Fun Maths Questions (1 Viewer)

[hyparzero]

Found the following questions quite interesting:

1. Find iⁱ where i = root(-1)

2. The number a is randomly selected from the set {0,1,2,3,...,98,99}. The number b is selected from the same set.

What is the probabilty that the number 3^a+7^b is an 16 digit number?

 

[Trev]

Just having a go at it, iⁱ:
Since e^ix=cisx, when x=pi; e^i.pi = -1.
ln[e^i.pi] = ln[-1]
So i.pi = ln[-1].

Let k=iⁱ
k²=i²ⁱ=(-1)ⁱ
ln[k²]=ln[(-1)ⁱ]
2ln[k]=i.ln[-1], since ln[-1]=i.pi then:
2ln[k]=i.i.pi
k=e^-pi/2
∴ iⁱ=e^-pi/2.
Yeah?

 

[Bobness]

I couldn't resist it ...

Maths is not fun

That is all

 

[hyparzero]

Just having a go at it, iⁱ:
Since e^ix=cisx, when x=pi; e^i.pi = -1.
ln[e^i.pi] = ln[-1]
So i.pi = ln[-1].

Let k=iⁱ
k²=i²ⁱ=(-1)ⁱ
ln[k²]=ln[(-1)ⁱ]
2ln[k]=i.ln[-1], since ln[-1]=i.pi then:
2ln[k]=i.i.pi
k=e^-pi/2
∴ iⁱ=e^-pi/2.
Yeah?

Yup absolutely correct. However it seems e^ix=cisx might have to be be proved using the power series beforehand.

Anyone want to take a shot at the question 2?

 

[Trev]

Power series?

 

[pLuvia]

e^iθ=cisθ

That's just one of Euler's identities