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• What about infinity times infinity?
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• Infinity times infinity Are all infinities the same? Is infinity plus one larger than infinity? Is infinity plus infinity larger than infinity? Is infinity times infinity larger than infinity?
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• A former Math 210 project on large numbers
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• Ordered pairs An ordered pair of numbers is simply two numbers, one listed before the other: (3,2), (3.14,2.71), (m,n) An ordered pair of elements of a set is simply two elements of the set, one listed before the other. For example, if the set is the alphabet then (a,b) is an ordered pair. (b,a) is a different ordered pair. Given any two sets A and B, the collection of all ordered pairs of elements, one from A then one from B, defines another set called the Cartesian product, denoted AxB
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• The number of rational numbers is equal to the number of whole numbers
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• Countable sets A set is countable if its elements can be enumerated using the whole numbers. A set is countable if it can be put in a one-to- one correspondence with the whole numbers 1,2,3,. Paradox: the Hilbert hotel
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• Any number between 0 and 1 can be represented by a sequence of zeros and ones
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• Any number between zero and 1 also has a decimal representation In this case each digit takes the value between 0 and 9. One divides [0,1] into 10 equal bins and assigns the digit correponding to which bin contains x, If x is not an endpoint then one repeats the process on 10(x - and so on. Example: Note: in this case. What about 0.99999.?
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• Binary representation of whole numbers Here Algorithm: Step 1: Find the largest power of 2 less than or equal to N. This is k. Step 2: If then done. Otherwise, subtract from N. Apply step 1 to stop when either the remainder is a power of two (possibly equal to one)
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• Example: Binary decomposition of N=27
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• Question: Is there any relationship between the binary decomposition of N and of 1/N? Example: compare 3 and 1/3.
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• The numbers between 0 and 1 are uncountable.
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• In search ofGeorg Cantor
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• Ordinal number: 0,1,2, etc Cardinal number: 2^N: number of subsets of a set of N elements Number of subsets of the natural numbers The Continuum hypothesis Aleph naught
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• Clicker question Cardinal numbers refer only to numbers worn on the jerseys of St Louis Cardinals players A True B - False
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• Clicker question Cardinal numbers can be infinite (larger than any finite number) A True B - False
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• Clicker question All infinite cardinal numbers are the same size A True B - False
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• Counting: some history The first recorded use of numbers consisted of notches on bones. Humans used addition before recorded history
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• Nowadays we use big numbers: Numbers are represented by symbols: 2 57,885,161 -1 has 17,425,170 digits17,425,170 To see: 57,885,161/3.32193=17,425,169.7 At 3000 characters per page, would take about 5000 pages to write down its digits. Very large numbers are represented by descriptions. For example, Shannons number is the number of chess game sequences. Very very large numbers are represented by increasingly abstract descriptions.
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• We use symbols to represent mathematical concepts such as numbers Some number systems facilitate calculations and handling large magnitudes better than others The symbols 0,1,2,3,4,5,6,7,8,9 are known as the Hindu arabic numerals
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• Some ancient number systems
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• Cuneiform (Babylonians): base 60
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• Mayans: Base 20 (with zero)
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• Egyptians: base 10
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• Greeks (base 10)
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• Romans (base 10)
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• Only the Mayans had a zero Babylonians: base 60 inherited today in angle measures. Used for divisibility. No placeholder: the idea of a power of 10 is present, but a new symbol had to be introduced for each new power of 10. Decimal notation was discovered several times historically, notably by Archimedes, but not popularized until the mid 14 th cent. Numbers have names
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• Base 10 2 3 4 5 6 7 8 9 10
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• Scientific notation Scientific notation allows us to represent numbers conveniently when only order of magnitude matters.
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• Powers of 10 Alt 1 More videos and other sources on powers of 10 More videos and other sources on powers of 10
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• Other cosmic questions
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• Orders of Magnitude Shannon number the number of atoms in the observable Universe is estimated to be between 4x10^79 and 10^81.
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• Some orders on human scales Human scale I: things that humans can sense directly (e.g., a bug, the moon, etc) Human scales II: things that humans can sense with light, sound etc amplification (e.g., bacteria, a man on the moon, etc) Large and small scales: things that require specialized instruments to detect or sense indirectly Indirect scales: things that cannot possibly be sensed directly: subatomic particles, black holes
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• These are a few of my least favorite things
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• Viruses vary in shape from simple helical and icosahedral shapes, to more complex structures. They are about 100 times smaller than bacteria Bacterial cells are about one tenth the size of eukaryotic cells and are typically 0.55.0 micrometres in length There are approximately five nonillion (.510^30) bacteria on Earth, forming much of the world's biomass.
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• Clicker question If the average weight of a bacterium is a picogram (10^12 or 1 trillion per gram). The average human is estimated to have about 50 trillion human cells, and it is estimated that the number of bacteria in a human is ten times the number of human cells. How much do the bacteria in a typical human weigh? A) < 10 grams B) between 10 and 100 grams C) between 100 grams and 1 kg D) between 1 Kg and 10 Kg E) > 10 Kg
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• How big is a googol?
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• Some small numbers 17 trillion: national debt 1 trillion: a partial bailout 314 million: number of americans 1 billion: 3 x (number of americans) (approx) 1 trillion: 1000 x 1 billion \$ 54,134: your share of the national debt Each month the national debt increases by the annual GDP of New Mexico
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• Visualizing quantities How many pennies would it take to fill the empire state building? Your share of the national debt
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• Clicker question If one cubic foot of pennies is worth \$491.52, your share of the national debt, in pennies, would fill a cube closest to the following dimensions: A) 1x1x1 foot (one cubic foot) B) 3x3x3 (27 cubic feet) C) 5x5x5 feet (125 cubic feet) D) 100x100x100 (1 million cubic feet) E) 1000x1000x1000 (1 billion cubic feet)
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• big numbers Small Numbers have names
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• How to make bigger numbers faster Googol: Googolplex:
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• Power towers
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• Power towers and large numbers
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• Number and Prime Numbers Natural numbers: 0,1,2,3, allow us to count things. Divisible: p is divisible by q if some whole number multiple of q is equal to p. Division allows us to divide the things counted into equal groups. Remainder: if p>q but p is not divisible by q then there is a largest m such that mq
• Proof: If Q is not prime then we can write Q=ab for whole numbers a, b where a>1 (and hence b1 (and hence d

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