 = 1 we have our answer (check with your calculator or realize that an even amount of -1 factors will always give you 1).\n\ni^300 = 1
493159
It does not have a real number value and is therefore defined as "i". i is an 'imaginary number'. The concept is a little too involved for me to be able to supply a complete answer describing imaginay numbers though. Hope that helps.
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.
567357
I can tell by the way you asked the question that you probably already know the answer. Right?  \n\nI like how you phrased the question.  "Got a 'complex' question...?"\n\nThe set of complex numbers are made up of Real and Imaginary numbers. Real numbers are numbers that we use in everyday life. Where as complex numbers could include imaginary numbers, such as the square root of a negative number.\n\nDid you know that a mathematician by the name of Leonard Euler (pronounced "oiler") was the one who came up with the imaginary number "i"? which represents the square root of a negative number? He also came up with the number "e" for 2.718...
19513
This seems to be a popular question. There is no real number that is the square root of -1.  However, it turns out that it is useful to develop as set of numbers that allows the square root of -1.    This is called "i" as is known as an imaginary number.  The set of numbers that includes i is called the set of complex numbers.
284143
j * sqrt(pi)\n\nj = i; which is imaginary number which is sqrt(-1)\nsqrt is "square root"
140502
The trigonometric identities have as their primary purpose to simplify calculations.  For example, why evaluate 2sinxcosx when you can evaluate sin2x.\n\nAs for the applications, oblique triangles are used in surveying and mapping.  For example, suppose you want to know your distance from an object on the opposite side of a river.  You can mark off a line of a fixed distance, say 100ft.  Then, go to the opposite ends of the line and use a transit to measure the angle between the object and the line you measured.  You can now determine all three sides with this information.
175089
i is an imaginary number and has no value.
735909
If you want an analytical method, use a Taylor series.\nIf you want an iterative method (successive approximation, to arbitrary precision), use the CORDIC algorithm.\nThere are surely heaps of other methods.\n\nTaylor series:\narcsin z = z + (1/2) z^3/3 + (1*2)/(3*4) z^5/5 + (1*3*5)/(2*4*6) z^7/7 + ...\nSee http://en.wikipedia.org/wiki/Arcsin#Infinite_series\n\nCORDIC algorithm:\nCORDIC is an iterative (not analytical) method involving shift-and-add/subtract, and some table lookup.\nIt doesn't even use multiplication! It's very efficient for both hardware and software implementations.\nhttp://en.wikipedia.org/wiki/CORDIC\n\nAs to gp4rts's comment, by "division" I presume you mean the hard kind, i.e. floating-point division esp. by a variable not an integer constant. 1/(2*4*6) is pretty trivial stuff.\nFor what it's worth both logarithms and square-rooting require either division and/or iterative methods (http://en.wikipedia.org/wiki/Division_%28electronics%29).\n\nSo the bottom line is if you want to avoid *all* division, even the easy integer stuff, use CORDIC. CORDIC is commonly used in chips today, it's a very compact implementation whose speed is good-enough for many applications, and can be pipelined.
942335
as you want an elaborate answer you can refer the following link:\n\n\nhttp://en.wikipedia.org/wiki/Computer_numbering_formats\n\nhttp://en.wikipedia.org/wiki/Integral_data_type
120160
http://mathworld.wolfram.com/ComplexNumber.html\n\nfind definitions of absolute value (||), power and product.\nIt should be straightforward from there.