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## Investment Returns and Geometric Mean What if an investment returned 5%, 20%, and 50% over three years?
Year 1:  $100 * .05 =$105
Year 2:  $105 * .20 =$126
Year 3:  $126 * .50 =$189

What was the annual return?  Can you just average the percentages?
Let’s try it.  The average of 5%, 20%, and 50% is 25%
Did you average 25% a year?

Year 1:  $100 * .25 =$125
Year 2:  $125 * .25 =$156.25
Year 3:  $156.25 * .25 =$195.31

No, 25% for 3 years gets you a different amount!

You calculate the average annual return via the geometric mean (not the arithmetic mean)

$$\sqrt{1.05 * 1.20 * 1.50} -1 = .236$$

Your annual return was actually 23.6%

Year 1:  $100 * .236 =$123.6
Year 2:  $123.6 * .236 =$152.77
Year 3:  $152.77 * .236 =$188.82 …

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## The Network Effect If you have the only cell phone in the world, it’s pretty useless, since you can’t call anyone. If there are 2 cell phones, there is one possible connection.  If there are 3 cell phones, you can make a total of 3 connections.  4 cell phones can have a total of 6 connections.  5 cell phones?  10 connections.   6 phones means 15 connections.

The more devices there are, the most connections you can make.  The more connections there are, the more useful the whole network becomes.  This is also called Metcalf’s Law.

Let’s look at the sequence of numbers generated above.

1, 3, 6, 10, 15, …

Can you see the pattern?   The number of connections can be represented by $$\frac{n(n-1)}{2}$$  where n is the number of nodes in the network.  Notice that this is very similar to $$\frac{n^2}{2} = \frac{1}{2}n^2$$.  So, the number of total possible connections is proportional to the square of the number of nodes in the network.

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## Gimmicky Bidding Websites (Series/Sequence) The way this site works is that if you bid, you have to pay that amount, even if you lose.  Bids are incremented by 1 cent.   Let’s say an item sells for 10 cents.  The guy who bid 1 cent still has to pay that, the guy who bid 2 cents still has to pay that, and so forth.   So, what does the auction site actually earn for selling that item for 10 cents?

Notice that 1+2+3+4+5+6+7+8+9+10  can be added up by grouping numbers from the opposite ends:  $$(1+10) + (2+9) + (3+8) + (4+7) + (5+6)$$   This is just $$11+11+11+11+11$$  or 11*5 = 55.  Note that when x = 10, and we ended up multiplying $$11*5$$ for the series sum.

So, the general formula is:

1 + 2 + 3 + … + n = $$\displaystyle\sum\limits_{x=0}^n x = (n+1)\frac{n}{2} = \frac{n^2+n}{2} = \frac{n(n+1)}{2}$$

Pop Quiz!  If the sunglasses in the photo end up selling for \$6.96, how much does the website make?  $$\frac{696*697}{2}$$ = $$\frac{485112}{2}$$ = $$242556$$ = $$\2,425.56$$   !!

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## Different kinds of Infinity

Which set is larger?  The set of all positive integers {1,2,3,4,…} or the set of positive even integers {2,4,6,8,…}  ?

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## Why do some habits catch on faster than others? • In the era of the iPhone, Facebook, and Twitter, we’ve become enamored of ideas that spread as effortlessly as ether. We want frictionless, “turnkey” solutions to the major difficulties of the world—hunger, disease, poverty. We prefer instructional videos to teachers, drones to troops, incentives to institutions. People and institutions can feel messy and anachronistic. They introduce, as the engineers put it, uncontrolled variability.
• Mass media can introduce a new idea to people. But, Rogers showed, people follow the lead of other people they know and trust when they decide whether to take it up. Every change requires effort, and the decision to make that effort is a social process.
•  As the (sales) rep had recognized, human interaction is the key force in overcoming resistance and speeding change.
• “Why did you listen to her?” I asked. “She had only a fraction of your experience.”
All the nurse could think to say was “She was nice.”
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## Fractals: Recursion & Iteration with Complex Numbers The essence of fractal geometry lies in recursive iteration.  What’s that?  It’s just a self-referring loop. Let’s start with a simple equation:  $$f(x)=2x+1$$

Let x=0 and plug it in, and you’ll get a 1:
$$f(0)=2(0)+1=1$$
(Now, take that 1 and plug it back into the same equation)
$$f(1)=2(1)+1=2$$        (Then, take this 2 and do the same thing)
$$f(2)=2(2)+1=5$$
$$f(5)=2(5)+1=11$$
$$f(11)=2(11)+1=23$$

And so forth.  You can keep doing this forever, and notice how this list of results (0,1,2,5,11,…) will tend towards infinity (This isn’t always the case)

The mother of all fractals, the Mandelbrot Set is defined by this deceptively simple equation:  $$f(z)=z^2+c$$ where c is some fixed constant.  If you do the same procedure above, you’ll get a series of numbers.  eg: Let c=5, and let’s start with z=0:

$$f(0)=0^2+5=5$$
$$f(5)=5^2+5=30$$
$$f(30)=30^2+5=905$$

At IBM, Benoit Mandelbrot used a complex number (a+bi) for that constant.  For example, let’s use $$c=1+i$$, as he did in his 1980 Scientific American article introducing fractals:

$$f(0)=0^2+(1+i)=1+i$$

$$f(1+i)=(1+i)^2+(1+i)=(1+i)(1+i)+(1+i)=(1+2i+i^2)+(1+i)=2i+(1+i)=1+3i$$

$$f(1+3i)=(1+3i)^2+(1+i)=(1+3i)(1+3i)+(1+i)=(1+6i+9i^2)+(1+i)=6i-8+(1+i)=-7+7i$$

…and so on. For the Mandelbrot set, the calculation is iterated until it’s clear whether the result is tending towards 0 or infinity. Based on this result, for every complex number, you plot a point on the complex plane either black or white.  (Or, it is colored based on how fast it tends towards infinity.)  For example, 1+i tends towards infinity when plugged into this equation, so a black point is plotted for 1+i on the complex plane. This process was then repeated for every complex number, and was only possible because of the advent of modern computers. Every single complex number gets a point (eg: .234234234 + .324325423i) The result is the deeply infinite, self-referential image you see above.  The image is much more complex than it appears.  For example, if you zoom in to a certain section, you will see the entire image repeat within itself, and then repeat within that zoom!  I can’t do it justice here, so if you want to learn how this Math models real life phenomena & situations, these 2 videos are a great primer for a layman:

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## Pullups and Logarithmic Decay So, I was talking to a friend about how pullups need more rest time between sets than other exercises:

Pullups need significant recovery time to not have logarithmic decay in the number of reps you can do.

I suggested waiting about 3 minutes between sets.  Of course, the next time I did them, I wanted to see just how logarithmic the decay is.  I plugged the numbers into Excel and did a log regression.  Wow, I wasn’t kidding, look at that correlation coefficient !! 