0-6-0, Usage, South Africa, Cape gauge
The Natal Government Railways placed a single locomotive in shunting service in 1880, later designated Class K, virtually identical to the Durban Harbour's John Milne and built by the same manufacturer.


0-6-0, Usage, South Africa, Cape gauge
In 1882, two 0-6-0 tank locomotives entered service on the private Kowie Railway between Grahamstown and Port Alfred. Both locomotives were rebuilt to a 4-4-0T wheel arrangement in 1884.


0-6-0, Usage, South Africa, Cape gauge
In 1890, the Nederlandsche-Zuid-Afrikaansche Spoorweg-Maatschappij of the Zuid-Afrikaansche Republiek (Transvaal Republic) placed six 18 Tonner 0-6-0ST locomotives in service on construction work.


0-6-0, Usage, South Africa, Cape gauge
In 1896 and 1897, three 26 Tonner saddle-tank locomotives were built for the Pretoria-Pietersburg Railway (PPR) by Hawthorn, Leslie and Company. These were the first locomotives to be obtained by the then recently established PPR. Two of these, named Nylstroom and Pietersburg, came into SAR stock in 1912 and survived into the 1940s.


0-6-0, Usage, South Africa, Cape gauge
In 1901, a single 0-6-0T harbour locomotive built by Hudswell, Clarke was delivered to the Harbours Department of Natal. It was named Edward Innes and retained this name when it was taken onto the SAR roster in 1912.


0-6-0, Usage, South Africa, Cape gauge
Two saddle-tank locomotives were supplied to the East London Harbour Board in 1902, built by Hunslet. Both survived until the 1930s, well into the SAR era.


0-6-0, Usage, South Africa, Cape gauge
In 1904, a single saddle tank harbour locomotive, named Sir Albert, was built by Hunslet for the Harbours Department of Natal. It came into SAR stock in 1912 and was withdrawn in 1915.


0-6-0, Usage, South Africa, Narrow gauges
In 1871, two 2 ft 6 in (762 mm) gauge tank locomotives, built by the Lilleshall Company of Oakengates, Shropshire in 1870 and 1871, were placed in service by the Cape of Good Hope Copper Mining Company. Named John King and Miner, they were the first steam locomotives to enter service on the hitherto mule-powered Namaqualand Railway between Port Nolloth and the Namaqualand copper mines around O'okiep in the Cape Colony.


0-6-0, Usage, South Africa, Narrow gauges
In 1902, Arthur Koppel, acting as agent, imported a single 0-6-0 2 ft (610 mm) narrow gauge tank steam locomotive for a customer in Durban. It was then purchased by the Cape Government Railways and used as construction locomotive on the Avontuur branch from 1903. In 1912, this locomotive was assimilated into the South African Railways and in 1917 it was sent to German South West Africa during the First World War campaign in that territory.


0-6-0, Usage, South West Africa
Between 1898 and 1905, more than fifty pairs of Zwillinge twin tank steam locomotives were acquired by the Swakopmund-Windhuk Staatsbahn (Swakopmund-Windhoek State Railway) in Deutsch-Südwest-Afrika (DSWA, now Namibia). Zwillinge locomotives were a class of small 600 mm (1 ft 11+5⁄8 in) Schmalspur (narrow gauge) 0-6-0T tank steam locomotives that were built in Germany in the late 19th and early 20th centuries. As indicated by their name Zwillinge (twins), they were designed to be used in pairs, semi-permanently coupled back-to-back at the cabs, allowing a single footplate crew to fire and control both locomotives.


0-6-0, Usage, South West Africa
The pairs of locomotives shared a common manufacturer's works number and engine number, with the units being designated as A and B. By 1922, when the SAR took control of all railway operations in South West Africa (SWA), only two single Illinge locomotives survived to be absorbed onto the roster of the SAR.


0-6-0, Usage, South West Africa
In 1907, the German Administration in DSWA acquired three Class Hc tank locomotives for the narrow gauge Otavi Mining and Railway Company. One more entered service in 1910, and another was obtained by the South African Railways in 1929.


0-6-0, Usage, South West Africa
In 1911, the Lüderitzbucht Eisenbahn (Lüderitzbucht Railway) placed two Cape gauge 0-6-0T locomotives in service as shunting engines. They were apparently no longer in service when all railways in the territory came under the administration of the South African Railways in 1922.


0-6-0, Usage, Switzerland
During the Second World War, Switzerland converted some 0-6-0 shunting engines into electric-steam locomotives.


0-6-0, Usage, United Kingdom
The 0-6-0 tender locomotive type was extremely common in Britain for more than a century and was still being built in large numbers during the 1940s. Between 1858 and 1872, 943 examples of the John Ramsbottom DX goods class were built by the London and North Western Railway and the Lancashire and Yorkshire Railway. This was the earliest example of standardisation and mass production of locomotives.


0-6-0, Usage, United Kingdom
Of the total stock of standard-gauge locomotives operating on British railways in 1900, around 20,000 engines, over a third were 0-6-0 tender types. The ultimate British 0-6-0 was the Q1 Austerity type, developed by the Southern Railway during the Second World War to haul very heavy freight trains. It was the most powerful steam 0-6-0 design produced in Europe.


0-6-0, Usage, United Kingdom
Similarly, the 0-6-0 tank locomotives became the most common locomotive type on all railways throughout the 20th century. All of the Big Four companies to emerge from the Railways Act, 1921 grouping used them in vast numbers. The Great Western Railway, in particular, had many of the type, most characteristically in the form of the pannier tank locomotive that remained in production well past railway nationalisation in 1948.


0-6-0, Usage, United Kingdom
When diesel shunters began to be introduced, the 0-6-0 type became the most common. Many of the British Railways shunter types were 0-6-0s, including Class 03, the standard light shunter, and Class 08 and Class 09, the standard heavier shunters.


0-6-0, Usage, United States
In the United States, huge numbers of 0-6-0 locomotives were produced, with the majority of them being used as switchers. The USRA 0-6-0 was the smallest of the USRA Standard classes designed and produced during the brief government control of the railroads through the USRA during the First World War. 255 of them were built and ended up in the hands of about two dozen United States railroads.


0-6-0, Usage, United States
In addition, many of the railroads (and others) built numerous copies after the war. The Pennsylvania Railroad rostered over 1,200 0-6-0 types over the years, which were classed as class B on that system. The United States 0-6-0s were generally tender locomotives.


0-6-0, Usage, United States
During the Second World War, no fewer than 514 USATC S100 Class 0-6-0 tank engines were built by the Davenport Locomotive Works, for use by the United States Army Transportation Corps in both Europe and North Africa. Some of these remained in service long after the war, having been purchased or otherwise adopted by the countries where they were used. These included Austria, Egypt, France, Iraq, the United Kingdom and Yugoslavia.


0-6-0, Usage, United States
The fourteen engines purchased by the Southern Railway in 1946 remained in service well into the 1960s. Designed to be extremely strong but easy to maintain, these engines had a very short wheelbase that allowed them to operate on dockyard railways.


