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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: cara.mel on November 10, 2007, 05:08:48 pm

Title: random question
Post by: cara.mel on November 10, 2007, 05:08:48 pm
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?
Title: random question
Post by: Collin Li on November 10, 2007, 05:20:09 pm
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.
Title: Re: random question
Post by: Fitness on November 10, 2007, 05:21:14 pm
Quote from: "cara.mel"
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) ...
Title: random question
Post by: Ahmad on November 10, 2007, 05:38:22 pm
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

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
Title: random question
Post by: Collin Li on November 10, 2007, 05:53:10 pm
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.
Title: random question
Post by: cara.mel on November 10, 2007, 05:57:37 pm
Thank you :)

What's sinh? =P
Title: random question
Post by: Ahmad on November 10, 2007, 06:02:01 pm
sinh