The duodecimal system is a positional notation numeral system using twelve as its base. The number twelve is instead written as "10" in duodecimal, whereas the digit string "12" means "1 dozen and 2 units". In duodecimal "100" means "1 gross", "1000" means "1 great gross", "0.1" means "1 twelfth". The number twelve, a superior composite number, is the smallest number with four non-trivial factors, the smallest to include as factors all four numbers within the subitizing range, the smallest abundant number; as a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers, duodecimal representations fit more than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system. Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a terminating representation in duodecimal.

In particular, the five most elementary fractions all have a short terminating representation in duodecimal, twelve is the smallest radix with this feature. This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, binary and hexadecimal systems. Although the trigesimal and sexagesimal systems do better in this respect, this is at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize. Various symbols have been used to stand for eleven in duodecimal notation. Using these symbols, a count from zero to twelve in duodecimal reads: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; these were implemented in Unicode 8.0, but as of 2019, most general Unicode fonts in use by current operating systems and browsers have not yet included them. A more common alternative is to use A and B, as in hexadecimal, this page uses "Ⅹ" and "Ɛ". In this section, numerals are based on decimal places. For example, 10 means ten, 12 means twelve.

Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu and the Nimbia dialect of Gwandara. Germanic languages have special words such as eleven and twelve in English. However, they come from Proto-Germanic *ainlif and *twalif, suggesting a decimal rather than duodecimal origin. Units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, the Babylonians had twelve hours in a day. Traditional Chinese calendars and compasses are based on the twelve Earthly Branches. There are 12 inches in an imperial foot, 12 troy ounces in a troy pound, 12 old British pence in a shilling, 24 hours in a day, many other items counted by the dozen, gross or great gross; the Romans used a fraction system based on 12, including the uncia which became both the English words ounce and inch. Pre-decimalisation and the United Kingdom used a mixed duodecimal-vigesimal currency system, Charlemagne established a monetary system that had a mixed base of twelve and twenty, the remnants of which persist in many places.

The importance of 12 has been attributed to the number of lunar cycles in a year, to the fact that humans have 12 finger bones on one hand. It is possible to count to 12 with the thumb acting as a pointer. A traditional finger counting system still in use in many regions of Asia works in this way, could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one hand counts to 12, displaying the number of iterations on the other, until five dozens, i. e. the 60, are full. In a duodecimal place system, twelve is written as 10, but there are numerous proposals for how to write ten and eleven. To allow entry on typewriters, letters such as A and B, T and E, X and E, or X and Z are used; some employ Greek letters such as δ and ε, or τ and ε. Frank Emerson Andrews, an early American advocate for duodecimal and used in his book New Numbers an X and ℰ. Edna Kramer in her 1951 book The Main Stream of Mathematics used a six-pointed asterisk ⚹ and a hash #.

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Chondropathy refers to a disease of the cartilage. It is divided into 5 grades, with 0-2 defined as normal and 3-4 defined as diseased. Achondroplasia: Reduced proliferation of chondrocytes in the epiphyseal plate of long bones during infancy and childhood, resulting in dwarfism. Cartilage tumors Costochondritis: Inflammation of cartilage in the ribs, causing chest pain. Osteoarthritis: The cartilage covering bones is thinned completely worn out, resulting in a "bone against bone" joint, resulting in pain and reduced mobility. Osteoarthritis is common, affects the joints exposed to high stress and is therefore considered the result of "wear and tear" rather than a true disease, it is treated by Arthroplasty, the replacement of the joint by a synthetic joint made of titanium and teflon. Chondroitin sulfate, a monomer of the polysaccharide portion of proteoglycan, has been shown to reduce the symptoms of osteoarthritis by increasing the synthesis of the extracellular matrix. Spinal disc herniation: Asymmetrical compression of an intervertebral disc ruptures the sac-like disc, causing a herniation of its soft content.