0-6-0+0-6-0
Under the Whyte notation for the classification of steam locomotives, 0-6-0+0-6-0 represents the wheel arrangement of an articulated locomotive with two separate swivelling engine units, each unit with no leading wheels, six powered and coupled driving wheels on three axles and no trailing wheels. The arrangement is effectively two 0-6-0 locomotives operating back-to-back and was used on Garratt, Double Fairlie, Meyer and Kitson-Meyer articulated locomotives. A similar arrangement exists for Mallet steam locomotives on which only the front engine unit swivels, but these are referred to as 0-6-6-0.


0-6-0+0-6-0
In the United Kingdom, the Whyte notation of wheel arrangement was also used for the classification of electric and diesel-electric locomotives with side-rod coupled driving wheels.


0-6-0+0-6-0, Overview
The 0-6-0+0-6-0 wheel arrangement was used on Garratt, Double Fairlie, Meyer and Kitson-Meyer locomotives, although in some cases Double Fairlies with this arrangement were also referred to as 0-6-6-0.


0-6-0+0-6-0, Overview, Garratt locomotives
The 0-6-0+0-6-0 was a rare Garratt model. Beyer, Peacock, the owner of the Garratt patent, only built two of this type to 2 ft 6 in (762 mm) gauge for the Buthidaung-Maungdaw Tramway in Burma. Belgian builder Société Anonyme St. Leonard of Liège constructed 31 for the Belgian Congo and two for the roadside tramways of the Belgian SNCV. Hanomag commenced the construction of a single locomotive, which was completed by Henschel for the Limburg Tramway in the Netherlands. This last was the only inside-cylinder Garratt to be built.


0-6-0+0-6-0, Overview, Kitson-Meyer locomotives
In 1894, Kitson and Company of Leeds built a modified Meyer articulated locomotive of this wheel arrangement for the Anglo-Chilean Nitrate and Railway Company. Thereafter, the Kitson-Meyer type was widely used in South America, particularly on the Colombian and Chilean railways. The four which were built for Southern Africa were not successful.


0-6-0+0-6-0, Usage, Belgian Congo
Between 1913 and 1926, Belgian locomotive builders Société Anonyme St. Leonard of Liège constructed 31 Garratt 0-6-0+0-6-0 locomotives for the 750 mm (2 ft 5 1⁄2 in) gauge Compagnie du C.F. du Congo in the Belgian Congo. The locomotives were delivered in four batches, one in 1913, twelve in 1920-21, eight in 1924-25 and the last ten in 1925-26.


0-6-0+0-6-0, Usage, Ireland
Córas Iompair Éireann no. CC1, generally known as the Turf Burner, was a prototype 0-6-0+0-6-0 articulated steam locomotive designed by Oliver Bulleid. The locomotive shared some of the characteristics of Bulleid's previous attempt to develop a modern steam locomotive, the Southern Railway's Leader class. The locomotive had a relatively short career and was never used in front-line service.


0-6-0+0-6-0, Usage, South Africa
In 1876, the Cape Government Railways (CGR) placed a single experimental Double Fairlie side-tank locomotive in service on the Cape Eastern system, working out of East London. Built by Avonside Engine Company, it was the first articulated locomotive to enter service in South Africa and also the first locomotive to be equipped with Walschaerts valve gear. After some shortcomings were brought to the attention of the locomotive builders, a second Double Fairlie which incorporated these improvements was delivered and placed in service in 1878.


0-6-0+0-6-0, Usage, South Africa
The Kitson-Meyer type was tried out by three railways in Southern Africa. In 1903, Kitson persuaded the CGR, the Beira and Mashonaland Railway (B&MR) and the Central South African Railways (CSAR) to try their new 0-6-0+0-6-0 Kitson-Meyer articulated steam locomotive. In 1903, one locomotive was delivered to the CGR and two to the B&MR and, in 1904, one to the CSAR. Unlike a Garratt, both engine units on these locomotives were arranged with the cylinders aft of the coupled wheels.


0-6-0+0-6-0, Usage, South Africa
All three railways found their Kitson-Meyers to be poor steamers and, as built, none of these locomotives had a long service life. The three CGR and B&MR locomotives were all scrapped by 1912. In 1906, the CSAR modified its Class M locomotive by reducing the diameter of the cylinders to bring them within the range of the boiler’s steam generating capacity. While this reduced the locomotive’s tractive effort, it improved its performance sufficiently to allow it to survive in service longer than the other three. In 1912, it was assimilated into the South African Railways and designated Class KM.


0-6-0+0-6-0, Usage, United Kingdom
The only steam locomotive example of this type of engine in the United Kingdom was the Leader. It was originally commissioned by the Southern Railway but it was completed by British Railways in 1949. The locomotive was a class of experimental articulated steam locomotive, produced in the United Kingdom to the design of the innovative engineer Oliver Bulleid. The Leader was effectively a Meyer locomotive since both sets of drivers were articulated. It was built in an attempt to extend the life of steam traction by eliminating many of the operational drawbacks associated with existing steam locomotives.


0-6-0+0-6-0, Usage, United Kingdom
The 0-6-0+0-6-0 configuration was also applied to diesel-electric locomotives when British Rail created the Class 13 in 1965. This was done by permanently coupling two Class 08 0-6-0DE shunting engines as "master and slave" units, the latter with its cab removed. In North American terminology, this is referred to as a "cow and calf" arrangement. The modification came about because of a need to provide more powerful shunting locomotives for the Tinsley Marshalling Yard.


0-6-2
Under the Whyte notation for the classification of steam locomotives, 0-6-2 represents the wheel arrangement of no leading wheels, six powered and coupled driving wheels on three axles and two trailing wheels on one axle. The type is sometimes known as a Webb or a Branchliner.


0-6-2, Overview
While some locomotives with this wheel arrangement had tenders, the majority were tank locomotives which carried their coal and water onboard.


0-6-2, Usage, Finland
Finland used two classes of 0-6-2T locomotive, the Vr2 and the Vr5.


0-6-2, Usage, Finland
The Vr2 class was numbered in the range from 950 to 965. Five of them are preserved in Finland, no. 950  at Joensuu, no. 951 at Tuuri, no. 953 at Haapamäki, no. 961 at Jyväskylä and no. 964 at the Veturimuseo at Toijala.


0-6-2, Usage, Finland
The Vr5 class was numbered in the range from 1400 to 1423. No . 1422 is preserved at Haapamäki.


0-6-2, Usage, Philippines
There were 30 Dagupan-type locomotives built between 1889 and 1890. All were tank locomotives, weighed 32 tonnes (71,000 lb) and were run a maximum speed of 33 km/h (21 mph). These were divided into two subclasses: the A subclass built by Neilson and Company and the B subclass built by Dübs and Company.


0-6-2, Usage, Philippines
Another 25 locomotives of the C class were built in 1906 by the North British Locomotive Company (which succeeded Dübs) and were regarded as distinct from the Dagupan class.


