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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 Etc… 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 Explanation: 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 Thermodynamics). 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 zero). 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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posted: | 11/27/2011 |

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