The hernia causes back pain. Relapsing polychondritis: a destruction autoimmune, of cartilage of the nose and ears, causing disfiguration. Death occurs by suffocation as the larynx collapses. Though articular cartilage damage is not life-threatening, it does affect the quality of life. Articular cartilage damage is the cause of severe pain, strong barriers to mobility and severe restrictions to the patient's activities. Over the last decades, however and biotech ventures have elaborated promising procedures that contribute to articular cartilage repair. However, these procedures do not treat osteoarthritis. American Orthopaedic Society for Sports Medicine The Steadman-Hawkins Clinic

Vancouver Barracks

Established in 1849, the Vancouver Barracks was the first U. S. Army base located in the Pacific Northwest. Built on a rise 20 feet above the Hudson's Bay Company trading station, Fort Vancouver, its buildings were formed in a line adjacent to the Columbia River about 2,000 yards from the water. It is now located within Washington. With the ratification of the Treaty of Oregon between Great Britain and the United States in 1846, the Oregon boundary dispute was settled; the two nations agreed to a partition of the Pacific Northwest along the 49th parallel, situating Fort Vancouver under U. S. jurisdiction. However, the agreement permitted Great Britain's Hudson's Bay Company to continue operation throughout the territory, including at Fort Vancouver; the Vancouver Barracks were established in direct response to Cayuse War. Congress wished to provide military power to facilitate the removal and control of the regions' native peoples and promote settlement of the Pacific Northwest by white Europeans.

The U. S. Army chose to build their base adjacent to Fort Vancouver because of the settlers and institutions in place there. By October 1849, a cross-country mobilization brought personnel and supplies to the Vancouver Barracks. Colonel William Loring led this brigade of mounted soldiers, accompanied by 700 horses, 1,200 mules and 171 supply wagons; as conflicts between indigenous peoples all around the Pacific Northwest and American settlers escalated and became violent, a number of wars broke out. This series of "Pacific Northwest Indian Wars," lasted from around 1848 until 1879. Forces from Fort Vancouver campaigned against the native peoples; the Vancouver Barracks was involved in nearly every operation against Native Americans throughout the Pacific Northwest. Major military conflicts administered through the Vancouver Barracks include the Cayuse War, Rogue River Wars, Snake River War, Klickitat War, Puget Sound War, Yakima War, Coeur d'Alene War, Paiute War, Snake War, Modoc War, Nez Perce War, Bannock War, Sheepeater Indian War.

These wars targeted a number of indigenous groups including the Cayuse, Tutuni, Nisqually, Yakama, Coeur d'Alene, Bannock, Nez Perce and Muckleshoot, among others. During these wars, the Vancouver Barracks functioned as an administrative center, station for troops, training ground, supply depot, prison. Forces from the Vancouver Barracks continued to intervene on behalf of settlers beyond this era of Indian Wars. Since the establishment of the Vancouver Barracks, the U. S. Army always maintained Guard House. Native Americans were forcefully imprisoned there as late as 1889. Groups of Native Americas were incarcerated as prisoners of war, in preparation of relocation to reservations, or as a precaution to protect white settlements; the U. S. Army targeted charismatic and spiritual leaders or used the threat of incarceration against mobilizing leaders; the extent of this incarceration of indigenous peoples caused some historical accounts to refer to the Fort as a reservation. The post remained in active service.

During World War I it was the home of the Army's Spruce Production Division under the command of Colonel Brice Disque. In the interwar years, the 5th Infantry Brigade was based there. Joseph E. Kuhn commanded the post and the 5th Infantry Brigade from October 1923 to July 1925. From 1936 to 1938, it was commanded by future Army Chief of Staff George C. Marshall. In World War II when Vancouver Barracks was used as a staging area for the Seattle Port of Embarkation, the post included 3,019 acres, had billeting space for 250 officers and 7,295 enlisted persons. After WWII, Vancouver Barracks became a sub-installation of Fort Lewis and maintained a small contingent of active duty troops; the majority of billeting space was transformed into military offices and became home to the 104th Division of the Army Reserve, plus Washington National Guard units as well. Vancouver Barracks closed in 2011, in accordance with the requirements of the Base Realignment and Closure Commission. A 2012 Memorial Day ceremony saw the south and east barracks turned over to the care of the National Park Service.

Because of its significance in United States history a plan was put together to preserve the location. The HBC Fort Vancouver was declared a U. S. National Monument on June 19, 1948, redesignated as Fort Vancouver National Historic Site on June 30, 1961; this was taken a step further in 1996 when a 366-acre area around the fort, including Kanaka Village, the Vancouver Barracks and the bank of the river, was established as the Vancouver National Historic Reserve maintained by the National Park Service. It is possible to tour the fort. Throughout its service as a U. S. Army station, Vancouver Barracks had several designations. At its foundation it was called Camp Vancouver but in 1850 it was renamed to Columbia Barracks; this name was used until 1853, when the station was renamed to Fort Vancouver, which lasted until 1879 when Vancouver Barracks was adopted. Fort Vancouver National Historic Site