0-6-2, Usage, Philippines
During the Manila Railroad era, they were replaced in mainline service by American tender locomotives such as the Porter 4-6-0 built in 1919 or the 4-6-2 Pacifics built by Baldwin Locomotive Works between 1926 and 1929.


0-6-2, Usage, Philippines
A B-class locomotive named Urdaneta (No. 17) remained in shunting service until 1963 and is one of only three steam locomotives preserved by the PNR. After its retirement, Urdaneta was first displayed in the Tutuban station. It is now on static display in Dagupan, Pangasinan. The rest were scrapped between 1917 and 1940.


0-6-2, Usage, South Africa, Tender locomotives
Between 1890 and 1898, four 0-6-2 tender locomotives were placed in service by the Cape Copper Company on its 2 ft 6 in (762 mm) gauge Namaqualand Railway between Port Nolloth and O'okiep in the Cape Colony. Acquired to meet the traffic needs of the upper mountainous section of the line, they became known as the Mountain type. The first three of these locomotives were later described as the Clara Class, while the fourth was included in this Class by some and included in the subsequent Scotia Class by others.


0-6-2, Usage, South Africa, Tender locomotives
Between 1900 and 1905, six more Mountain type 0-6-2 tender locomotives were placed in service by the Cape Copper Company. Later described as the Scotia Class, they were similar to the earlier Clara Class locomotives, but with longer boilers, longer fireboxes and larger firegrates.


0-6-2, Usage, South Africa, Tank locomotive
In 1892 and 1893, the Nederlandsche-Zuid-Afrikaansche Spoorweg-Maatschappij of the Zuid-Afrikaansche Republiek (Transvaal Republic) placed twenty 3 ft 6 in (1,067 mm) Cape gauge 0-6-2T locomotives in mainline service. Since the railway classified its locomotives according to their weight, these locomotives were known as the 40 Tonners.


0-6-2, Usage, South West Africa
Three classes of 600 mm (1 ft 11+5⁄8 in) gauge 0-6-2 locomotives were supplied to German South West Africa between 1904 and 1908.


0-6-2, Usage, United Kingdom
In the United Kingdom, the type was only ever used for tank engines and was first used by William Barton Wright of the Lancashire and Yorkshire Railway in 1880.


0-6-2, Usage, United Kingdom
The arrangement was soon afterwards used by F.W. Webb of the London and North Western Railway on his famous Coal Tanks of 1881–1897. Many locomotives of this type were also used to haul coal in the South Wales Valleys by the Great Western Railway and its predecessors.


0-6-2, Usage, United Kingdom
Several railways around London later used the type for heavy suburban passenger trains, notably the following:


0-6-2, Usage, United Kingdom
Gresley later improved upon the GER class with various versions of his London and North Eastern Railway (LNER) N7 class, built between 1925 and 1928.


0-6-2, Usage, United States
In the United States, 0-6-2 locomotives were largely 2-6-0 type locomotives which had been rebuilt with a larger firebox and therefore required greater weight distribution near their backs. The leading wheels were therefore relocated to the rear as trailing wheels. Nearly all of these locomotives were assigned to switch locomotive workings or used on branch lines.


0-6-2, Usage, United States
Many 0-6-2 types were found in the state of Hawaii on sugar cane railroads across the state. Most notable were the 0-6-2T’s of the Mcbryde Sugar Company of Kauai, 3 of which survive and are currently the only original steam engines operating in Hawaii.


0-6-2+2-6-0T
A 0-6-2+2-6-0, in the Whyte notation for the classification of steam locomotives by wheel arrangement.. The only examples were forms of the Meyer. articulated locomotive.


0-6-2+2-6-0T
It is best known for its use in the French du Bousquet locomotives by Gaston du Bousquet.


0-6-2+2-6-0T
The wheel arrangement was used by the Ferrocarril de Antofagasta a Bolivia in Chile, and by the Chemin de Fer du Nord in France


0-6-4
Under the Whyte notation for the classification of steam locomotives, 0-6-4 represents the wheel arrangement of no leading wheels, six powered and coupled driving wheels on three axles, and four trailing wheels on two axles.


0-6-4, Overview
The 0-6-4 wheel arrangement appears to have only been used on tank engines and Single Fairlies. The earliest known example was the Moel Tryfan narrow gauge locomotive, built for use on the North Wales Narrow Gauge Railways. It was a Single Fairlie type, built by the Vulcan Foundry near Manchester in 1875. It was followed by the R class and S class, built by the Avonside Engine Company of England for the New Zealand Railways Department between 1878 and 1881.


0-6-4, Usage, Australia
The South Australian Railways K class locomotives were introduced in 1884, designed by William Thow. They were noted to run more smoothly bunker-first. After the electrification of the Mersey Railway in England, four of its 0-6-4T locomotives were sold to J. & A. Brown of New South Wales, Australia, where one, number 5, is preserved at the New South Wales Rail Transport Museum, Thirlmere, New South Wales.


0-6-4, Usage, Australia
Three members of New Zealand's S class were also sold to the Western Australian Government Railways in 1891.


0-6-4, Usage, New Zealand
New Zealand’s R class and S class Single Fairlies were popular with crews and capable of all duties from express passenger trains to shunting tasks. The S class were limited to the Wellington Region when they were introduced, but the R class were distributed throughout the country. All were withdrawn by 1936, but R class no. 28 is preserved as a static exhibit in a Reefton park.


0-6-4, Usage, South Africa, Netherlands-South African Railway Company
This wheel arrangement provided the bulk of the motive power for the 3 ft 6 in (1,067 mm) Nederlandsche-Zuid-Afrikaansche Spoorwegmaatschappij (NZASM) in the Zuid-Afrikaansche Republiek (ZAR). Between 1893 and 1898, 175 46 Tonner 0-6-4T steam locomotives were placed in service, built by the Maschinenfabrik Esslingen in Germany.


0-6-4, Usage, South Africa, Netherlands-South African Railway Company
In 1899, twenty more were ordered from the Nederlandse Fabriek van Werktuigen en Spoorwegmaterieel (Werkspoor) in the Netherlands, of which only two were delivered by the time the Imperial Military Railways (IMR) took over all railway operations in the ZAR during the Second Boer War. The other eighteen locomotives in this order were delivered directly to the IMR, who diverted two of them to Lourenço Marques in Mozambique.


0-6-4, Usage, South Africa, Netherlands-South African Railway Company
At the end of the war, the survivors of these locomotives were taken onto the roster of the Central South African Railways (CSAR) and designated Class B, while the two in Mozambique were taken onto the roster of the Caminhos de Ferro de Mocambique (CFM). In 1912, the remaining CSAR locomotives were assimilated into the South African Railways (SAR).


0-6-4, Usage, South Africa, Mozambique
The CFM eventually had at least thirty 46 Tonner locomotives in service. Between 1897 and 1898, some 46 Tonners were sold by the NZASM to the CFM. The two locomotives which were delivered after the outbreak of the war and diverted to Lourenço Marques upon arrival, were also taken onto the CFM roster at the end of the war. Later, between 1911 and 1920 during the CSAR and SAR eras, six more were sold to the CFM.


