why don't imaginary numbers make so much sense?

PG1995

Joined Apr 15, 2011
818
Hi

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

THE_RB

Joined Feb 11, 2008
5,438
You can't divide by zero in the real world because infinity simply doesn't exist in the real world.

Infinity only exists conceptually, as a CONCEPT, to make the work of mathematicians easier as it allows them to use the inverse of zero in their calcs.

Wendy

Joined Mar 24, 2008
22,332
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Whew! For a moment there I thought you were going to discuss the √-1 , or j1.

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russ_hensel

Joined Jan 11, 2009
825
Actually, in a way you can divide by 0 if the numerator is also and you are willing to take a limit. Consider x/sin(x) and x = 0. Result is 1.

t_n_k

Joined Mar 6, 2009
5,455
At the heart of complex number representation of sinusoidal quantities is the ease with which their representative complex phasor "equivalents" can be manipulated algebraically. Rather than having to handle time domain trigonometric functions we can use simple algebraic relationships involving complex numbers in the frequency domain. Complex number representation is the key to the two-way door between the time domain world & the frequency domain world.

The notion stems from the brilliant Swiss mathematician Leonhard Euler's work in the 18th century.

True - it goes against our intuition to think of √-1 as having any use in the physical world. Yet it does and it works very nicely. Keep in mind that many accepted mathematical concepts are relatively modern. You mentioned division by zero. The idea of 'zero,' is "relatively" new in human history and it took some time for it to be accepted without reservation.

Do you believe in parallel universes or more than three physical dimensions? Many people do and with some justification.

BillO

Joined Nov 24, 2008
990
Actually, in a way you can divide by 0 if the numerator is also and you are willing to take a limit. Consider x/sin(x) and x = 0. Result is 1.
Not altogether true. x/sin(x) is undefined at x=0

Sure;

$$\stackrel{lim}{\small{x\rightarrow 0}}\ \ \ \frac{x}{sin(x)}\ =\ 1$$

But;

$$\frac{x}{sin(x)}\ \neq\ 1, \ \ \ \ where\ x\ = \0$$

russ_hensel

Joined Jan 11, 2009
825
Not altogether true. x/sin(x) is undefined at x=0

Sure;

$$\stackrel{lim}{\small{x\rightarrow 0}}\ \ \ \frac{x}{sin(x)}\ =\ 1$$

But;

$$\frac{x}{sin(x)}\ \neq\ 1, \ \ \ \ where\ x\ = \0$$
You seemed to have missed "if your are willing to take a limit". Moderen physics, to say nothing of mathematics, is full of methods of dealing with nominal infinities. Consider "removable singularities" and "quantum renormilization". Yes none of them let you evaluate 1/0 as a simple division with a finite result.

atferrari

Joined Jan 6, 2004
4,250
At the heart of complex number representation of sinusoidal quantities is the ease with which their representative complex phasor "equivalents" can be manipulated algebraically. Rather than having to handle time domain trigonometric functions we can use simple algebraic relationships involving complex numbers in the frequency domain. Complex number representation is the key to the two-way door between the time domain world & the frequency domain world.

The notion stems from the brilliant Swiss mathematician Leonhard Euler's work in the 18th century.

True - it goes against our intuition to think of √-1 as having any use in the physical world. Yet it does and it works very nicely. Keep in mind that many accepted mathematical concepts are relatively modern. You mentioned division by zero. The idea of 'zero,' is "relatively" new in human history and it took some time for it to be accepted without reservation.

Do you believe in parallel universes or more than three physical dimensions? Many people do and with some justification.
Intuition has nothing to do with Maths other than leaving you perplex and maybe unhappy when you fail to grasp concepts.

BillO

Joined Nov 24, 2008
990
You seemed to have missed "if your are willing to take a limit". Moderen physics, to say nothing of mathematics, is full of methods of dealing with nominal infinities. Consider "removable singularities" and "quantum renormilization". Yes none of them let you evaluate 1/0 as a simple division with a finite result.
No, I didn't miss it. The issue I have is saying, or writing what you wrote:

Consider x/sin(x) and x = 0. Result is 1.
In taking a limit you are allowed to arbitrarily approach 0. You are not allowed to actually be at 0, which is what you wrote.

The correct way to write it is;

$$\stackrel{lim}{\small{x\rightarrow 0}}\ \ \ \frac{x}{sin(x)}\ =\ 1$$

Or, if you wish;

$$\frac{x}{sin(x)}\ =\ 1\ \ \ \stackrel{lim}{\small{x\rightarrow 0}}$$

Just me being picky.

