CHAPTER ONE Real numbers   Definition 1. A set F is called a field if it has two operations defined on it, addition x+y and multiplication xy, and if it satisfies the following axioms. (A1) If x F and y F, then x+y F. (A2) (commutativity of addition) x+y = y+x for all x,y F.

(A3) (associativity of addition) (x+y)+z = x+(y+z) for all x,y, z F. (A4) There exists an element 0 2 F such that 0+x = x for all x F. (A5) For every element x F there exists an element -x 2 F such that x+(-x) = 0. (M1) If x F and y F, then xy F . (M2) (commutativity of multiplication) xy = yx for all x,y F.

connection to even perfect* numbers  In the 4th century bc, Euclid  proved that if  Mp  is a Mersenne prime then 2p-1 ⋅  (2p-1 ) or  Mp(Mp + 1)/2 is an even perfect number. Ex. Take M7 Then 26 ⋅  (27-1) is perfect Perfect number, a positive integer that is equal to the sum of its proper divisors. smallest perfect number is 6=1+ 2+ 3. also 28, 496, and 8,128 are perfect. 

It is not known whether odd perfect numbers exist or not! Prime numbers in binary 2 10 3 11 5 101 7 111 11 1011 13 1101

10001 10011 10111 11111 Mersenne Prime numbers in binary: 11, 111, 11111, 1111111, 1111111111111, 11111111111111111, 1111111111111111111, 1111111111111111111111111111111, 111111111111111111111111111111111111111111111111111111111111

even perfect number is represented in binary form Example. 610 = 22 + 21 = 1102 2810 = 24 + 23 + 22 = 111002 49610 = 28 + 27 + 26 + 25 + 24 = 1111100002 and 812810 = 26 × (26 + 25 + 24 + 23 + 22 + 21 +1) = 11111110000002.

Fibonacci's Numbers متتالية فيبوناتشي أو أعداد فيبوناتشي  Fibonacci’s numbers) نسبة إلى ليوناردو فيبوناتشي، هي الأعداد في المتتالية التالية: n:0 1 2 3 4 5 6 7 8 9 10 11 12 ... F(n):0 1 1 2 3 5 8 13 21 34 55 89 144 … 1,1,2,3,5,8,13,21,34,55,89,144,… وتحسب بالقاعده: Fn= Fn-1+ Fn-2

Patterns in the Fibonacci Numbers The Final Digits 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ... Is there a pattern in the final digits? 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, ... The cycle length is 60, then it repeats the same sequence again.  

Fibonacci Numbers and Primes We notice that Every k-th Fibonacci number is a multiple of F(k). Or F(nk) is a multiple of F(k) for all values of n and k =1,2,3,… So [If k is composite then so is F(k) with one exception (if k=4)].

Any Fibonacci number that is a prime number must also have a subscript that is a prime number. Thus F(k) prime implies k is prime. But the converse is not always true. For example: 19 is prime but F(19)=4181= 113x37 is not prime.

Golden ratio construct: to draw a rectangle with the Golden Ratio: Draw a square (of size "1") Place a dot half way along one side Draw a line from that point to an opposite corner (it is √5/2 in length) Turn that line so that it runs along the square's side Then you can extend the square to be a rectangle with the Golden Ratio.

Note that 1 / 1 = 1, 2 / 1 = 2, 3 / 2 = 1.5, and so on up to 144 / 89 = 1.6179…. The resulting sequence is: 1, 2, 1.5, 1.666..., 1.6, 1.625, 1.615…, 1.619…, 1.6176…, 1.6181…, 1.6179…

Golden Mean or The GoldenRatio نسبة فابيوناتشي الرئيسية وهي%61

Using The Golden Ratio to Calculate Fibonacci Numbers we can calculate any Fibonacci Number using the Golden Ratio: The answer always comes out as a whole number, exactly equal to the addition of the previous two terms. By using   Xn=[(φn-(1- φ)n]/21\2

Example 8= X6=[(φ6-(1- φ)6]/21\2 Where φ= 1.618034…

تستخدم أرقام فيبوناتشي في تحليل الأسواق المالية، في استراتيجيات مثل ارتداد فيبوناتشي وفي خوارزميات االكمبيوتر مثل تقنية فيبوناتشي للبحث    وهي تظهر أيضا في الترتيبات البيولوجية، مثل تفريعات الأشجار، ترتيب الأوراق على الساق وطرف الثمرة من الأناناس  

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