0-6-4, Usage, United Kingdom
Other than examples for export, 0-6-4T locomotives enjoyed a brief vogue in the United Kingdom prior to the First World War, but were not widely used. Nine locomotives of this type were supplied by Beyer, Peacock and Company for the opening of the Mersey Railway in 1886.


0-6-4, Usage, United Kingdom
William Dean built three crane tanks in 1901, and Kitson & Co. of Leeds supplied nine locomotives to the Lancashire, Derbyshire and East Coast Railway in 1904.


0-6-4, Usage, United Kingdom
Other examples included the Midland Railway 2000 Class of 1907, the Highland Railway Drummond 0-6-4T Class of 1909, the SECR J class of 1913 and the Metropolitan Railway G Class of 1915. The type was eventually superseded by the popular 2-6-4T locomotive.


0-6-4, Usage, United States
In the United States, the 0-6-4 locomotive was largely built only for use in railyards, essentially as an adaptation of an 0-6-0 switch engine with an extended firebox, or a 4-6-0 reconstructed with a larger firebox which necessitated the relocation of the leading wheels to the rear to support the firebox. Some Mason Bogie locomotives used this wheel arrangement.


0-6-6
Under the Whyte notation for the classification of steam locomotives, 0-6-6 represents the wheel arrangement of no leading wheels, six powered and coupled driving wheels on three axles and six trailing wheels on three axles.


0-6-6, Overview
The 0-6-6 wheel arrangement was usually found on Single Fairlie or Mason Bogie locomotives. The Fairlie locomotive was invented and patented in 1864 by the Scottish engineer Robert Francis Fairlie. The first Fairlie locomotives later became known as Double Fairlies.


0-6-6, Overview
A variation of the original Fairlie concept was the Single Fairlie, also known as the Mason Fairlie. The Single Fairlie design was essentially half a Double Fairlie. It retained the ability to negotiate sharp curves and, while it abandoned the bidirectional nature of the Double Fairlie, it regained the ability of conventional locomotives to have a large water and coal bunker behind the cab and to use a trailing tender if necessary.


0-6-6, Overview
Most Single Fairlies were tank locomotives and early models were similar in general appearance to conventional tank engines with side tanks and a coal bunker aft of the cab, all mounted on a single rigid frame. The pivoting engine unit was mounted under the boiler and the unpowered bogie under the cab and bunker.


0-6-6, Overview
Later models were similar in appearance to conventional tender locomotives. It was developed by William Mason in the United States, where the type became known as the Mason Bogie. It had one boiler at the front, a cab in the centre and a water-and-coal bunker at the rear end, all mounted on a single rigid frame, with a single engine unit under the boiler and an unpowered bogie under the bunker.


0-6-6, Usage, United States
The first known 0-6-6T locomotive was built for the 3 ft (914 mm) gauge New Bedford Railroad by Mason Machine Works in May 1874. It was apparently not numbered, but bore the name Wm. Mason. The locomotive later went to the Boston, Clinton & Fitchburg as its no. 23, then to the Old Colony as its no. 108 and finally to the New York, New Haven & Hartford as its no. 708.


0-6-6, Usage, United States
More 0-6-6T locomotives were produced by Mason between 1875 and 1881. Many of them were subsequently rebuilt to a 2-6-6T wheel arrangement.


0-6-6-0
Under the Whyte notation for the classification of steam locomotives, a 0-6-6-0 wheel arrangement refers to a locomotive with two engine units mounted under a rigid locomotive frame, with the front engine unit pivoting and each engine unit with six coupled driving wheels without any leading or trailing wheels. The wheel arrangement was mostly used to describe Mallet locomotive types.


0-6-6-0
A similar wheel arrangement exists for Double Fairlie, Meyer, Kitson-Meyer and Garratt articulated locomotives, but on these types it is referred to as 0-6-0+0-6-0 since both engine units are pivoting.


0-6-6-0, Overview
The 0-6-6-0 wheel arrangement was used mostly on Mallet locomotives, on which the engine units were mounted either in tandem or facing each other.


0-6-6-0, Usage, Canada
The only compound Mallets to operate in Canada were the R1 class 0-6-6-0 Vaughan design locomotives, with the cylinder ends of the engine units facing each other. The class was owned by the Canadian Pacific Railway and served on the Big Hill in British Columbia, which had a 4.1% grade. Five locomotives were built between 1909 and 1911. A sixth one was built, but it was a simple expansion Mallet with two sets of high-pressure cylinders. All the locomotives in this class were later converted to 2-10-0 types and were used as shunting and transfer engines in Montreal.


0-6-6-0, Usage, Germany
The Saxon Class XV HTV was a class of goods train tank steam locomotive operated by the Royal Saxon State Railways, which had been conceived for hauling trains and acting as banking engines for routes in the Ore Mountains. The two CCh4v locomotives were built in 1916 at the Sächsischen Maschinenfabrik, formerly Hartmann. In 1925, the Deutsche Reichsbahn grouped them into their DRG Class 79.0. The locomotive was of unusual design with two fixed six-coupled engine units with a central double cylinder on each side, each with a high-pressure cylinder for the rear and a low-pressure cylinder for the front drive.


0-6-6-0, Usage, United States
The first Mallet locomotive in North America was built in the United States and was of this type, the Baltimore and Ohio Railroad Class O no. 2400. Nicknamed Old Maude after a cartoon mule, it had a 71,500 pounds-force (318 kilonewtons) tractive effort and was a great success despite a top speed of only 21 miles per hour (34 kilometres per hour).


0-6-6-0, Usage, United States
The Kansas City Southern used the type as freight engines, with pilots, and had the most of them with twelve locomotives. The 0-6-6-0 wheel arrangement was also used to a limited extent on logging railroads and in mountain terminals.


0-6-6-0, Usage, United States
The Western Maryland Railway had a small fleet of 2-6-6-2 locomotives which, at one time, were the heaviest locomotives in the world, weighing 264 Tons. They were all converted to 0-6-6-0 locomotives for heavy switching.


0-8-0
Under the Whyte notation for the classification of steam locomotives, 0-8-0 represents the wheel arrangement of no leading wheels, eight powered and coupled driving wheels on four axles and no trailing wheels. Locomotives of this type are also referred to as eight coupled.


0-8-0, Overview
Examples of the 0-8-0 wheel arrangement were constructed both as tender and tank locomotives. The earliest locomotives were built for mainline haulage, particularly for freight, but the configuration was later also often used for large switcher (shunter) types.


0-8-0, Overview
The wheel arrangement provided a powerful layout with all engine weight as adhesive weight, which maximised the tractive effort and factor of adhesion. The layout was generally too large for smaller and lighter railways, where the more popular 0-6-0 wheel arrangement would often be found performing similar duties.


