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Math Club Workshop Danville Senior Center May 5, 2016 The plan…sort of… A seminar, not a class. I have an “agenda”, but we can ignore it. But let’s start with introductions-and maybe include your math background and interests. The plan…sort of… My “agenda”, but we can ignore it Starting with very basic numbers and counting Move on to Rationals, Reals, and Complex numbers On to operations on numbers Then on to Algebra on numbers Then “other” Algebras: Vectors, Matrices, and Groups And if we have time/interest -Geometry The plan…sort of… Along the way we’ll Solve some Linear Equations Derive the Quadratic Formula Do a Geometric proof And for extra credit, calculate the value of ii Numbers 7/15 2 π X i 697 0 Integers Integers Integers Symbols Integers Integers Integers Integers Integers Group Symbols Integers Integers Integers Roman Numerals V X L Integers LXXXIII Integers M D C L XVI MMXVI ? Integers MC L X V MD C L X V I Etruscan X X˄ I ↓ Egyptian Integers ? Integers Zero 0 Integers The number zero as we know it arrived in the West circa 1200, most famously delivered by Italian mathematician Fibonacci (aka Leonardo of Pisa), who brought it, along with the rest of the Arabic numerals, back from his travels to north Africa. But the history of zero, both as a concept and a number, stretches far deeper into history. http://www.scientificamerican.com/article/history-of-zero/ Integers "There are at least two discoveries, or inventions, of zero," says Charles Seife, author of Zero: The Biography of a Dangerous Idea(Viking, 2000). "The one that we got the zero from came from the Fertile Crescent." It first came to be between 400 and 300 B.C. in Babylon, Seife says, before developing in India, wending its way through northern Africa and, in Fibonacci's hands, crossing into Europe via Italy. The second appearance of zero occurred independently in the New World, in Mayan culture, likely in the first few centuries A.D. "That, I suppose, is the most striking example of the zero being devised wholly from scratch," Kaplan says. Integers Decimal Numbering System 0 1 2 3 4 5 6 7 8 9 I II III IV V VI VII VIII IX Groups of 10 10 100 1000 … X C M Integers 10 11 12 13 100 101 102 423 1000 1001 7135 945,389,234,703,662 Integers A prime number (or a prime) is an Integer greater than 1 that has no positive divisors other than 1 and itself. 2,3,5,7,11,… You test for prime number “P” by trying to divide P by every Integer between 2 and 𝑃 . As of January 2016, the largest known prime number has 22,338,618 decimal digits. Primes are used in several routines in information technology, such as publickey cryptography (RSA), which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Source: Wikipedia Integers Binary, Octal, and Hexadecimal Binary 0 1 Groups of 2 10 100 1000 2 4 8 1000000000 1024 100000000000 4096 Octal 0 1 2 3 4 5 6 7 Groups of 8 10 8 100 64 1000 512 10000 4096 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F Groups of 16 10 16 100 256 1000 4096 Integers How many Integers are there? Infinite number of them: no largest Integer. A particular kind of infinite named “countably infinite” Countably infinite is the smallest kind of infinite. The symbol for a countably infinite set is (Pronounced Aleph Null) Are there bigger kinds of infinite? More Numbers Integers 0 1 697 Rationals (Fractions) 1/3 7/15 34/15 (includes Integers) Reals Complex 2 5 7 i 3+5i π e (includes Rationals) 𝑒 2+3𝑖 (includes Reals) Rationals (Fractions, including Integers) 1/10000 1/3 7/15 34/15 100089/5872 Are there more Rationals than Integers? Rationals Are there more Rationals than Integers? No. Rationals All the possible Rationals Rationals All the possible Rationals Reals Are there numbers that are not Rationals? Reals The set of Real numbers includes the Rationals, but also has numbers that are NOT fractions. Examples are: π e 2 5 3 … 17 … Any Prime number … Reals Example: 2 is NOT a Rational Proof: Suppose 2 is Rational Then there exist integers P and Q such that P/Q = 2 And P, Q in lowest common denominator (in particular, both NOT even) Then P2/Q2 = 2 P2 = 2Q2 So P2 must be even But if P2 is even, then P must be even But then P2 must be divisible by 4 But that means Q2 must be even Which means that Q must be even Which means both P and Q are both even But they can’t be both even, so there is no such P/Q Therefore 2 is NOT a Rational Reals There are more Reals than Rationals or Integers. Reals are a bigger kind of Infinite than Countably Infinite The “Continuum Hypothesis” says that there is no kind of infinity between that of the Integers and the Reals. The Cardinality of the Reals is Aleph One 1 Are there bigger infinities than the Reals? Yes, but that is a little bit more “advanced” topic. Complex numbers i = −1 i2 = -1 Complex Numbers are usually written as either x + iy Or 𝑒 𝑥+𝑦𝑖 { equal to 𝑒 𝑥 (cos[y] + i sin[y] ) } Where both x and y are Real Numbers Complex numbers Are there more Complex Numbers than Rationals? Complex numbers Are there more Complex Numbers than Reals? No. Complex numbers Example: Take the Complex Number 123.45 + 67.89i Map that to a Real number by stating at the decimal point and alternating between the digits in either half of the Complex Number 123.45 + 67.89i 16273.4859 Then you can get back to Complex number by stripping off the digits in reverse order, starting from the decimal point. So there are just as many Complex Numbers as Reals Complex numbers Extra credit Who can calculate i i Complex numbers Extra credit Who can calculate i i 𝑒 𝑖𝑥 = cos(x) + isin(x) So 𝑒 𝑖π/2 = cos(π/2) + isin(π/2) But cos(π/2) = 0 And isin(π/2) = i So 𝑒 𝑖π/2 = i Substituting back into ii We get (𝑒 𝑖π/2 ) i = 𝑒 −π/2 = .207… Large Numbers Often represent large numbers in exponential notation, some number raised to the power of another number. 100 = 1 101 = 10 102 = 100 103 = 1000 104 = 10,000 : 10100 = a Googol (More than the number of atoms in the observable Universe) : 10Googol = a Googolplex (Largest number I know) Numbers M 7/15 1 𝑒 π X 2 𝑥+𝑦𝑖 X i 697 0