Are Complex Numbers Truly Magical?

[Telos]

Are complex numbers "magical?"

So I'm reading Penrose, and all of sudden he explodes into excitement like a schoolgirl, fawning over complex numbers because they are "magical" and perform "miraculous" things, further spilling exclamation marks in the surrounding paragraphs about how he's only scratched the surface of "number magic!"

What do you think? Are complex numbers "magical?" Do they perform "miracles?"

Can someone show me a complex number equation that applies to the real world and performs a "miracle?"

 

[whozum]

By de Moivre's formula:

exp(pi) ~ 23.14
exp(i*pi) = -1 where i = -1^(1/2)

Thats pretty magical to me.

 

[Telos]

Please elaborate.

 

[HallsofIvy]

Define "miracle"!

 

[Zurtex]

Yes complex numbers are magical, I love complex numbers, I will be proposing to them next week I love them that much. The amount of times my lecturer has made an assumption or supposedly proven stuff and I put my hand up and said "but what about in complex numbers" to show they had not rigourously enough defined their proof is just magical .

 

[hypermorphism]

Please elaborate.

Since you insist on physical world magic, two words, contour integration.
From a perspective of pure connectivity, I have more fun playing with another pair of words, Riemann sphere.

 

[mathwonk]

they are magical because they frighten timid people and reserve for the initiates a special status as wizards. when wonks become wizards, some kind of miracle has definitely occurred.

 

[Crosson]

Some results from calculus in the complex plane (complex analysis):

z is a complex number, a + bi ,where a and b are real.

We say that a function is f(z) is analytic if it has a first derivative defined by:

$$\frac {df}{dz}(z_0) = lim_{h->0} \frac{f(z_0 + h) - f(z)}{h}$$

Theorem: if a function is analytic in a region, it can be represented by a convergent taylor series in the region (it has derivatives of all orders automatically, unlike in real calculus).

We can also define line integrals over paths in the complex plane the same way we define them in multivariable calculus.

Theorem: The integral of an analytic function over a closed loop is zero.

$$\oint f(z) dz = 0$$

You know when functions have a division by zero? These are classified as "poles". Depending how severe the divide by zero is, you have higher and higher order poles. If it is infinitely severe, we call it a singularity. Functions are not analytic at poles.

All poles leave behind a certain residue. This is calculated by a simple limit, but it is a little hard to explain (read a book on complex analysis).

One of the most important theorems in complex analysis is this:

$$\oint f(z) dz = 2\pi i \Sigma Residues$$

which can be used to evaluate normal integrals such as:

$$\int_{-\infty}^{\infty} \frac{sin(x)}{x} = \pi$$

 

[Icebreaker]

It's magical, for we are mathemagicians.

$$\mathbb{MATHEMAGICS}$$

 

[Telos]

Thank you all. You've provided very good food for thought. Crosson, thank you for going through the trouble of writing all of that. It is intriguing to see imaginary numbers succeed where real numbers fail.

Icebreaker, you beat me to my next question. ;) If numbers can be so magical, why don't we just be serious about it and call mathematicians "magicians." Really - let's be serious about it.

magic n.
The art that purports to control or forecast natural events, effects, or forces by invoking the supernatural.

Imaginary numbers are supernatural numbers.

supernatural adj.

1. Of or relating to existence outside the natural world.
2. Attributed to a power that seems to violate or go beyond natural forces.
...
5. Of or relating to the miraculous.

I say if we get enough people in the right universities, we can stage a worldwide magicians coup.

 

[Icebreaker]

Math is the OTHER kind of magic -- where the hand is quicker than the eye; no supernatural forces are invoked. That is, if you don't understand how it's done, then it will seem magical to you, let it be pulling rabbits out of a hat or complex numbers.

 

[matt grime]

The "imaginary numbers" is simply an algebraic structure on pairs of reals (we need never mention i but it is a useful thing) and nothing magical at all. Its unfortunate christening as imaginary and the others as real has almost caused more harm than good, as this thread shows.