0-8-0, Usage, Austria
Two 0-8-0 locomotives were delivered from Andre Koechlin & Cie in Mulhouse to the Austrian Southern Railway in 1862. They were later sent to Italy and worked over the Apennines between Bologna and Pistoja.


0-8-0, Usage, China
In 1952, the Chrzanów works in Poland supplied 81 750mm gauge locomotives, which were later versions of the Russian P24 class. By 1958, China was building their own copies resulting in such classes as the C2, YJ, ZM-4, ZG and ZM16-4.


0-8-0, Usage, Christmas Island
Peckett & Sons of Bristol built a 0-8-0 tender locomotive for the Christmas Island Phosphate Co.'s Railway in 1931.


0-8-0, Usage, Germany
Freight engines with an 0-8-0 wheel arrangement were once very popular in Germany. The Prussian state railways had several types of 0-8-0s that were all classified as G7, G8 and G9.


0-8-0, Usage, Germany
The latest of these, the Prussian G 8.1, was the most numerous German state railway locomotive with over five thousand examples being built between 1913 and 1921. They remained in service with the Deutsche Bundesbahn until 1972.


0-8-0, Usage, Germany
The narrow gauge Heeresfeldbahn class HF 160 D were developed for wartime service during the Second World War. The engines were also classified as Kriegsdampflokomotive 11 (Military steam locomotive 11 or KDL 11). After the war, the locomotives were put to use for civilian purposes.


0-8-0, Usage, Russia
The first steam locomotives 0-8-0 in Russia — Е class have been produced since 1858. In Russia, the 0-8-0 type locomotives were represented by the various O-class (Osnovnoj-mainline) freight locomotives. They were built from the end of the 19th century until the 1920s. They were commonly called the Ovechka (Sheep) and were the most common freight locomotives in Tsarist Russia. Some are still preserved in working order.


0-8-0, Usage, Russia
One-thousand of the 750mm gauge standard design, also known as class 159, were built between 1930 and 1941. They were poor performers, so the Kolomna works built an improved version of these locomotives, known as the P24 class. Nine were built before 1941.


0-8-0, Usage, South Africa
On the South African Railways (SAR), shunting was traditionally performed by downgraded mainline locomotives. When purpose-built 3 ft 6 in (1,067 mm) Cape gauge shunting locomotives were eventually introduced in 1929, the SAR preferred to adhere to the American practice of using tender locomotives for shunting, rather than the European practice of using tank locomotives. Three classes of 0-8-0 shunting steam locomotives were introduced between 1929 and 1952.


0-8-0, Usage, South Africa
In 1929, fourteen Class S locomotives were placed in service. They were built by Henschel & Son in Germany, designed to SAR specifications. The top sides of the tender’s coal bunker were set inwards and the water tank top was rounded to improve the crew’s rearward vision.


0-8-0, Usage, South Africa
The second type, the Class S1, was designed by MM Loubser, chief mechanical engineer of the SAR from 1939 to 1949. Twelve of these locomotives, a heavier version of the Class S, were built at the Salt River workshops in Cape Town with the first being delivered in October 1947. A further 25 Class S1 locomotives were ordered from the North British Locomotive Company. Glasgow in 1952 and delivered in 1953 and 1954. The Class S1 was noted for its efficiency and economy and could cope with block loads of up to 2,000 long tons (2,032 tonnes).


0-8-0, Usage, South Africa
To meet the need for shunting locomotives with a light axle load for harbour work, these were followed in 1952 and 1953 by 100 Class S2 locomotives, built by Friedrich Krupp in Essen, Germany. To adhere to the specified weight limit, the Class S2 was built with a small boiler, with the result that it had the appearance of a Cape gauge locomotive with a narrow gauge boiler, particularly when viewed from the front. Also to reduce the axle load, it had Type MY1 tenders which rode on Buckeye three-axle bogies.


0-8-0, Usage, Sweden
0-8-0 was wheel arrangement on some Swedish freight locomotives in the early 20th century. The most well known is probably the E class of steam locomotives as many of them survived in the strategic reserve until the 1990s, when all steam engines were removed from the strategic reserve. The E class of locomotives entered production in 1907 and many were modified to a 2-8-0 configuration with a name change to E2. The locomotives were intended as mixed locomotives in northern Sweden with its steeper inclines and for heavy freight in southern Sweden where the landscape is flatter.


0-8-0, Usage, Sweden
Another locomotive was the Prussian G8.1 named the G class in Sweden which was ordered by the national railway company during WWI in 1916, but delivery was delayed until 1918.


0-8-0, Usage, United Kingdom
Two examples of 0-8-0T tank locomotives were built by Archibald Sturrock of the United Kingdom’s Great Northern Railway in 1866, but the design was not perpetuated. A tender locomotive version was introduced on the Barry Railway Company in 1889 to haul coal trains.


0-8-0, Usage, United Kingdom
Francis Webb of the London & North Western Railway (LNWR) built 282 examples of a compound 0-8-0 locomotive between 1892 and 1904. A further 290 examples of a simple expansion version were built by his successor between 1910 and 1922.


0-8-0, Usage, United Kingdom
In 1929, Richard Maunsell of the Southern Railway designed and built eight Z class side tank engines.


0-8-0, Usage, United Kingdom
In 1902, John Robinson of the Great Central Railway introduced his Class 8A tender engines, which were designated the Q4 class under the London & North Eastern Railway. From 1934, the class was replaced by the Robinson 2-8-0's and their withdrawal and scrapping began, but between 1942 and 1945 Edward Thompson converted thirteen into side-tanks, designated LNER Class Q1.


0-8-0, Usage, United Kingdom
Under the grouping of 1923, the LNWR became part of the London, Midland & Scottish Railway (LMS). Henry Fowler designed an inside cylinder engine in 1929 to replace the LNWR examples, but they proved to be unsatisfactory and ended up having shorter lives than the LNWR locomotives.


0-8-0, Usage, United Kingdom
In 1914, Manning Wardle of Leeds built a side-tank engine called Katharine for the Bridge Water Collieries system. On the Kent & East Sussex Railway, the Hecate was built for Colonel Stephens by RW Hawthorn, Leslie & Company in 1904, but the branchline for which it was built was never completed and since the engine was too big for his other railways, it was exchanged in 1932 for a smaller engine from the Southern Railway. The engine Hecate ended up as a motive power depot shunter at Nine Elms Locomotive Works and was scrapped in 1950.


0-8-0, Usage, United Kingdom
Southern Railway, Richard Maunsell's Z class were first built in 1929 as a heavy shunting engine, 8 were built by 1930.


0-8-0, Usage, United States
The 0-8-0 wheel arrangement appeared early in locomotive development in the United States, during the mid-1840s. The configuration became popular and was more commonly constructed as a tender locomotive. It saw extensive use as a heavy switcher and freight engine.


0-8-0, Usage, United States
Beginning in 1844, Ross Winans developed a series of 0-8-0 types for the Baltimore & Ohio Railroad (B&O), starting with a vertical-boiler design where the crankshaft was directly above and geared to the rear driving wheel. With a horizontal boiler, this became the Mud Digger class of engines on the B&O, of which twelve were built. In late 1847, the B&O moved to abandon geared drives and, in 1848, Baldwin delivered the first of a series of 0-8-0 freight engines.


