why don't imaginary numbers make so much sense?

[PG1995]

Hi :smile:

One of founding pillars of mathematics is that you cannot divide by zero; it's utterly absurd. Therefore, if one starts claiming that division by zero isn't absurd because it pops up while solving some problems. I think then it would cast a lot of doubt on the credibility of mathematics and give rise to disbelief in mathematics.

Another one of the founding principles of mathematics is that when you multiply any two 'real' numbers you will get a +ve number;no matter even if the numbers were -ve. Now when we write sqrt(-1), we are trying to do something which isn't allowed. I think I don't have much problem with writing swrt(-1) and calling it iota. Obviously, I would have serious problem accepting the result if it was said that sqrt(-1) equals some real number. So, I think my problem only lies in the fact that how come we end up with sqrt(-1) expression while solving other 'normal' problems. How does nature make use of such 'nonsense' expression? Could you please give me some simple example where iota is used and we can make some sense out of it? I don't think nature can make much use iota when one can't even tell which one the two or more imaginary numbers is greater; e.g. you can't tell whether 4i is greater than 2i or not! Please don't use more math to explain math. Thank you.

Regards
PG

 

[andre_luis]

PG,


The reason of existence of imaginary numbers is realised just when you use them like phasor operators.
The SQRT(-1) value is not applicable itself, and neither exists, in fact.
However, operators and transformators allow us to reduce complexity in certain domain, once migrating to another.

For instance, take the multiplication of 2 phasors :
**broken link removed**

Taking this sight, we can consider the complex numbers like a bridge between algebra and Geometry.

+++

 

[Prof78]

Consider an (x) axis along which we can plot all the numbers, starting from zero at the origin and with positive numbers to the right and negative to the left as shown in figure 1:

Consider an (x) axis along which we can plot all the numbers, starting from zero at the origin and with positive numbers to the right and negative to the left as shown in figure 1:



Consider a positive number say 3 (any number will do) and it will be represented by the vector 0A (a vector has magnitude and direction). Now how do we change this number 3 into -3. A simple way would be to subtract 6 from 3 to give 3, but this means that there is a special operation for every number rather than one single type of operation that would apply to any number. Any negative number can more particularly be written as -1×3but we normally take the -1 as understood. So we can look on × (-1) as an ‘operator’ that operates on any number to make it into the negative. The idea of an ‘operator’ is common in mathematics as for example d/dx the differential operator.
So going back to figure 1 we can say that the operator ×(-1) operates on our number to rotate it (the vector) from 0A to 0B to give -3 i.e. the operator rotates the vector by 180° (by convention the sense is taken as anticlockwise). Now if we multiply -3 (0B) by -1, then there will be a further rotation by 180° (anticlockwise) to bring us back to our original vector 0A i.e. multiplying two negative numbers together gives us a positive number: (-3)×(-1) = +3. This demonstrates the rule for multiplying signs.

The operator view of ×(-1) as causing a rotation by 180° can give us an introduction to the field of complex numbers, the more general form of numbers. Consider figure 2 with the same vectors (0A and 0B) we used in figure 1. We ask the question, what operator will rotate 0A by 90° (to give 0C) rather than 180° ? We are now operating in a two dimensional domain. The form of the operator, say ‘Op’, is not readily evident but we can say that two successive applications of Op will have the same effect as the operator ×(-1). Thus:

Op ×Op ×0A= -1×0A = 0B or Op×Op= -1 and hence Op= √-1 or i² =-1
and so Op ×0A = i ×0A = 0C

