Dividing by zero creates a Big Bang, and from this any and all values arise. And then some kid jerks off in a shower and goes to hell. Or something.

Here's an advanced one for rob with complex numbers:

i = i

That's obviously true. In other words

Sqrt( -1 ) = Sqrt( -1 )
Sqrt( -1/1 ) = Sqrt ( 1/-1 )
Sqrt(-1)/Sqrt(1)=Sqrt(1)/Sqrt(-1)
Sqrt(-1)=1/Sqrt(-1)

Now we multiply by i (or Sqrt(-1))

Sqrt(-1)Sqrt(-1)=1

-1 = 1

Hey, that's sneaky!

The dodgy step has to be sqrt(1/-1) = sqrt(1)/sqrt(-1) because it's not true!

Hmm...I've never come across this before! Splitting the square root clearly isn't as simple as with real numbers. Of course sqrt(1) can be -1 as well, or sqrt(-1) can be -i, which would balance the equation above. There must be a more robust way to address it though. Always have to be careful with roots.

Yeah, exactly one sneakily moves to a different "sheet" of the square root in the complex plane. There is a less obvious version of that with exponentials that I couldn't find right away.

Nice!

I've had people say to me before, "I don't see the point of algebra. We already have lots of numbers, why do we need letters too?"

They have no idea quite how ridiculous they sound. I wonder what they would make of complex numbers?

I've always found them fascinating. They are almost like a mathematical supernatural force; they don't exist in reality, but by jumping in and out of their imaginary space you can prove results about real numbers.

Having said that, negative numbers don't exactly exist in reality either! Neither do numbers...

And down the rabbit hole we go.

Meh, complex numbers are just pairs of numbers with a funny multiplication rule
People always make them sound way too magical if you ask me. In one of Dan Browns books, the protagonist tells the skeptical physicist - well you *believe* in things too, for example in imaginary numbers"
A typical Dan Brown howler. But that's because they are always sold as something mysterious, when it's just an efficient method to calculate with pairs of numbers.

But it is strange that they seem to be built into the fabric of reality somehow. After all, the schroedinger equation wouldn't be the same without them (instead you'd have two coupled real equations)

It is bizarre! You're right, they are no more mysterious than regular numbers, they just have their own rules.

In some ways things make more sense in the complex plane, because then any order n polynomial has n roots instead of anything from 0 to n roots. It's... complete? Is that the word? Been a while since my studies It complements the real numbers to provide roots to all polynomials, not just ones that behave themselves!

Ah, yes. Algebraically closed is what we're looking for!

What I found interesting is that as you add dimensions, you keep losing properties. For example, the real numbers are a nice field with an order. Pairs of real numbers can be made into a field ©, but one loses a well defined ordering. With four numbers (quaternions) you lose commutativity, AB=BA. I don't even know what's next. Octonions I suppose.I forgot whether one can build a ring with three real numbers, maybe not.

(June 28, 2015 at 5:35 am)Alex K Wrote: Meh, complex numbers are just pairs of numbers with a funny multiplication rule
People always make them sound way too magical if you ask me. In one of Dan Browns books, the protagonist tells the skeptical physicist - well you *believe* in things too, for example in imaginary numbers"
A typical Dan Brown howler. But that's because they are always sold as something mysterious, when it's just an efficient method to calculate with pairs of numbers.

But it is strange that they seem to be built into the fabric of reality somehow. After all, the schroedinger equation wouldn't be the same without them (instead you'd have two coupled real equations)

 Yeah that's deep.  How / in what way could i (small i lol) be represented in physical reality? How does reality "know" how to express situations where it's hiding behind the surface somewhere?

Just think of something in reality with two distinct properties. The real part can then describe one of them, the imaginary part the other. For example the amplitude of a wave and its phase can be described by 1. the absolute value of the complex number, and 2. the ratio of imaginary and complex part, respectively