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#1 2016-03-29 04:07:02

Hariharan
Guest

What does this equation say about ‘i’?

Greetings! I am Hariharan. I have been having this problem for quite a while. I know that something in this equation I've come up with is wrong, but I do not know where I've gone wrong and how to rectify it.

Let us consider René Descartes’ construction for square root extraction.
Draw a line AB of measure x and add 1cm length to it. Let this line be AC. Keeping AC as diameter draw a semicircle with centre O. Now, draw a perpendicular to AC at B, meeting the semicircle at D. We observe that m(BD) = sqrt(x).

Join OD. In triangle BOD, let angle BOD be θ.
Using the cosine function of this, we can say that cos(θ) = OB/OD. Substituting values of OB and OD as (x-1)/2 and (x+1)/2 respectively, we get cos(θ) = (x-1)/(x+1). This implies that θ = arccos{(x-1)/(x+1)}.
Similarly, sine of the angle θ will be BD/OD, which is sqrt(x)/{(x+1)/2}.
Substituting the value of θ in this equation, we get,
sqrt(x)= sin{arccos(x-1/x+1)} * (x+1)/2.

Let x be any positive number.
Thus, sqrt(-x) = sqrt(x) * i                                            .......(1)
From my equation,
Sqrt(x) = sin{arccos(x-1/x+1)} * (x+1)/2                     ........(2)     
Also, sqrt(-x) = sin{arccos(-x-1/-x+1) * (-x+1)/2
                     = sin{arccos(x+1/x-1) * (-x+1)/2            ........(3)


Substituting eqs.(2) and (3) in eq.(1), we get
sin{arccos(x+1/x-1)} * (-x+1)/2 = [sin{arccos(x-1/x+1)} * (x+1)/2] * i
=> i = [[sin{arccos(x+1/x-1)] / [sin{arccos(x-1/x+1)}]] * (-x+1)/(x+1)

Now, let’s look into i in terms of the Argand plane. Argand plane (or the complex number plane) is based on the fact of rotation of real numbers into a different dimension. Let me make this clear.
i = sqrt(-1)
=> i^2 = -1
=> 1* i*i = -1
“This means that i is some transformation that when applied two times turns 1 to -1. So, the only way is a 90 degree rotation to the left.”
This means that the magnitude of i is the same as 1. The only difference is that 1 and i lie in different dimensions. I guess I am right till here...
Now, this is where my question begins. My equation gives me weird answers for i. What’s wrong with my equation(s)? Why doesn't my equation bring out the above fact? Please help me to sort this problem out.
Thanks!!!

#2 2016-03-29 04:46:17

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: What does this equation say about ‘i’?

Hi;

Also ...(2) sqrt(-x) = sin{arccos(-x-1/-x+1) * (-x+1)/2

That is not working.

Sqrt(x) = sin{arccos(x-1/x+1)} * (x+1)/2

Seems to be true only for x > 0


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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