Hi everyone. I've had a look a some of Maxwell's equations, and I've also looked into equations relating to the reflection of waves at a dielectric. I've used complex numbers before, but only in pure maths, and I'm curious as to how imaginery and complex numbers fit into real life. I understand that they are very important, but what is it about the properties which means that they have to be used? Thanks, Sparky
It is hard to give a satisfying answer to this question. The fundamental reason for complex numbers is completeness. Without them there would be a whole class of polynomials that had no solutions. Complex numbers are required so that every polynomial has a number of roots equal to the degree of the polynomial. In linear algebra they are required for finding eigenvalues and eigenvectors for a large class of matrices with real elements. Another analogy is that multiplication by j is actually a counter clockwise "rotation" operator. Dividing by j is a clockwise rotation operator. [Edit] The really cool thing about this is, that rotation is baked into the algebra with nary a sine, cosine, or arctan anywhere in sight. Euler's identity relating e, pi, and -1 also requires the imaginary unit. http://en.wikipedia.org/wiki/Euler%27s_identity
Eulers identity has brushed my feathers backwards, in such a way, to a degree, where this thread has made me realise; simple math is beautiful.
There is nothing imaginary about imaginary numbers. They are "real" and they exist. A complex number is a mathematical expression for combining into one expression two pieces of information from two separate planes. Don't let the words "imaginary" and "complex" put you off.
The most intuitive answer is that Imaginary quantities in electronics designate STORAGE of energy, while the REAL part designates DISSIPATED, or LOST energy. The term "Imaginary" is rather unfortunate, because it has a very real significance...(No, not a REAL significance!). Or even simpler....REAL components get hot....imaginary components do not. Eric
It's all becomming clearer. So is it a resonable way to look at "imaginary numbers", is that they are simple a number which is split into two distinct parts in order to solve equations which have no "real" answers with standard, one-parted numbers?
I had an epiphany a few years ago. Imaginary numbers are a sign, similar to + and -. + * + = + + * - = - + * j = +j - * - = + - * j = j- j * j = - The last statement above is why I came to the conclusion. Since imaginary numbers are absolutely required to describe real properties of real objects (filters) I concluded they are real a long time ago. To me they illustrate math is a human concept, a human invention, used to help describe the world we live in. If we ever run into aliens it will be interesting if they used something like math, but not the same one we came up with. My feeling is there are so many internal biases of how our rules operate them we simply can not see it. We are locked in somewhat by how our brains are wired.
Have a care with some of these homespun interpretations. They often contain part truths and mixed up definitions, which will lead you into trouble if you progress further. Strictly imaginary numbers require one piece of information, not two. So √-7 is an imaginary number. Meaning simple that there is no solution to the equation x = √-7 to be found in the real number system. We find that we can represent all imaginary numbers by expanding √-7 = (√-1) * (√7) So we only require one new symbol for all the imaginary numbers viz (√-1) This is usually given one of the symbols i , j or k. So an imaginary number has the form jp where p is any real number. A complex number requires two pieces of information and is formed by combining a real and an imaginary number {a+jb} Is such a 'complex' number. Real numbers and imaginary numbers may have a sign. Complex numbers may not. Are you with me so far?
True, "imaginary" is a misnomer. "imaginary" components are no different from "real" components. It is just a way of distinguishing one component from another. "imaginary" components contain energy and can make things get hot.
So you disagree with my statements? I do believe the base definition of Square Root of a -1 was understood.
Is this a guilty conscience speaking, Bill. There have been several rather lax statements in this thread by various sources. I am not trying to single anyone out. Most people get the general idea but... It is important to realise that there is a difference between an imaginary number and a complex number. It is important to realise that there are pitfalls associated with regarding ib as a rotation. OK so we say the iy part is a rotation when we write x+iy, but why do we write not ix+iy? After all we write ix+jy+kz in Maxwells equations. All I am trying to do is lead sparky49 along good paths. But it is a very complicated subject, I think best tackled in bite sized morsels.
Wookie has described himself as a math brick. I am in the same situation overall, I have forgotten a lot of what I once knew, and I wasn't that proficient with higher math to begin with. Certain maths I have hung on to because they are useful. I don't consider them to be very advance, but to someone out side the field I would guess they are esoteric enough. Which is a long winded way of saying I'm not too secure in my math skills.
You clearly distinguished between complex and imaginary numbers here, but contradicted this in post#6. Do you wish to continue my step by step approach? No criticism is intended.
Yes, step by step is great. Post 6 was a bit of a mess up, I was too lazy to think about what I was actually referring to. Sorry about that. I definately understand the difference between complex and imaginary numbers.
You mentioned Maxwell's equations. Does this mean you are familiar with differential equations? If so how about Laplace transform methods? The motivation behind most of engineering maths is to replace a difficult problem with one that is easier to solve.
Differential yes, but I'm not sure about Laplace. Quiet often I know how to do stuff, but not their names.
OK one answer your original question. Continuing with the engineering idea of making maths easier: A transform is a method where we change a difficult equation or expression into one that is easier to handle. Sometimes we have to accept that the transform results in two (or more) easier equations replacing one difficult one. Then we manipulate or solve the easier system. Finally we apply the reverse transform to convert the easier system back to the original. This is a bit like a lever. We can apply more force with a lever, but at the expense of having to travel further with the crank. A Laplace transform is this sort of method. We take the transform of a differential equation. This turns the differential equation into a simpler algebraic one. After solving the algebraic one we transform back. Laplace transforms are based on multiplying by an 'integrating factor' which contains an imaginary number
A simple example of a transform that was used extensively in the days before calculators was the logarithmic transform for multiplication. To multiply two numbers A and B Take the log of A and the log of B Add log(A) + log(B) Take the antilog to find AB = antilog{ logA + logB} This did not involve complex or imaginary numbers. But addition is a much simpler operation than multiplication, especially for 10 digit numbers.
I've thought the same thing. We call numbers like Pi "irrational" because we can't reconcile it with ouir man-made nuimerical system, when, in reality, Pi is the most rational identity in the universe. Perhaps we need a numnerical system based on Pi! Eric
Except the words "rational" and "irrational" in the mathematical sense have nothing to do with the common definition of those words. A rational number is one that is formed from a ratio of two integers. An irrational number is one that cannot be expressed as the ratio of two integers. These definitions have nothing to do with behavior.