HONRS 296 Fall 2010 Lecture Notes 2

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HONRS 296 Fall 2010 Lecture Notes 2 Powered By Docstoc
					HONRS 296                                              Fall 2010
Lecture Notes 2                                        Dr. Blackwell

How to Conceptualize large Numbers
103 = One Hundred
106 = One Million
109 = One Thousand 106ths
1012 = One Million 106ths
1018 = One Billion 106ths

Orders of Magnitude
10-31 = Mass of an Electron (in kg)
10-27 = Mass of a Proton or mass of a Neutron (in kg)
10-16 = Probability of rolling snake-eyes 10 times in a row
10-11 = Radius of a Hydrogen atom (in meters)
10-9 = Odds of Winning the Powerball (1 in a billion)
10-3 = Odds of being dealt a Full House
104 = Number of Genes in the Human Body (c. 20,000)
105 = Number of words in War and Peace, or Strands of Hair on your Head
106 = Number of Named Species on our Planet
107 = Number of Websites on the Internet
108 = Number of Books in the Library of Congress
109 = Population of the World
1010 = Number of Bacteria in Human Body
1011 = Total number of Homo Sapiens who have ever lived
1012 = Number of Ants on the Earth, or Maximum amount of External Hard Drive Space.
1013 = Number of Bits on a Hard Disk
1014 = Number of Cells in the Human Body
1018 = Number of Ways to pick the NCAA Bracket
1021 = Grains of sand in the world, or Number of possible Sudoku grids
1022 = Number of Stars in the Universe
1027 = Number of Atoms in the Human Body, or the Distance of the Visible Universe
1030 = Number of Bacteria on Earth
1036 = Number of Atoms in every Human Body on Earth
1067 = Number of ways to Order a 52-Card Deck (or 52!)
1079 = Number of Protons in the Universe (The Eddington Number)
10120 = Number of Possible Chess Moves
Timeline of the Universe from the beginning to the end (and back again)

0 Seconds: The Big Bang (Everything after this I refer to as ‘After the Big Bang’ (ABB))

10-43 Seconds ABB: Planck Epoch
        All Four Fundamental Forces are unified (S + W + E + G)
10-36 Seconds ABB: Grand Unified Epoch (1028 K degrees (Room temperature = 300K))
        Gravity Separates from the other three Forces (S + W + E) / (G)
10-32 Seconds ABB: Electro-Weak Epoch
        Strong Force separates from Electro-Weak Force (E + W) / (S) / (G)
10-6 Seconds ABB: Quark Epoch (Temp = 250x hotter than the Sun)
        All four Forces separated
1 Second ABB: Hadron Epoch
        Formation of first Subatomic Particles
10 Seconds ABB: Lepton Epoch
        Formation of Leptons (like electrons and positrons)
20 Minutes ABB: Nucleosynthesis begins
        Subatomic Particles combine into Atoms
70,000 Years ABB: Matter solidifies
        Atoms begin to form Compounds
377,000 Years ABB: Recombination of 1H and 2He in Stars
150 Million Years ABB: Star and Galaxy Formation begins
8 Billion years ABB: Solar System Forms
4.57 Billion years ABB: Our Sun forms
4.54 Billion Years ABB: Our Earth Forms

5771 BCE: Year of Creation according to the Jewish Calendar

Total Age of Universe to Date= 13.7 Billion Years

9 Billion Years ABB: Sun becomes Red Giant (Earth probably eaten by the Corona)
5 x 1011 ABB: Earth and Moon Tide locked (If they are still around)
4 x 1012 ABB: Actinide used up: Bismuth is heaviest trace element in universe
1012 ABB: Star Formation Ceases
1014 ABB: No more visible Stars / Lifetime of the Brahmin in Hindu Mythology
1015 ABB: Outer planets ejected from their orbits
1020 ABB: Earth ejected from its orbit (assuming it is still here)
2.2 x 1024 ABB: Half-Life of Tellurium 128
4 x 1028 ABB: One complete universe cycle according to the Mayans
1032 ABB: Proton Half-Life
1034 ABB: All Nucleons Begin to Decay
1065 ABB: Chemical bonds break apart (only atoms again)
1066 ABB: Black Holes Decay (assuming Protons decay)
10106ABB: Supermassive Black Holes Decay (assuming Protons decay)
101500 ABB: All Atoms decays to 56Fe

