What practical applications do imaginary numbers have?
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).
425367
I think math is a convenient tool that can be used to facilitate operations in many other fields.\n\nIn computer science's programming, boolean algebra is used widely for logical operations. At the low machine level, native operations such as OR, XOR, NOT, etc are used to manipulate bits of information. At the high level, numerical analysis can be used to solve complex problems. Databases can use statistical methods to perform analysis.\n\nMath is an idealized (perfect) model that is coherent within itself. Hence it provides a very reliable framework for representation and modeling. Of course, the interpretation of the numbers is left to the more intelligent humans.\n\nComputer Science uses some aspects of Math. Math has alot more to offer.
291373
I have no idea what you're trying to ask!  This looks a little like a math question.  Does the "i" in your question stand for an imaginary number?
613651
Real numbers are what we use every day. They include all numbers other than immaginary numbers, which are the result of the sqare root of a negative number. The main disadvantage of the real number system is that almost all real number can only be approximated using the decimal system. Pi for example can only be approximated. 1/3 writen in decimal form can only be approximated as .333333333....
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).
1043176
That's what they call the imaginary number, i.  i is being used greatly in complex numbers.  Complex numbers are those 4+7i type of thing.  And complex numbers are being used greatly in physics or applied math.  I still remember those days when I had to take that Complex Variables class (I was a math major) in my senior year in university.
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.
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.
232073
Pi is what is know as an "irrational" number because it cannot be written as a ratio of two integers.  However, it can be easily defined as the ratio of the circumference of a circle (the distance around the circle) to the diameter of the circle (the circle's width).  This makes it an extremely useful number to "approximate" (since it cannot be expressed fully with a 100% accurate number) when dealing with anything that is cyclical (planetary orbits or repeating patterns for example).  I've included a link with more fun facts.
755470
You didn't answer this correctly. if the number inside the square root (-320) is negative, then it's an imaginary number.\n\nYou also shouldn't have a plus or minus sign to the left of the square root at this point of the equation if you've done it correctly.\n\nThere are two answers to this question once you've done it correctly, unless the question asks you to measure something, in which case, the only correct answer is the positive value. Try posting the actual question you were given and what you came up with. Perhaps I can help you there.\n\nGood luck
621340
The first one is (x-2)(x+9)(x-7).\nHowever, for the second part, everyone else has forgotten that imaginary zeroes come in CONJUGATE PAIRS.\nSo there are actually 5 zeroes to part 2, 4i, -5, -2i, 2i, -4i.\n\nSo the solution would be (x-4i)(x+4i)(x+5)(x-2i)(x+2i)\n(x^2+16)(x+5)(x^2+4)\nx^5 + 5x^4 + 20x^3 +100x^2 + 64x + 320\nViola...
119291
First, you won't have a complete understanding of how i is applied in real life until you study complex analysis. Note, the form of a complex number is: a + bi, where a,b are real numbers. i means a is 0 and b is 1.\n\n     i is used in the applications (conformal mapping) to solve a number of physical problems in heat conduction, electrostatic potential, and fluid flow, to name a few.\n\n     In the study of heat conduction, the FLUX across a surface within a solid body at a point on that surface is the quantity of heat flowing in a specified direction normal to the surface per unit time per unit area at the point. Flux is usually measured in calories per second per square centimeter. The stove in your kitchen has this. Remember the FLUX capacitor in the movie, 'Back to the Future'?\n\nIn an electrostatic force field, the field intensity at a point is a vector representing the force exerted on a unit positive charge placed at that point. The elctrostatic potential is a scalar function of the space coordinates such that, at each point, its directional derivative in any direction is the negative of the component of the field intensity in that direction. Gravitational potential is another example in physics.\n\nComplex harmonic function functions play an important role in hydrodynamics and aerodynamics. We let vectors be in complex form. That is, V = p + iq would represent the velocity of a particle of a fluid at any point. Note, fluid could be liquid or air around the surface, as in aerodynamics.\n\nHope this helps.
938958
No, it would have to be the square of something like ax+b, whose square is a^2x+2abx+b^2.  You would have to have a=1 or -1 and b=i or -i, but then 2ab would be imaginary, and not -2.
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.
42408
Something that calculates numbers in rapid succession.
461173
Maybe in mathematics sqrt of -1 = i, but in electrical engineering it is j.\nIn electrical engineering and related fields, the imaginary unit is often written as j to avoid confusion with a changing current, traditionally denoted by i.\n\nThis represents the imaginary component of electronics which cannot be ignored. Phasor diagrams may help to explain this phenomenon. A phasor is a constant complex number representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. It is usually expressed in exponential form. Phasors are used in engineering to simplify computations involving sinusoids, where they can often reduce a differential equation problem to an algebraic one.
926495
it can be factored over the complex numbers. \nThe complex roots are:\n(1+i)/sqrt(2), (1-i)/sqrt(2), (-1-i)/sqrt(2), (-1 +i)/sqrt(2)\ni.e.\nx^4+1 = ( x-(1+i)/sqrt(2))( x- (1-i)/sqrt(2)) (x - (-1-i)/sqrt(2) )( x-  (-1 +i)/sqrt(2) )\n'