the golden ratio is a term often erroneously ascribed to the

the golden ratio is a term often erroneously ascribed to the ancient Greeks. For each y<= 20, find the integer x making absolute value(x-y gamma) as small as possible. Which rational number x/y with y <= 20 most closely approximates gamma?

Solution

golden ratio is between 1.6 and 1.7, approx 1.61

3/2 = 1.5, 4/2 = 2

5/3 = 1.6666 , 6/3 =2

6/4 =1.5 7/4 = 1.65

8/5 = 1.6 , 9/5 = 1.8

9/6 = 1.5 ,10/6 =1.6666

11/7 = 1.57 12/7 = 1.7

12/8 = 1.5 13/8 = 1.625

14/9 = 1.5555 15/9 = 1.6666

16/10 = 1.6 17/10 = 1.7

17/11 = 1.5454 18/11 = 1.6363

19/12 = 1.5833 20/12 = 1.6666

21/13 = 1.615 22/13 = 1.69

22/14 = 1.57 23/14 =1.64

24/15 = 1.6 25/15 = 1.666

25/16 =1.5625 26/16 = 1.625   

27/17 = 1.58 28/17 = 1.64

28/18 = 1.555 29/18 = 1.61111

30/19 = 1.57 31/19 = 1.631

32/20 =1.6 33/20 = 1.605

two competitors are

21/13 and 29/18

21/13 = 1.61538461538

29/18 = 1.6111111111

golden ratio = 1.61803398875

it is close to 21/13