An old sailor, by the name of John, was in his attic with his bright young great-nephew Stewart, showing him all the old tools they used to use while at sea.
Stewart was fascinated by all the trinkets. One item caught his eye, which was a giant compass like one would find on a table inside the ship. It was beautifully constructed, with hand-printed markings for north, east, south and west. Halfway between those markings were four more markings for the 45° (1/8) bearings of north-east, south-east, south-west, and north-west. Halfway again between those markings were eight markings for the 22.5° (1/16) bearings corresponding to north-north-east, east-north-east, and so on.
But what was special about this giant compass is that the markings went further down. There were sixteen markings for each of the 1/32 bearings, starting from north-north-north-east. And between those were thirty-two markings for the 1/64 bearings, starting from north-north-north-north-east. And it also had the 1/128 bearings and the 1/256 bearings, and so on down until they were so small that Stewart couldn't see them anymore.
Stewart asked his great-uncle, "Uncle John, how did you keep track of all these directions? Didn't you use degrees like we do today?"
The uncle laughed. "Of course we used degrees. These directions are just for decoration on the compass. But we did learn how to interpret them. Each direction is named the combination of the two directions it's between. Just like north-east is between north and east, and east-north-east is between east and north-east."
Stewart decided to challenge the old sailor on this. "So what would north-east-east-north-east be?"
John immediately said, "5/32 from north. Check for yourself." Stewart located the arrow pointing 5/32 of the way around the circle, and sure enough, it read north-east-east-north-east.
Stewart then decided to ask, "What about east-east-east-north-north-east-north?"
John said, "That's 9/64 from north. You can name any number of 'north's and 'east's as you like, and I can tell you what fraction it is."
"That's the same as the last one, 9/64 from north," explained Stewart. "Only the number of norths and easts matters, not their orders."
"How about south-east-east-south?" asked Stewart.
"That's just south-east. You can't use a number of 'south's and 'east's with a common factor," explained John. "They'd all be the same."
"Okay then," said Stewart, frowning in thought. He then blurted out, "How about west-west-south-south-south-west-west-south-west-west-west-west-south-south-west-west-west?"
John had his fingers out counting how many 'south's and 'west's his young pupil had uttered out (six and eleven, respectively). "That would be 351/512 from north."
"What about north-west-south-east?" asked Stewart smartly.
"It'll take more than that to outfox me," laughed John. "You can only name directions with two of the four cardinal directions that are next to each other. But I assure you, I can tell you the direction of any combination of two adjacent cardinal directions you like."
"That's cool," said Stewart, now turning his attention to a wind-up clock.
Is the old sailor's claim true? Can you come up with some combination of "north"s and "east"s (or some other combination of two adjacent cardinal points) that doesn't correspond to a point on the infinite compass rose?