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Hallucinogenic Maths – The 1-in-10 Gay Problem Concluded

Visualising imaginary forms…

complex_circular_vectors

This post builds further on the concepts discussed in the original 1-in-10 Gay Problem post and its sequel, Find a way to my Heart, and delves deeper into the abstract to discover an impressive 3d form hidden in our function.  It will be assumed that the reader has studied these preceding posts (10 minutes’ fascinating reading!)

A quick recap: The 1-in-n problem reads, “If the chance of somebody having a specified characteristic is 1-in-n, and there are n people in a room, what is the probability that there is at least 1 person in the room having that characteristic?

This may be mathematically paraphrased to say:

P(there is at least one person in the room with the characteristic)
= 1 – P(nobody in the room has that characteristic)

And the probability is hence modelled with our function:

eqn1_2

In Find a way to my Heart there were two occasions in particular where we needed to limit the scope of the article to keep it concise.  The first of these concerns another interesting feature of Graphs 1 & 2, and the second delves into imaginary numbers!

Limit as n tends to infinity

There is more to Graphs 1 & 2 than just the jump in the middle between n=0 and n=1.  What about the other extremes?  Focus on graph 1 for a moment.

Graph 1

Graph 1

What happens as n gets very large indeed? Does the probability of at least one person having the characteristic get smaller and smaller forever?  Does it ever reach zero?  For example, if the chance of you winning the lottery on any given occasion is 1-in-a-million, and you play the lottery a total of a million times, what’s the probability that you will win the lottery? Is it close to zero?  Is it close to 1 (certainty)?  Graph 1 might suggest it is close to zero, but in maths you can’t just plug a big number into the equation and “see what happens” – No! – because computation can be astronomical and for even bigger numbers the function might behave differently and unexpectedly.  To answer this question properly we need to find “the limit of our probability function as n tends to infinity”, which means as n gets bigger and bigger and becomes infinitely huge.

In this case we may apply a useful tool called L’Hôpital’s Rule to find the limit.  It’s a bit too advanced to include in the body of this post but you may view the steps involved by clicking here.  It turns out that, although the probability does continue to decrease forever as n increases, the amount by which it decreases shrinks and shrinks so that there is actually a limit and it is not zero.
Rather, as n tends to infinity, the probability approaches Fullscreen capture 15082009 154510, which is approximately equal to 0.632 (just under two-thirds).  This is called an asymptote, which is a line to which the graph gets closer and closer forever but which it never quite reaches.

new Graph 1

This is an interesting result.  e, sometimes called Euler’s Number, is prominent in logarithms and calculus, and is similar to π in that it’s an irrational and transcendental universal constant with many important properties.  As is evident here, it has a habit of appearing when you least expect it!

Visualising imaginary roots

In Find a way to my Heart we said that taking square roots (or 4th roots or other even roots) of negative numbers was not allowed, which meant that between 0 and 1 we were only able to work with values of n that could be written as a fraction with an odd denominator.  However this is not strictly true, and things are about to get weird…

getreal

There are square roots of negative numbers, but they are imaginary.  There is actually a whole underworld of imaginary numbers and it is with these that we are about to play.

When you take a square root of a number, there are always two answers.  For example, the square roots of 9 are 3 and -3, because 3×3=9 and (-3)x(-3)=9 (because a negative number multiplied by a negative number makes a positive number), and the square roots of 1 are 1 and -1.  When you take square roots of a positive number like 1 or 9, both answers are “real” because they both exist in the number line.  But when you take square roots of a negative number, both answers are imaginary. They exist, but only in our imagination.  This is a strange concept to get you head round.  Picture an “imaginary unit” that you can square to reach -1.  We call that unit “i” (or sometimes “j” in engineering).  (See Suggested Reading below for more info on imaginary numbers.)

When you take cube roots, you get three answers, one of which is real (with which we dealt in the last post) but you also get two imaginary answers.  With 4th roots there are four answers, at least 2 of which are imaginary.  There are five 5th roots, six 6th roots, and so on and so forth.

What’s interesting is when, instead of graphing just the real answers (which we did to plot graph 3 in Find a way to my Heart), we plot all these answers, imaginary and real.  We take a value of n between 0 and 1 and plug it into our function, and plot every single answer it gives in a 3-dimensional graph with the real part of each answer on the y-axis (height) and the imaginary part of each answer on the z-axis (depth).

And this is what it looks like:

bluegif3

To get a better feel for where this shape sits, click the animation to view it with the three axes: Value of n (x-axis); Real part of function (y-axis); and Imaginary part of function (z-axis).

Isn’t it pretty?  It almost makes you want to reach into the computer screen to touch it.  Furthermore, if we stare down the open end of this vase-like Riemann surface we can identify a number of cardioid curves reminiscent of the heart shape we discovered in the last post.

red jpeg cardioids

I titled this post “Hallucinogenic Maths” not only to grab your attention but also because to hallucinate, in a broad sense, means to see things which aren’t there, and that’s exactly what we do when we create shapes in imaginary space; the shape doesn’t “exist” in real life but we can still play with it.

Conclusion

We have seen the original problem – of calculating the probability of there being at least one gay person in the room – extrapolated to a general case (the 1-in-n problem), then tweaked to find hidden solutions and finally abstracted into the world of imaginary numbers to deliver a most aesthetically pleasing 3d form.

Finding the limit as n tends to infinity shows how the 1-in-n problem probability can never drop below 0.632, which means we can unquestionably state, without actually having to do any more calculating, that, if the chance of you winning the lottery on any given occasion is 1-in-a-million, and you play the lottery exactly 1 million times, the probability that you will win the lottery is just over 63.2% (not even two-thirds).

The bottom line is that we have shown quite convincingly how a simple idea can lead to the most unexpected hidden beauty.

Suggested Reading

1. Imaginary Unit – Wikipedia – http://en.wikipedia.org/wiki/Imaginary_unit

  1. rik
    August 19th, 2009 at 09:12 | #1

    this is biology, maybe a fractal, not euclidian, approach makes more sense….?

  2. rik
    August 19th, 2009 at 09:08 | #2

    absolute=n+1, obviously…..

    good luck babe,

    W

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