Math 1 - January 22, 2008 Numbers 1, 2,3,4,5 . (natural...

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Add factors: 1+2+3+4+6= 16 16 12 Add factors: 1+2+3= 6 6=6 ***6 is a perfect number 6 C B A 1 1 1 a Every rational number has a decimal expansion with a repeating pattern of digits. Irrational numbers have no pa **Odds square to odd numbers and evens square to even numbers. Irrational numbers guarantee no repetitions Elements of the sets (members) N= Natural Numbers, whole numbers Q= Rational Numbers, can be written as a ratio = subset ε = membership { } = set U = union = intersection U = universal set # = number First element Second element Third element Fourth element all positive fraction (Rational numbers) covered by this scheme. First element Second element Third element Fourth element Fifth element Sixth element Exterior angle Interior angle Pan (-2,1) (1,0) (0,0) (1,1) (0,1) (x 1 ,x 2 ) 0 1 0 1 Totally there are four possibilities 4 vertices (1,0,1) (1,0,0) (0,0,0) (1,1,0) (1,1,1) (0,1,1) (0,0,1) (0,1,0) (3,4) 3 (-3,1) (1,2) Real Numbers Imaginary Numbers January 22, 2008 Numbers – 1, 2,3,4,5 … (natural numbers) | | -pattern Spotting Patterns 1. 2,4,6,8… (even numbers) 2. 2,4,8,16,32… (powers of 2) 3. 2, 3,5,7,11,13… (prime numbers) 4. 1, 1,2,3,5,8,13… (Fibonacci Numbers – adding two previous numbers to get the next) Prime Numbers Prime Numbers- are only divisible by 1 and itself ex. 5 is a prime number but 6 is not. 1. Compare Even – 2, 4, 6,8,10… Prime – 2,3,5,7,11,13,17.19,23,29… *Primes cannot be predicted they must all be figured out. No known formula. No easy way to tell if a number is prime. 2. Goldbach’s conjecture (“guess”) Every even number greater than 2 is the sum of two primes 12=5+7 32=13+19 32=3+29 3. How many primes? 2,3,5,7,11,13,17,19,23,29,31,37,41,43… There are infinitely many. 4. Twin primes Successive odd numbers that are prime
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Infinite many twin primes? Unknown. January 24, 2008 Infinitely Many Primes Strategy: Suppose there is a last prime, show that it is impossible. If there is a last prime number, you can list all the primes. P 1, P 2, P 3 …..P L First, second, third….last 2, 3, 5 Make a new number N= P 1, P 2, P 3 …..P L (2x3x5x7….P L ) +1 Claim: N is not always divisible by any of the primes P 1, P 2, P 3 …..P L . There will always be a remainder of 1. Now N is either prime itself or divisible by some prime other than P 1, P 2, P 3 …..P L . -OR- Some prime other than P 1, P 2, P 3 …..P L . Either way you have a new prime that is not on the list. Therefore – no finite list of primes can include all primes, there are infinitely many primes. 35 + 7 35 ÷7 takes longer to calculate Decisions about whether or not: 326780127378820037 is prime takes longer to calculate Each digit takes about 3x as long to figure out… 2 digits 9x, 3 digits 15x etc…
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Are there alternative methods to find primes? There are some, the Sieve of Eratosthenes.
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This note was uploaded on 04/07/2008 for the course MTH 001 taught by Professor Borde during the Spring '08 term at Long Island U..

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Math 1 - January 22, 2008 Numbers 1, 2,3,4,5 . (natural...

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