Discovering A Queer Property of Golden Ratio

It was another of those hectic college days. I was hanging out in the corridor, trying to catch some air before the next lecture. A question crossed my mind -
Is there a number, whose reciprocal is the fractional part of the number? Eg: If the number is 1.f, then the reciprocal of this number will be 0.f.

Instantly, the golden ratio 1.618033988...popped up in my mind. I knew, at that moment, that the answer has to be the golden ratio. However, I didn't want to credit myself with the answer, until I have verified it. I made a mental note to check the reciprocal of the golden ratio, once I am done with the lecture. Surprisingly, I found that the golden ratio does satisfy the rule.
1 / 1.618033988... = 0.618033988...
If you have seen or read the Da Vinci Code, you are already familiar with the beauty of the golden ratio and the bountiful ways it's exhibited in the nature. For those who are new to the golden ratio, here are a few links so you can read up some interesting properties of the golden ratio & why it is named so.

After finding the fractal number, the next step was to figure out the "why" & write it down in the form of an equation. Coming up with an equation isn't very difficult.
If x is the number we seek, which has an integral part (denoted by [x] & a fractional part), then we can write,
1/x = x - [x]

To give an example of what this equation means, lets say we are seeking a number 1.f such that,
1/1.f = 0.f
We can also write 0.f at the RHS as
1/1.f = 1.f - 1
which is same as writing,
1/1.f = 1.f - [1.f]           { since the integral part of 1.f is 1 }

In general, we can say,
1/x = x - [x]
This boils down to,
x^2 - x[x] - 1 = 0

How do we solve this equation? What are we supposed to do with [x]? We can't simply multiply x and [x].
What we have to do next, is to make a choice...well...the blue pill or the red pill? Kidding.
We have to make a simple choice of [x]. Since [x] is the integral part of x, [x] is always a whole number.
Which whole number, you may ask.
If we are looking for a number of the form 1.f, then [x] is 1. If we are looking for a number of the form 2.f, then [x] is 2, and so on.

In general, [x] can be replaced by any number n. The solution we thus obtain, will be a number of the form n.f whose reciprocal will be 0.f

Lets work it out!

We were originally looking for a number of the form 1.f whose reciprocal will be 0.f. The integral part of the number 1.f is 1. Which means, our choice of [x] will be 1.
The equation we formed was
x^2 - x[x] - 1 = 0
Sustituting [x] = 1 in the equation, it turns out to be
x^2 - x - 1 = 0.
If you solve this quadratic, you will find
x = 1/2(sqrt(5) + 1)
For now, lets ignore the other root 1/2(sqrt(5)-1)

Arent you curious to find out what the value of 1/2(sqrt(5)+1) is?
Well, its      1.6180339887498948482045868343656...
And what about its reciprocal?
Its              0.6180339887498948482045868343656...

Just what we wanted. Exicting, isnt it?!

(Remember we ignored the other root 1/2(sqrt(5)-1)? Want to find out what it comes out to be?)

So far, so good. It isn't time yet to put our feet up on the table & rest on our laurels.

Lets dig a little bit deeper.

Now, the number we found was of the form 1.f.
The next question is, is this the only number of this kind? Can we find a number of the form 2.f, whose reciprocal is 0.f.
Yes! We can find it the same way we found the number of the form 1.f.
Lets use our original equation.
Since the integral part of 2.f is 2, just substitute [x] as 2 in the equation we had derived.
The equation x^2 -x[x] -1 = 0 thus becomes
x^2 - 2x - 1 = 0.
Solving this equation for x, we find
x = 1 + sqrt(2)        [Ignoring 1 - sqrt(2)]
It turns out...
x =       2.4142135623730950488016887242097
And how about its reciprocal?
It is     0.4142135623730950488016887242097

Cool. Now we are getting somewhere. This means, we can also find numbers of the form 3.f, 4.f, etc. with the same property. In general we can find numbers of the form n.f, whose reciprocal is 0.f.

Now this is exciting. What we have done here is, we have discovered a whole new class of numbers! Numbers whose reciprocal is fractional part of the number!

The time has come to generalize our equation. Lets say we are trying to find a number n.f. The fractional part of n.f is n.

Re-writing our original equation using n, we get
x^2 - nx - 1 = 0
Solving this equation for x, we find
formula for solving x in quadratic equation
Substituting n as 1,2,3,4....etc., we get a whole set of numbers whose reciprocal is the fractional part of those numbers. Cool, isn't it?
I am listing first 10 numbers of the form n.f, whose reciprocal is 0.f. These are upto first 10 decimal places.
If you have a calculator within your reach, feel free to try out a few of them.

n (n + sqrt(n^2 + 4))/2 Reciprocal
1  1.6180339887  0.6180339887
2  2.4142135623  0.4142135623
3  3.3027756377  0.3027756377
4  4.2360679774  0.2360679774
5  5.1925824035  0.1925824035
6  6.1622776601  0.1622776601
7  7.1400549446  0.1400549446
8  8.1231056256  0.1231056256
9  9.1097722286  0.1097722286
10 10.0990195135  0.0990195135

Are we done yet? Well now, the choice is really the red pill or the blue pill.
We can be satisfied with our discovery, light up our pipe, puff up some nicotine & lose ourselves in the swirling ectoplasmic eddies until something else strikes us.
Or we can take the blue pill & "Show ourselves the truth"; ask ourselves some more questions, find some more interesting properties of the set of numbers we just discovered.
What questions can we ask ourselves? Well, thats another good question.
Here are a few...
What about numbers whose reciprocal is TWICE their fractional parts? What about THRICE? In general, k times their fractional parts. What is the sum of the squares of reciprocals of these numbers?
There are several other questions we can ask ourselves. There are a few interesting ones, which I will be posting in a short while along with their answers. Meanwhile, if you find something interesting about these intriguing numbers, do write to me.
I will be glad to post your findings here, along with your credits.
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