Chain Fraction

Natural e

@e@ = @e0@ + 1/( @e1@ + 1/( @e2@ + 1/( @e3@ + 1/( @e4@ + 1/( @e5@ + 1/( @e6@ + 1/( @e7@ + 1/( @e8@ + 1/( @e9@ + 1/( @e10@ + 1/( @e11@ + 1/( @e12@ + 1/( @e13@ + 1/( @e14@ + 1/( @e15@ + 1/( @e16@ + 1/( @e17@ + 1/( @e18@ + 1/( @e19@ + ... ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

Golden Ratio

@g@ = @g0@ + 1/( @g1@ + 1/( @g2@ + 1/( @g3@ + 1/( @g4@ + 1/( @g5@ + 1/( @g6@ + 1/( @g7@ + 1/( @g8@ + 1/( @g9@ + 1/( @g10@ + 1/( @g11@ + 1/( @g12@ + 1/( @g13@ + 1/( @g14@ + 1/( @g15@ + 1/( @g16@ + 1/( @g17@ + 1/( @g18@ + 1/( @g19@ + ... ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

Square root of 3

@r@ = @r0@ + 1/( @r1@ + 1/( @r2@ + 1/( @r3@ + 1/( @r4@ + 1/( @r5@ + 1/( @r6@ + 1/( @r7@ + 1/( @r8@ + 1/( @r9@ + 1/( @r10@ + 1/( @r11@ + 1/( @r12@ + 1/( @r13@ + 1/( @r14@ + 1/( @r15@ + 1/( @r16@ + 1/( @r17@ + 1/( @r18@ + 1/( @r19@ + ... ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

Your x

@x@ = @n0@ + 1/( @n1@ + 1/( @n2@ + 1/( @n3@ + 1/( @n4@ + 1/( @n5@ + 1/( @n6@ + 1/( @n7@ + 1/( @n8@ + 1/( @n9@ + 1/( @n10@ + 1/( @n11@ + 1/( @n12@ + 1/( @n13@ + 1/( @n14@ + 1/( @n15@ + 1/( @n16@ + 1/( @n17@ + 1/( @n18@ + 1/( @n19@ + ... ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )