Complex Numbers (maybe to complex?)

[kurt.physics]

Complex Numbers (maybe to complex?)

I just don't get how this branch of mathematics can exist. How is it that we can use "i" or √-1, its not even real! The question I am trying to ask is, what is the use of i, how can we multiply, add, subtract e.t.c with it, doesn't that make the whole statement not true?

Could anyone help please

 

[D H]

The square root of -1 is not "real" in the sense that is not a real number. Big deal. It is a very, very useful concept and is, in a sense, very "real". Many calculations made in physics regarding very real events are not possible without complex analysis.

One can argue that none of mathematics is "real". Show me a one. Not one apple or one electron, or one of anything. I want you to show me a one. You can't. Even the counting numbers are abstract human inventions. The same goes for the rationals and the reals. Mathematics is an invention. Physicists pounce on those mathematical inventions that describes something physical. The complex numbers are one of those inventions.

The mathematics of complex numbers is very well defined. A pair of complex numbers can be added to form a sum, subtracted to form a difference, multiplied to form a product, or divided to form a ratio. One huge difference between complex numbers and the reals is in the area of the roots of a polynomia. For example, x²+1=0 has solutions in the complex numbers but not in the reals. This is the tip of the iceberg. The roots of any polynomial with real or complex coefficients are complex numbers. This fact is so important that it is called the fundamental theorem of algebra.

 

[arildno]

Is 1 "real"?
Where does it live?
Does it have a phone number?

What about 23000000000? Or 3.5?

 

[neutrino]

I just don't get how this branch of mathematics can exist.

You have the answer right there. It's a branch of mathematics. But surprisingly (or maybe not so), it has applications in other subjects, as well.

 

[arildno]

Ask yourself:
Do coordinates in the (x,y) plane exist?

I am inclined you'd answer yes for some reason!

Now, since they "exist", we may "do" stuff with them, right?

Let us make rules for coordinate "addition" and "multiplication"

Given two points, X=(a,b), Y=(c,d), we DEFINE the sum of these two be
X+Y=(a+b,c+d), where a,b,c,d are real numbers, and the plus signs within the parenthesis are normal plus between real numbers.

Now, to have a little bit of fun, we DEFINE multiplication between X and Y as follows:

X*Y=(a*c-b*d,a*d+b*c), where all signs within the parenthesis are normal number operations.


Now, let for starters b=d=0.

Then, we have:
X*Y=(a,0)*(c,0)=(a*c,0)
That is, X and Y multiplication is basically indinguishable from multiplication between real numbers.
We may even identify a real number "a" with the coordinate beast (a,0), if we like!

Let us now consider X=(0,1)=Y, and compute:
X*X=(0,1)*(0,1)=(0*0-1*1,0*1+1*0)=(-1,0)

That is, the square of (0,1) equals (-1,0) which we already have identified with the real number -1!

Thus, you can regard the complex numbers as coordinate points in a plane, that acan be subject to combinations we choose to call "addition" and "multiplication".
It is in this sense that (0,1) is the square root of the number "-1".

 

[matteo16]

also when the pitagoreans have discovered the irrational numbers it seems an unthinkable idea because they only used to think with the rational numbers.
it's the same thing
the complex numbers seem unthinkable because we think by a real conception
but it doesn't so
the complex numbers are most used for physcs analysis