random question

[cara.mel]

Ok, you know how ages ago they told us the square root of negative numbers doesn't exist, but this year we learnt they do?

Do other numbers they told us don't exist, exist as well? like log -1, arcsin 1.1 etc?

[Collin Li]

Nope. The realm of complex numbers is as far as the number system goes. Those numbers are simply undefined. There's no number for 1/0 either

I think in some maths subjects, you will treat infinity as a number (with special rules, obviously), and 1/0 can give infinity.

[Fitness]

Do other numbers they told us don't exist, exist as well? like log -1, arcsin 1.1 etc?

I want to say no. But thinking about it, it only sounds as stupid sqrt(-1) ...

[Ahmad]

They do exist, but they're just complex numbers. Log is really a multi-valued function, so there's more than one answer.

x = log[-1]
e^x = -1 = e^(i (pi + 2pik))

=> x = i(pi + 2pik), k in Z

For arcsin[1.1]:
y = sinh

• = -isin[ix]

x = arcsin[iy]/i = arcsinh[y] = log[sqrt[1+y^2]+y]
=> arcsin[iy] = i log[sqrt[1+y^2]+y]
iy = 1.1 => y = -1.1i
then you get x in terms of log which is again multi-valued

[Collin Li]

Wow! I shouldn't have answered, because I wasn't aware of the fundamental definition of those elementary functions. I know that there is no number field larger than the complex ones, at least.

[cara.mel]

Thank you

What's sinh? =P

[Ahmad]

sinh

• is defined as (e^x - e^-x)/2