Melody

$$\frac{\sqrt{27}}{3}=
\frac{\sqrt{9}\sqrt 3}{3}=
\frac{3\sqrt 3}{3}=\sqrt3$$

i can just be a pronumeral like any other but it can also mean

$$i=\sqrt{-1}$$

Noe this has no real solutions - you cannot find the square root of a negative number but there are other numbers called imaginary or complex numbers.  i is the basis of that number system.   

$$-9n\times -15=135n$$

     $$2+\frac{1}{\sqrt2}
=2+\frac{\sqrt2}{2}
=\frac{4+\sqrt2}{2}$$

Well that would depend on how often the money is compounding but to make it easy I will say it is compounding annually.

P(1.06^20)=1000000

p=1000000/(1.06^20)

$${\frac{{\mathtt{1\,000\,000}}}{\left({{\mathtt{1.06}}}^{{\mathtt{20}}}\right)}} = {\mathtt{311\,804.726\: \!886\: \!084\: \!594\: \!591\: \!2}}$$

A million dollars in 20 years is equivalent to    $311 804  today.

Yes there was no reason to delete it.

I was just showing that a penthouse in sydney is up for sale at 30 million dollars.

It is just one floor of a building!   Plus the strata fees (compulsory fees for upkeep) are $100,000 anually!

There is some formula that you are supposed to know but I can never remember it.

so I'll do it the long way.

$$\\let\quad \theta=atan(x/a)\\
tan\theta=\frac{x}{a}\\\\
x=a*tan\theta\\\\
\frac{dx}{d\theta}=a*sec^2\theta\\\\
\frac{d\theta}{dx}=\frac{cos^2\theta}{a}\\\\
\frac{d\theta}{dx}=\frac{\frac{a^2}{x^2+a^2}}{a}\\\\
\frac{d\theta}{dx}=\frac{a^2}{a(x^2+a^2)}\\\\
\frac{d\theta}{dx}=\frac{a}{a^2+x^2}\\\\$$

I expect it is more sensible to remember the formua  LOL

it is 

atan

Alan, do you like classical music?

I'm enjoying this.

Yea apples are fickled things - they fall with the wind.     LOL