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What is the Yosemite Decimal System?
The Yosemite Decimal System (YDS) is one way to quantify the approximate difficulty of a climb. It's almost exclusively used in North America. As a rock climber you'll usually see routes graded in a format of 5.6, 5.9, 5.12b, and similar. The "five-dot" prefix references the class 5 portion of the scale pertaining to vertical movement such as you see in a climbing gym. The numbers and letters that succeed the dot are the most informative to the climbing that you'll be doing.
Most gym climbing begins at a difficulty of 5.5 or 5.6. You can imagine the difficulty of movement in this range to be similar to climbing a tall ladder with every other rung removed.
Following this is the grade of 5.7, 5.8, and 5.9, with each increment reflecting a more strenuous, awkward, or puzzling nature of the climb. There's no logical method for determining the grade; it's simply arrived at through consensus among the routesetters at the gym.
After 5.9 comes the grades of 5.10, 5.11, 5.12, 5.13 and so forth. These grades are actually a union of even smaller subdivisions, expressed as e.g. 5.10a, 5.10b, 5.10c, and 5.10d, with each subdivision intended to reflect a noticeable increase in difficulty such as you would feel in the progression from 5.7 to 5.8.
The 5.11 series of grades is subdivided similarly: 5.11a, 5.11b, 5.11c, and 5.11d. This continues up to 5.15c, which is — at least at present — the hardest in the world.
If you're thinking this is all unnecessarily complicated, you're totally right. It's a fuzzy system that has evolved over the years and picked up a lot of cruft along the way.