Joined Jul 7, 2009
1,583
I suspect everyone, including the OP, can profitably read Feynman's chapter 22 in his volume 1 of the Feynman Lectures on Physics. The topic is basic algebra and the whole can be understood with an exposure to basic high school math. It is a remarkable exposition by a master teacher and provides a rather profound view of these topics. In particular, for the OP's benefit, it discusses complex numbers, their origin, and has the underlying threads of abstraction and generalization running through the text. If, after reading this piece by Feynman, you're still confused by complex numbers, you should at least be confused on a higher level (with apologies to Fermi).

If you've never read it, do so -- you'll be delighted with the presentation and the topics. He ends with Euler's jewel, which relates algebra and geometry -- and Feynman considers this the most remarkable equation in mathematics. It would be hard to disagree.

To whet your appetite, here's the last paragraph of the 10 pages of text:
When we began this chapter, armed only with the basic notions of integers and counting, we had little idea of the power of the process of abstraction and generalization. Using the set of algebraic "laws," or properties of numbers, Eq. (22.1), and the definitions of inverse operations (22.2), we have been able here, ourselves, to manufacture not only numbers but useful things like tables of logarithms, powers, and trigonometric functions (for these are what the imaginary powers of real numbers are), all merely by extracting ten successive square roots of ten!

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PG1995

Joined Apr 15, 2011
818
Thank you very much, everyone. You guys are really nice and helpful.

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

Wendy

Joined Mar 24, 2008
22,332
Electronics especially. I mentioned the imaginary number of the square root of -1. It is commonly used for filter design, you could not do the math without it. Of course, you also have positive and negative number with analog computers, not to mention they (analog computers) do calculus in real time. During WWII they were absolutely vital for things like bomber sights.

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ErnieM

Joined Apr 24, 2011
8,126
In his classic "One, Two, Three...Infinity" (1947, revised 1961) George Gamow demonstrates how to find a pirate treasure using sqr(-1).

BillO

Joined Nov 24, 2008
990
... Now coming to the main point. Suppose, we end up with a complex solution, i.e., solution which contains complex number.
Unless you need the result to be used in another complex function or relation, then, in some cases you would just truncate the imaginary term. In other cases it might be the magnitude of the complex number that is of interest, or possibly it's real square, as in QM. Either way, the results used are real numbers.

Papabravo

Joined Feb 24, 2006
15,530
The imaginary unit comes from a nearly trivial polynomial equation
Rich (BB code):
x^2 + 1 = 0
It is obvious from both an algebraic and geometrical point of view that this equation has no real solutions. The solution requires an abstraction. This abstraction allows all polynomial equations to have solutions. It is the usefulness of such things. You can accept them or not as the mood moves you, and the rest of us could care less which way you decide.

Papabravo

Joined Feb 24, 2006
15,530
Thank you very much, everyone. You guys are really nice and helpful.

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
If you take a second order system composed of a spring, a mass and a damper and give it an initial displacement the the roots of the characteristic equation may be complex numbers. The real part, which should be negative, corresponds to a decaying exponential envelope. A positive real part would correspond to a system whose response will increase exponentially, without bound, until the system destroys itself.

The imaginary part corresponds to a periodic signal (sines and cosines) which is then multiplied by the decaying exponential envelope.

That is the real physics behind the mathematics of complex numbers. The purpose of Mathematics is not numbers, but understanding. How you feel about these things is quite simply irrelevant.

Wendy

Joined Mar 24, 2008
22,332
There may also be some confusion about imaginary numbers. In mathematics I was taught that the term imaginary number has a definition of square root of -1.

I notice the OP is referring to something else, but it isn't an imaginary number.

Papabravo

Joined Feb 24, 2006
15,530
I reread the original post and the notation was sqrt(-1). It is a common notation in programming languages that I understand to represent the square root of negative 1.

justtrying

Joined Mar 9, 2011
432
The use of imaginary numbers - it all depends on the original problem and context, doesn't it? Could be discarded, could be split into real and imaginary part with real part being one defined quantity and imaginary part being something else (think vectors). They are really useful.
I think someone already mentioned that math is not invented but discovered, so true. Now how about discussing what is 0/0, that has been bugging me for a long time

THE_RB

Joined Feb 11, 2008
5,438
... Now how about discussing what is 0/0, that has been bugging me for a long time
1. Zero is real, it exists, so rules apply that apply to other existing things.
2. Nothing can be smaller than zero so there cannot be multiple items contained within zero. Hence the answer cannot be greater than 1.
3. That means there would be the ONE original zero contained within the zero.
4. As a real world proof eliminating math prejudices, there would be one dog contained within dog, so dog/dog = 1, and this applies to all things so 0/0 = 1.