0-8-0, Usage, United States
The USRA 0-8-0 was a USRA standard class, designed by the United States Railroad Administration during World War I. This was the standard heavy switcher of the USRA types, of which 175 examples were built by ALCO, Baldwin and Lima for many different railroads in the United States. After the dissolution of the USRA in 1920, an additional 1,200 examples of the USRA 0-8-0 were built.


0-8-0, Usage, United States
In the 1920s the Pennsylvania Railroad wanted the best motive power possible to handle the switching chores at their yards and interchanges. Built in its own Juniata Shops, the Pennsylvania Railroad class C1, at 278,000 lb, was the heaviest two-cylinder 0-8-0 switcher ever produced. The calculated tractive effort was 76,154 lb.


0-8-0, Usage, United States
The last steam locomotive to be built in the USA for a Class I railroad was 0-8-0 no. 244, a Class S1 switch engine erected by the Norfolk & Western's Roanoke Shops in December 1953. It was retired in March 1958.


0-8-2
Under the Whyte notation for the classification of steam locomotives, 0-8-2 represents the wheel arrangement of no leading wheels, eight powered and coupled driving wheels on four axles, and two trailing wheels on one axle (usually in a trailing truck).


0-8-2
Other equivalent classifications are:UIC classification (also known as German classification and Italian classification): D1, French classification: 041,Turkish classification: 45,Swiss classification: 4/5.


0-8-2, Usage
This has been a relatively unusual wheel arrangement on mainline railways


0-8-2, Usage, United Kingdom
In the United Kingdom, a number of tank locomotive designs were built of the 0-8-2 type, including the London and North Eastern Railway (LNER) R1 class, designed by Henry A. Ivatt and built originally for the Great Northern Railway as their class L1. These locomotives were intended for suburban passenger service, but did not prove satisfactory, so they ended up on freight service.


0-8-2, Usage, United Kingdom
Other examples include the LNWR 1185 Class and the Port Talbot Railway 0-8-2T (Cooke) and  Port Talbot Railway 0-8-2T (Sharp Stewart).


0-8-2, Usage, Australia
Dorrigo Steam Railway and Museum has preserved Avonside 0-8-2 Tank Locomotive number 14, formerly operated by South Maitland Railways collieries line in the Hunter Valley of N.S.W. Number 14 is operational, it was built in 1909 in Bristol. The design combination of this 0-8-2 tank Locomotive and the N.S.W. Government Railways 50 class 2-8-0, produced the South Maitland Railways 10 class 2-8-2 Tank Locomotives. Number 14 is the only 0-8-2 in Australia.


0-8-2, Usage, North America
The 0-8-2 was not a common wheel arrangement. In North America, its use was confined to 2-8-2 "Mikado" types assigned to switcher roles; the lead truck was often removed to give more weight on drivers, a guiding truck not being needed at slow speed. However, the Illinois Central rebuilt some of their 2-8-2s into 0-8-2s for use in Markham Yard in Chicago, Illinois.


0-8-2, Usage, Narrow-gauge railways
The arrangement has proved a little more popular on narrow-gauge and minimum-gauge lines, where the lack of leading wheels was less important due to the relatively slow operating speeds.


0-8-2, Usage, Narrow-gauge railways
0-8-2 locomotives operate, for example, on the Zillertalbahn in Austria. The X class locomotives of the metre-gauge Nilgiri Mountain Railway in the Nilgiri Hills of southern India has a class 0-8-2T rack and pinion compound locomotives built by the Swiss Locomotive and Machine Works, Winterthur.


0-8-2, Usage, Minimum-gauge railways
One Ravenglass and Eskdale Railway, 0-8-0T locomotive was rebuilt in 1927, as an 0-8-2 tender locomotive named River Irt. It has remained in traffic on passenger duties ever since and is now the oldest working 15 inch gauge locomotive in the world.


0-8-4
"0-8-4" is the second episode of the first season of the American television series Agents of S.H.I.E.L.D. Based on the Marvel Comics organization S.H.I.E.L.D., it follows Phil Coulson and his team of S.H.I.E.L.D. agents as they travel to Peru to investigate an object of unknown origins. It is set in the Marvel Cinematic Universe (MCU) and acknowledges the franchise's films. The episode was written by showrunners Maurissa Tancharoen, Jed Whedon, and Jeffrey Bell, and was directed by David Straiton.


0-8-4
Clark Gregg reprises his role as Coulson from the film series, starring alongside Ming-Na Wen, Brett Dalton, Chloe Bennet, Iain De Caestecker, and Elizabeth Henstridge. The episode is set in Peru, featuring guest star Leonor Varela as a member of the Peruvian military. Filming took place in July 2013 at the Los Angeles County Arboretum and Botanic Garden, with some location shooting in Peru. Composer Bear McCreary used ethnic instruments to support this setting. Tesseract technology is carried over from the films to tie-in with the conflict, while special guest star Samuel L. Jackson also reprises his film role of Nick Fury in a cameo appearance.


0-8-4
"0-8-4" originally aired on ABC on October 1, 2013, and was watched by 13.17 million viewers within a week. The episode received a mostly positive critical response, with Jackson's appearance considered a highlight by many, but also seen as unearned.


0-8-4, Plot
Beginning immediately after "Pilot", "0-8-4" sees Skye accept Agent Phil Coulson's offer to join his S.H.I.E.L.D. team as a consultant. Though agents Melinda May and Grant Ward oppose this due to her hacktivist background and lack of S.H.I.E.L.D. training, Coulson believes that Skye can be an asset.


0-8-4, Plot
The team travels to Peru to investigate a reported 0-8-4 (the S.H.I.E.L.D. designation for "an object of unknown origin"). They find the object within an ancient Incan temple, and agents Leo Fitz and Jemma Simmons determine that it is Hydra made: powered by the Tesseract and extremely volatile. The national military arrives to claim the weapon for the Peruvian government, led by Camilla Reyes, a former colleague of Coulson's. When they are all attacked by local rebels, the S.H.I.E.L.D. agents and soldiers escape with the weapon to the plane that serves as the agents' mobile base.


0-8-4, Plot
En route to a classified S.H.I.E.L.D. facility, tensions among the agents are high due to poor communication during the fight. This concerns Reyes, who decides to double-cross Coulson and secure the 0-8-4 for her government. Together, the agents devise a plan to activate the weapon, blowing a hole in the Bus. The drop in pressure opens the interior doors, allowing the agents to subdue the soldiers. At the facility, Reyes and her men are incarcerated and the 0-8-4 is launched into the sun in a rocket. The team watch the launch together, celebrating their combined efforts, while Ward agrees to supervise Skye's S.H.I.E.L.D. training. Skye secretly confirms her allegiance to the hacktivist group the Rising Tide.


