Complex numbers imaginary part

[MikeSv]

Hello everyone.

Iam reading about complex numbers at the moment ad Iam quite confused.
I know how to use them but Iam not getting a real understanding of what they actually are :-(

What exactly is the imaginary part of a complex number? I read that it could in example be phase...

Thanks in advance,

best regards,

Mike

 

[mfb]

What exactly is the imaginary part of a complex number?

What exactly is the number 5.6?

Real numbers can be displayed as points on a line. Complex numbers can be displayed as points on a plane. The imaginary part is the vertical coordinate of the point.

 

[MikeSv]

Hi and thanks for your reply!

I found this link here:
https://www.st-andrews.ac.uk/~jcgl/Scots_Guide/info/signals/complex/cmplx.html

So as far as I understand, the imaginary parts describes a kind of rotation (or change in phase) which is represented as the vertical coordinate?
Mike

 

[mfb]

Forget that website if you are interested in the mathematical side of complex numbers.
That website discusses a specific application of them. You first have to understand the concept before you can use it in applications.

 

[MikeSv]

Forget that website if you are interested in the mathematical side of complex numbers.
That website discusses a specific application of them. You first have to understand the concept before you can use it in applications.

Hi again.

Could you recommend any good tutorial on complex numbers?

/Mike

 

[member 587159]

We can write every complex number as $z = a + bi$, where $i$ is the imaginary unit satisfying $i^2 = -1$. The number $b$ is called the imaginary part of the complex number $z$.

 

[fresh_42]

I think you should read
https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/

$i$ is mathematically simply one of the two solutions of the real equation $x^2+1=0$.
Or a bit more sophisticated: one representative of the two equivalence classes of $\mathbb{R}[x]/(x^2+1)$.
That's it.

Now you might object, that this doesn't help a lot. But in the end, it is that easy. People faced similar problems (ancient Greeks), when the had to imagine $\sqrt{2}$. It is no number. Not even finite in its decimal expression. It wasn't easy for them, as numbers always were either something to count with or a relation between two numbers, a quotient. But $\sqrt{2}$ is neither. However, they could draw it, namely as the diagonal of a square. It's not as easy with complex numbers, but in the end we found a way: the plane of complex numbers with axis $\{1\}$ and $\{i\}$. So things are a bit more complicated with complex numbers as they are with irrationals like $\sqrt{2}$, but therefore we can do an awful lot more with them. And as $\sqrt{2}$ solves $x^2 -2 = 0$, $i$ solves $x^2+1=0$. Finally it comes down to this point.

If you like, you can go back even further in time. The Babylonians needed negative numbers - nothing you can find in nature. But they did their accounting and book keeping with them. They were used for economic reasons and people didn't bother whether they are natural or not. They simply helped them doing their calculations. I'm almost sure, they did not write $-3$ but used two sides of a balance sheet, one for positive and one for negative numbers. However, this is a matter of syntax only.

And my personal most favorite achievement by mankind dates back even longer. The Indians started to use $0$ for the first time. (IIRC some seven thousand years ago.) It was by no means a trivial discovery. Someone decided to count nothing! They must have taken him as crazy. To count something that isn't there. It couldn't even be drawn. There is nothing to draw. Nevertheless, it turned out to have some practical benefits.

It was a long way to go from old India to Gauß. And in all cases, it was the benefits in calculations, that founded their popularity, not their property of being imaginable.

 

[jedishrfu]

Khan Academy videos are usually pretty good:

https://www.khanacademy.org/math/al...plex-numbers-algebra-2/v/complex-number-intro

I think there are 5 parts to this sequence.

Let us know what you think and if they helped.

 

[MikeSv]

Thanks for all the comments and the link to the video lectures!

But what about sinusoidal signals in example.

What is the difference between a completely real valued sinusoidal and a complex one? What does the imaginary part do to the sinusoid?
Cheers,

Mike

 

[Ronald Channing]

The imaginary part of a complex number $\large z = a + bi$ is defined as
$\large I am ( \, z \, )= b$ and where $\large Re( \, z \,) = a$

Does this help? A complex number is defined as:
$\large a + bi$ where $a,b \in \mathbb{R}$ and $\large i = +\sqrt -1$ where $i$ is called the imaginary unit. So it "consists" of a "real" part and "imaginary" part. This is a definition. You cannot prove the definition but can prove some abstract object corresponds to the definition, of course definitions must be consistent with each other.
So $z = a + b\sqrt-1$The magnitude of a complex number uses the "real" part and the "imaginary" part:
$| z | = \sqrt{Re(z)^2 + Im(z)^2}$
and in other operations (addition, subtraction, multiplication etc).

Of course there are many complicated ways to construct them, please look at some real analysis books.
What they are? They are the same as other mathematical abstract objects, constructed out of axioms and definitions.
The real analysis books give some definitions based on order pairs whereas other application focused books simply give this definition. Yes complex numbers are not easy to use and grasp, you must practice loads of questions.

It is a definition, there are many in mathematics, axioms, definitions and theorems are what the body of knowledge can be somewhat structured into.
Definitions of such abstract objects state what the class of such things are, and exclude all others.

 

[Svein]

Thanks for all the comments and the link to the video lectures!

But what about sinusoidal signals in example.

What is the difference between a completely real valued sinusoidal and a complex one? What does the imaginary part do to the sinusoid?
Cheers,

Mike

It does not do anything, but it can make calculations easier.

Example: $e^{i\varphi}=cos(\varphi)+i \cdot sin(\varphi)$does not seem very useful, but some operations are much easier in exponents than in trigonometry. I do not think anybody remembers all trigonometric formulae, but the complex version makes it fairly easy to find them.

Example: What is the formula for the cosine and sine of a sum of two angles? The complex version goes $$cos(\varphi+\vartheta )+i\cdot sin(\varphi+\vartheta) =e^{i(\varphi+\vartheta})=e^{i\varphi}\cdot e^{i\vartheta}$$$$=(cos(\varphi)+i\cdot sin(\varphi))\cdot(cos(\vartheta )+i\cdot sin(\vartheta))$$$$=cos(\varphi)\cdot cos(\vartheta )-sin(\varphi)\cdot sin(\vartheta)+i(sin(\varphi)\cdot cos(\vartheta )+cos(\varphi)\cdot sin(\vartheta))$$

 

[FactChecker]

As @Svein shows, e^iφ=cos(φ)+i⋅sin(φ) is a great way to work with angles.

It is also the perfect way to work with rotating objects. The real and imaginary parts keep track of the coordinated X-Y position of any rotation in the XY plane. Also, the definition of the complex derivative makes d/dφ(e^iφ) = ie^iφ = -sin(φ) + i cos(φ). So it keeps track of the coordination of the X and Y derivatives of a rotating object. This extends to higher derivatives.

There are many additional physical properties that this applies to. The trade-off of potential and kinetic energy of a pendulum is easy to represent as a rotation. Likewise for the Bernoulli Principle.

Some people consider e^iφ=cos(φ)+i⋅sin(φ) to be the most important equation in mathematics / physics.