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Find a rational between 'a' and 'b'." }}{PARA 0 "" 0 " " {TEXT 211 128 "Remark: We note the function f below. When x gets lar ge, arctan(x) gets close to Pi/2. So a and b should be close to each o ther." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x->1/arctan(x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(f(x),x=-10..10,y=- 10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=20;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a:=evalf((1/arctan(101)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "b:=evalf((1/arctan(100)) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 20 "We want to find the " } {TEXT 208 8 "smallest" }{TEXT 211 81 " possible integer n so that (b-a )>1/n, which exists by the Archimedian Principle." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "solve((b-a)>1/x,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x>1/(b-a),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "n:=24609;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 194 "W e consider the set S=\{x is integer and x/n >a\}. Since S is unbounded above so S is non-empty. Also, since S is bounded below, S has the sm allest member. We want to find such member as follows. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve((x/n)>a,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "m:=15766;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(m/n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf((m/n)-a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ev alf((m/n)-b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(((m+ 1)/n)-a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(((m+1)/n )-b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(((m-1)/n)-a) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 211 110 "Remark: We note that the i nteger m indeed is the smallest possible integer such that m/n >a sinc e (m-1)/n