0-8-4, Plot
In an end tag, S.H.I.E.L.D. Director Nick Fury scolds Coulson for the damage caused to the plane during the fight, and expresses his doubts over Skye's loyalty.


0-8-4, Production, Development and design
ABC announced in May 2013 that it had ordered a full season of Agents of S.H.I.E.L.D. based on the pilot episode. That September, Marvel Television revealed that the second episode in the series was titled "0-8-4", and had been written by the series' showrunners Maurissa Tancharoen, Jed Whedon, and Jeffrey Bell, with David Straiton directing. The 0-8-4 device was created by prop master Scott Bauer to be "stylistic" yet "timeless" and its design was inspired by Art Deco.


0-8-4, Production, Casting
Marvel confirmed in September 2013 that the episode would star main cast members Clark Gregg as Phil Coulson, Ming-Na Wen as Melinda May, Brett Dalton as Grant Ward, Chloe Bennet as Skye, Iain De Caestecker as Leo Fitz, and Elizabeth Henstridge as Jemma Simmons. They also announced the episode's guest cast, which includes Leonor Varela as Camilla Reyes and Carlos Leal as an archaeologist. Samuel L. Jackson makes a surprise cameo appearance at the end of the episode, reprising his role of S.H.I.E.L.D. director Nick Fury from the MCU films.


0-8-4, Production, Casting
Jackson had previously expressed interest in appearing in the show in June 2013. Executive producer Jeph Loeb said there were "a number of places that we thought Nick Fury would have a big impact on the show," ultimately settling on a cameo in this episode as a way to "kind of christen the show, legitimize it in its own way." Bell highlighted Marvel's security team and the dedication of the cast and crew to keeping Jackson's cameo a surprise despite "this age of tweets and spoilers".


0-8-4, Production, Filming
Filming for the episode occurred from July 17 to July 29, 2013. The Peru setting was filmed at the Los Angeles County Arboretum and Botanic Garden, which has gardens dedicated to specific areas of the world, with some location shooting also occurring in Peru. The sequence in which a hole is blown in the side of the Bus was filmed in one day. The final shots utilized stunts and wirework, rigged explosives, wind machines, and green screen. Much of the Nick Fury scene was scripted, but Gregg ad-libbed the line about a proposed fish tank for the Bus.


0-8-4, Production, Music
Composer Bear McCreary had a larger orchestra to work with on "0-8-4" than he did with the pilot, allowing him to compose a much more traditional and "grandiose" score. He also opted to expand his synthesizer use to be "beefier and more aggressive". The South American setting allowed McCreary to add an ethnic component to the score, with his frequent collaborators M.B. Gordy and Chris Bleth playing tribal-sounding drums and ethnic woodwinds, respectively. Guitarist Ed Trybek also recorded various South American guitars for the episode, including timple and charango.


0-8-4, Production, Marvel Cinematic Universe tie-ins
The titular weapon is powered by the Tesseract, the macguffin of the MCU films Captain America: The First Avenger (2011) and The Avengers (2012). The "0-8-4" was made by Hydra, a fictional organization that also appeared in Captain America: The First Avenger. It is stated in the episode that the last object of unknown origin that S.H.I.E.L.D. encountered was "a hammer", referring to Thor's weapon Mjolnir which Coulson discovered on Earth during the events of the film Thor (2011). Coulson also refers to Skye as a consultant, which is how S.H.I.E.L.D. classified Tony Stark during Iron Man 2 (2010), The Avengers, and the Marvel One-Shot short film The Consultant (2011).


0-8-4, Release, Broadcast
"0-8-4" was first aired in the United States on ABC on October 1, 2013. It was aired alongside the US broadcast in Canada on CTV, while it was first aired in the United Kingdom on Channel 4 on October 4, 2013. It premiered on the Seven Network in Australia on October 2, 2013.


0-8-4, Release, Home media
The episode, along with the rest of Agents of S.H.I.E.L.D. 's first season, was released on Blu-ray and DVD on September 9, 2014. Bonus features include behind-the-scenes featurettes, audio commentary, deleted scenes, and a blooper reel. On November 20, 2014, the episode became available for streaming on Netflix.


0-8-4, Reception, Ratings
In the United States the episode received a 3.3/10 percent share among adults between the ages of 18 and 49, meaning that it was seen by 3.3 percent of all households, and 10 percent of all of those watching television at the time of the broadcast. It was watched by 8.66 million viewers. The Canadian broadcast gained 1.83 million viewers, the fourth highest for that day and the twelfth highest for the week. The United Kingdom premiere had 3.08 million viewers, and in Australia, the premiere had 2.8 million viewers, including 1.3 million timeshifted viewers. Within a week of its release, the episode was watched by 13.17 million U.S. viewers, above the season average of 8.31.


0-8-4, Reception, Critical response
MTV.com gave a positive review, saying "If tonight's installment is any indication, the cast will soon be able to support their own weight, make Coulson proud, and audiences sit up and pay attention", and comparing it positively to Tarzan, Beastmaster, and Mutant X. Terri Schwartz of Zap2it also gave a particularly positive review, praising both the connections to the films, including Jackson's cameo, and the internal development of the show, namely that of the character Skye and the team as a whole.


0-8-4, Reception, Critical response
Dan Casey of Nerdist called "0-8-4" "a strong second episode, [which] managed to course-correct from some of the missteps of the pilot". He praised the "solid mix of action, character development, and humor" and concluded that the episode was "genuinely enjoyable television". Eric Goldman of IGN scored the episode 7.5 out of 10, comparing it positively to The A-Team and Indiana Jones, praising its self-awareness, Jackson's cameo, and the development of Coulson's character, but criticizing the lack of development for other characters.


0-8-4, Reception, Critical response
Oliver Sava of The A.V. Club called the episode "an adequate hour of action-adventure television, but the first 59 minutes are missing the spark of the final post-credits scene", seeing room for improvement for all the cast members, and concluding that the show falls "somewhere between Firefly and Dollhouse on the spectrum of Whedon TV influences". Graeme Virtue of The Guardian called Gregg "Agents of S.H.I.E.L.D. 's greatest asset", finding the Jackson cameo to be a "thrill", but that "plot-wise, things perhaps still feel a little inconsequential".


0-8-4, Reception, Critical response
The Hollywood Reporter's Marc Bernardin praised the scale of the episode, describing it as coming "out of the gate like a blockbuster", but criticized its ambitions, asking "Shouldn't this show be, well, nuttier? ... Agents of S.H.I.E.L.D. needs to unhinge itself, but good, and not just be a procedural." He also singled out Skye and May as being unfocused and underdeveloped, respectively, as characters, and he felt the Jackson cameo "gave the whole thing a charge that, in truth, it didn't really earn." Jim Steranko, known for his work on Nick Fury, Agent of S.H.I.E.L.D., found the episode to be "smoother [than 'Pilot'], although more formulaic". He criticized the plot and characters, but praised Jackson's cameo as "an electrifying reminder of what the series could and should be."