and so we discover the form of the square root of minus 1 and what it does. In the field of ordinary or ‘real’ numbers the square root of a negative number has no meaning, but in a two dimensional space it is quite simple. This is such an important matter that root minus one is given its own symbol ‘i’. However, in electrical applications where ‘i’ is the traditional symbol for current we usually use ‘j’ instead, but they are the same thing. So 0C is the number j×3 or usually just j3, the × being understood. Such a number was named an ‘imaginary number’ but this should just be taken as a name or label without some mystical meaning. A number such as 3+j4 is called a ‘complex number’ with a real part 3 and imaginary part 4. In figure 3 this is plotted as the vector 0D with x-coordinate (real axis) 3 and y-coordinate (imaginary axis) 4. If the complex number was instead 3j4 then we would have the vector/number 0E instead. Complex numbers can, just like real numbers, be added, subtracted, multiplied, divided, raised to powers and have roots; real numbers are just special cases of complex numbers. As an example Figure 4 shows the addition and subtraction of two complex numbers Z1 and Z2. The magnitude of a complex number is the length of the vector, so in the example in the post, 4j is greater than 2j. If the vectors are not (as in the previous case) along the same direction then the magnitude of a complex number (a+jb) is found from the square root of the sum of the squares of a and b. Contrary to the view expressed in the post by PG1995 complex numbers make a great deal of sense and analysis of circuits would be enormously more difficult without their use. If you do not study and understand their use you greatly restrict your ability to analyse and understand circuits.
The effect of j in rotating a vector by 90degree is very convenient in dealing with the analysis of reactive circuits where I presume you already understand the current and voltage are out of phase by 90degree, so that (for sinusoidal time variation) by expressing the impedance of an inductor L as jωL and a capacitor C as –j/ωC we take care of phase and amplitude simultaneously and so can write down the impedance of a circuit containing resistive and reactive elements easily. But what do you do if the time variation is not sinusoidal? Then these expressions no longer hold and we must seek an alternative means of analysis of the circuit. This is most readily achieved by the introduction of the concept of a complex frequency s, usually written as:
s = σ + jω
i.e. the ‘frequency ‘ has a ‘real’ part σ and the ‘sinusoidal’ part ω. To analyse circuits one can now write down impedances (e.g. sL and 1/sC }and transfer functions, the ratio of output form to input form and then make use of the powerful technique of the Laplace Transform (in effect a complex variable version of the Fourier Transform). I would urge you to become familiar with the complex variable if you intend to progress in electronics. Try:

Scott Hamilton 2003 An Analog Electronics Companion, Cambridge University Press . ISBN 978-0-521-68780-5

Prof78.

 

[PG1995]

Thanks a lot, Andre, Prof.

Some may say mathematics is invented but I'm of the position that mathematics is discovered and development of most of mathematics has been driven by humans' struggle to mathematize the real life phenomena around them. We give interpretation to many of the concepts of mathematics. For instance, depending on the context we can interpret what a negative number mean. e.g. we understand what negative temperature mean, what negative velocity mean, what negative displacement mean. So, if we end up with a negative solution at the end in a certain problem then depending on the context we can figure out what that negative solution means. Now coming to the main point. Suppose, we end up with a complex solution, i.e., solution which contains complex number. What would it imply? Can you please give me some simple real life example which elaborate this? I hope you understand what I'm after.

Regards
PG

 

[FvM]

At first sight, "real life" according to your question seems to refer to the primary field of human experience, not involving any kind of theoretical abstraction? But we are talking about numbers, nothing that has been used by a caveman, as far as we know. Although they appear as a natural thing these days, they are an abstraction as well. Negative numbers are demanding even more mental work. But once you become familiar with it's operations, you can apply them to "real life" problems like counting borrowed money.

Imaginary numbers aren't related to "real life" in the same way. There's some theory in between. But the imaginary and complex numbers are an important tool to solve equations which are describing life, e.g. a mechanical, optical, electrical problem. Physicists and engineers are considering them real for this reason.

 

[rberek]

Well, if you want a concrete example, with minimal math, the instantaneous state of an RF signal can be described by an amplitude and a phase. Complex numbers are used for this, where the real part is the amplitude and the phase is the imaginary part. Though in the RF system, the terms real and imaginary are not used, "In-phase" and "Quadrature" are used instead. And when this RF signal is digitized for signal processing, each data point becomes two 16-bit signed values which are used throughout the digital system. Even though different terms are used to describe these two numbers, they represent a complex number. This is a very simplified description.

So every time you talk on your cell phone, your voice is converted to complex numbers.

r.b.