       Light elements undergo nuclear fusion and heavy elements undergo nuclear
       fission release energy as their nucleons bind more tightly over time, and the
       resulting nuclei approach the minimum total energy per nucleon, which occurs at
       iron-56. As the Universe ages, more of the matter is converted into extremely
       tightly bound nuclei of iron-56. This progression of matter toward iron is one of
       the phenomena responsible for the heat death of the universe (the second Law of

10 (10*26) ABB: All Matter collapses into Black Holes (Assuming protons do Not decay)
10(10*50) ABB: Boltzmann Brain Forms
         Spontaneously arising self-aware entity that arises out of Chaotic Fluctuations
  (10) (10) (10*2.08)
10                    ABB: Poincare Recurrence Time

Example: The Reality Behind a Very Large Number
Take the number 40! (called the factorial of 40), which is easy to understand
mathematically: 40! = 40 x 39 x 38 x… x 1. But when you start to do this calculation,
you find that after just three iterations, you are into the millions (if you start from the
top), so you decide to switch to a good scientific calculator. There, you find that 40! is
equal to about 8.159 x 1096, which again, is pretty easy to understand mathematically: 8
followed by about 96 zeros. But what if you want to calculate the actual number all the
way out, not just the computational estimation of the range of this number? This would
be putting the mathematical (or abstract) sense into a real, human scale. We think, Ok,
let’s use a computer to actually calculate it. If we started our computer on this task and
the average computer can calculate about 1 billion strings per second (109), it would take
the computer about 6 x 1079 years to complete. When you figure this against the actual
age of the universe to date (about 1.37 x 1010 years), you find that it would take 1069
times the age of our universe to finish this calculation! That would put us well into a
black hole somewhere by then, right?

So we think that we may be able to cut that time way down by adding more computers,
all working in series, right? Let’s use a billion mainframe computers (and there are not a
billion computers on the planet, by the way). We would cut our time down, but only by a
factor of 109 years, still leaving us with 1070 years to complete our calculation! How
about trying to do the calculation with a billion computers on a billion other Earths? It
would still take 1061 years! You see how small a billion is on this scale.

A number like 1079 years seems absolutely incomprehensible to us, but it would
definitely not take forever. It wouldn’t even take long, cosmically speaking. Let’s look
at that number in relation to the future of our universe. Eventually, we know that all
matter in the universe will collapse into black holes, as long as our universe keeps
expanding at the rate it does now (and assuming that protons do not decay).
Cosmologists estimate that this will happen in about 1010 ^ 26 years (remember that taking
off 13 billion years—the current age of our universe—from this number is statistically

With this new perspective in place, we find that such a time frame for our calculation is
not that long at all. Using our one computer on our planet (again, assuming we are even
around for this long as a planet), we could calculate the exact answer of 40! around 1025
times before we fell into a black hole!

Ok, now ready for a mind blow (as if you haven’t had a few already, right)? Doing the
math shows that it doesn’t matter whether we use one computer on our planet or a billion
computers on a billion planets around a billion suns in a billion different galaxies to do
this calculation. It will still take about the same amount of time, since we are only taking
out a factor of 1036 (a billion-billion-billion-billion) from the number 1010 ^ 27, which,
again, is statistically zero. In the ‘human’ scale, this is like taking away 36 away from
the number 1026, which is still 1026, since the exponent does not change at all until we
reach a number in the exponent range (remember that 36 are the last two numbers: tens
and hundreds of this very long number).

Now think about how long it will be before our universe resets and has another Big
Bang…That’s why we say math makes very complicated ideas extremely simple.

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