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# decimal system

decimal system [Lat.,=of tenths], numeration system based on powers of 10. A number is written as a row of digits, with each position in the row corresponding to a certain power of 10. A decimal point in the row divides it into those powers of 10 equal to or greater than 0 and those less than 0, i.e., negative powers of 10. Positions farther to the left of the decimal point correspond to increasing positive powers of 10 and those farther to the right to increasing negative powers, i.e., to division by higher positive powers of 10. For example, 4,309=(4×103)+(3x102)+(0×101)+(9×100)=4,000+300+0+9, and 4.309=(4×100)+(3×10-1)+(0×10-2)+(9×10-3)=4+3/10+0/100+9/1000. It is believed that the decimal system is based on 10 because humans have 10 fingers and so became used to counting by 10s early in the course of civilization. The decimal system was introduced into Europe c.1300. It greatly simplified arithmetic and was a much-needed improvement over the Roman numerals, which did not use a positional system. A number written in the decimal system is called a decimal, although sometimes this term is used to refer only to a proper fraction written in this system and not to a mixed number. Decimals are added and subtracted in the same way as are integers (whole numbers) except that when these operations are written in columnar form the decimal points in the column entries and in the answer must all be placed one under another. In multiplying two decimals the operation is the same as for integers except that the number of decimal places in the product, i.e., digits to the right of the decimal point, is equal to the sum of the decimal places in the factors; e.g., the factor 7.24 to two decimal places and the factor 6.3 to one decimal place have the product 45.612 to three decimal places. In division, e.g., 4.32 /12.8 where there is a decimal point in the divisor (4.32), the point is shifted to the extreme right (i.e., to 432.) and the decimal point in the dividend (12.8) is shifted the same number of places to the right (to 1280), with one or more zeros added before the decimal to make this possible. The decimal point in the quotient is then placed above that in the dividend, i.e., 432 /1280.0 zeros are added to the right of the decimal point in the dividend as needed, and the division proceeds the same as for integers. The decimal system is widely used in various systems employing numbers. The metric system of weights and measures, used in most of the world, is based on the decimal system, as are most systems of national currency.
The Yosemite Decimal System is a numerical system for rating the difficulty of walks, hikes, and climbs, primarily used for mountaineering in the United States and Canada. The rock climbing (5.x) portion of the scale is the primary climb grading system used in the US and Canada.

The system was initially developed as the Sierra Club grading system in the 1930s to rate hikes and climbs in the Sierra Nevada range. Previously, hikes and climbs were described relative to others ("harder than X, but easier than Y"), but this made it difficult for those who hadn't done the other hikes or climbs to compare climbs, so the numerical grading system was developed to codify climbs on a single scale.

Currently, according to the climbing textbook Mountaineering: The Freedom of the Hills, the system divides all hikes and climbs into five classes:

• Class 1: Hiking.
• Class 2: Simple scrambling, with possible occasional use of the hands.
• Class 3: Scrambling, a rope can be carried but is usually not required.
• Class 4: Simple climbing, with exposure. A rope is often used. Natural protection can be easily found. Falls may well be fatal.
• Class 5: Technical free climbing. Climbing involves rope, belaying, and other protection hardware for safety.

The original Sierra Club grading system also had a Class 6, for artificial, or aid climbing. This sort of climbing uses ropes and other equipment for progress (e.g. climbing a rope up a sheer face with no holds). Class 6 is no longer widely used, however, and artificial climbs today are graded on a separate scale from A0 through A5.

Note that the exact definition of the classes is somewhat controversial.

The increasing technical difficulty of Class 5 climbs led to the same relative-grading problem that had caused the initial development of the system; as a result, Class 5 was subdivided in the 1950s. Initially it was based on ten climbs of Tahquitz Rock in Idyllwild, California, and ranged from "the Trough" at 5.0, a relatively modest technical climb, to "the Open Book" at 5.9, considered at the time the most difficult unaided climb humanly possible. This system was developed by members of the Rock Climbing Section of the Angeles Chapter of the Sierra Club.

Ratings between indoor gym climbing, sport climbing and traditional climbing can also vary quite a bit, depending on location and history. For example, a 5.8 climb in a New Jersey gym can correspond to a 5.10c climb in a California gym. A 5.6 climb in a California gym may correspond to a 4th class climb at Yosemite.

Advances in techniques and equipment since then have led to harder climbs being completed. The first such climb was given the rating 5.10; the second the rating 5.11. It was later determined that the 5.11 climb was much harder than 5.10, leaving many climbs of varying difficulty bunched up at 5.10. To solve this, the scale has been further subdivided above the 5.9 mark with a-d suffixes. As of 2005, several climbs are widely agreed to be at the 5.15a difficulty. Akira, by Fred Rouhling, has been claimed as a 9b (French grade) which translates to 5.15b. Chilam Balam by Bernabé Fernández was graded as 9b+/5.15c. Both are controversial.

See SACIN for tables comparing 16 different climbing grading systems and a list of the hardest climbs