0-8-4T
Under the Whyte notation for the classification of steam locomotives, 0-8-4 represents the wheel arrangement of no leading wheels, eight powered and coupled driving wheels on four axles, and four trailing wheels on two axles (usually in a trailing bogie).


0-8-4T, Examples
All examples of this wheel arrangement were tank locomotives; there are no 0-8-4 tender locomotives recorded.


0-8-4T, Examples, United Kingdom
The tank locomotives were themselves rare. Two separate classes were built in the UK, by two different railway companies. Both of these had their origins with an 0-8-0 tender design. Both classes were designed as powerful, but slow-speed, locomotives for heavy shunting. They did not require high speed or long range, so had no need for a leading truck or the greater coal capacity of a tender. Other than this though, they were quite distinct.


0-8-4T, Examples, United Kingdom
The first example was the Great Central Railway Class 8H of 1907. These were designed for hump shunting and so required high tractive effort, good adhesion and traction for starting from rest. Although developed from the 8A tender class, and having some interchangeable parts in their running gear, they also had three cylinders rather than two. The three cylinder tank locomotive was in fashion at this time, as a means of achieving good acceleration from rest, owing to their more even power delivery and the reduced risk of wheelslip.


0-8-4T, Examples, United Kingdom
This three-cylindered pattern had begun with Holden's Decapod of 1902 and carried through into Worsdell and Raven's fast passenger tank locomotives of 4-6-2T and 4-4-4T layout for the North Eastern Railway in 1910 and 1913. Worsdell also designed a comparable heavy shunter of his own, the Class X, although this used the 4-8-0T layout with a leading bogie, rather than trailing.


0-8-4T, Examples, United Kingdom
The class was considered a success, although highly specialised, and developments were rebuilt and built new by the LNER. The rebuilt locomotive trialled a new outside-framed bogie, fitted with a booster engine, the LNER being one of the few UK railways to favour these. Two further locomotives were also built by the LNER. Six were built in total.


0-8-4T, Examples, United Kingdom
The second example was LNWR 380 Class. These were a simple stretched version of the inside-cylindered 0-8-2T 1185 Class, which had been derived from the LNWR's numerous 0-8-0 freight locomotives, with a larger coal bunker. The class was intended for both shunting, banking and as mineral engines for the heavy coal or iron train use in the South Wales coalfield. The enlarged bunker made them more suitable for these longer workings. Thirty were built.


0-8-6
An 0-8-6, in the Whyte notation for the classification of steam locomotives by wheel arrangement, is a locomotive with no leading wheels, eight driving wheels (4 axles) fixed in a rigid frame, and six trailing wheels (normally mounted in a trailing truck). Examples of this type of locomotive were built by Wilhelm von Engerth.


0-8-6-0T
An 0-8-6-0, in the Whyte notation for the classification of steam locomotives by wheel arrangement, is an articulated locomotive with no leading wheels, eight driving wheels (4 axles) fixed in a rigid frame, six driving wheels (3 axles) and no trailing wheels. In the UIC system, this would be described as a DC't arrangement.


0-8-6-0T, Examples
Six locomotives with this wheel arrangement were built by Kitson & Co. as Kitson Meyers for the Transandine Railway, three of which survive today.


0-8-8-0
In the Whyte notation for classifying the wheel arrangement of steam locomotives, an 0-8-8-0 is a locomotive with two sets of eight driving wheels and neither leading wheels nor trailing wheels. Two sets of driving wheels would give far too long a wheelbase to be mounted in a fixed locomotive frame, so all 0-8-8-0s have been articulated locomotives of the Mallet type, whether simple or compound. In the UIC classification, this arrangement would be, refined to Mallet locomotives,  (D)D.  The type was sometimes called Angus in North America.


0-8-8-0
Other equivalent classifications are:UIC classification: DD (also known as German classification and Italian classification)French classification: 040+040Turkish classification: 44+44Swiss classification: 4/4+4/4


0-8-8-0
The lack of leading and trailing wheels to assist the tracking and stability of the locomotive means that the 0-8-8-0 type is not suited to high speeds. The vast majority have seen use as very heavy switchers (generally for hump yard work), transfer locomotives for hauling cuts of cars between rail yards, or pushers for assistance on grades.


0-8-8-0
Most locomotives of this arrangement were built and served in North America, but there were exceptions. The Bavarian State Railways (K.Bay.St.B) built a total of 25 0-8-8-0T tank locomotives of class Gt 2x4/4 between 1913 and 1923, classified after unification of Germany's railway systems as class BR96. These worked trains over heavily graded stretches of line, mostly as bankers (US: pushers) and were the largest locomotives in Europe when introduced.


0-8-8-0, Failures
In 1915, the ? purchased an 0-8-8-0 numbered 2000 but proved to be very unreliable and also very costly to maintain as the engine steamed poorly and derailed often. Eventually in the 1930s, the engine was split into two locomotives. The result was a useful 2-8-0 freight engine and a barely functional 0-8-0T switcher engine.


0-8-8-0, Failures
The switcher engine was later sold to the N&W where they decided to convert their W2 2-8-0's to the newly designated W6 class 0-8-0t's based on the aftermath of the ? 0-8-8-0 that earlier was cut into two engines because at the time they were looking for a yard goat engine that could work in the yards of Roanoke.


0-8-8-0, Failures
This would keep the N&W W6 class engines busy working near and around the area of Roanoke like bringing engines in for servicing, taking the engines to the coal elevators, water towers, the ready tracks and so on while the successful N&W S1 and S1a's would handle the major switching tasks and servicing the local industries nearby. The original 0-8-0 was sold for scrap by 1950 while the W6 copies lasted until their retirement after 1958. The 2-8-0 and 0-8-0 locomotives that were from the 0-8-8-0 locomotive chassis do not exist anymore.


0-8-8-0, Popular culture
Used in Lionel O gauge only in Erie and Pennsylvania road names.


0-8-8-0, Popular culture
The Bavarian locomotives have been modelled in HO gauge by Marklin/Trix and by Rivarossi.


0-D Beat Drop
0-D Beat Drop is a 2009 rhythm-puzzle video game developed by Cyclone Zero and JAMS SOFT and published by Aksys Games and Arc System Works. With elements of Puyo Puyo and Lumines, the game has players organizing colored shapes onto a playing field in order to match three pieces and clear them using the game's titular Beat Drop, which requires the player to listen to the techno soundtrack. It also sponsored a Japanese K-pop band Sweat Vacation and the difficulty for this game was added based on it.


0-D Beat Drop, Gameplay
0-D Beat Drop is designed as combat puzzle game for two or more players. As the players match up differently shaped pieces in various colors, a group of blocks is stored in the bank. When a chain is completed, a certain number of pieces fall down on the opponent's playing field. The pieces drop from the top of the screen, and they need to be placed so that the colors match. Matches formed from three consecutive colors can be vertical, horizontal, or go around angles.


0-D Beat Drop, Gameplay
For a block to form a match, it has to be dropped within the beat of the techno track in the background. A meter on the side of the screen helps the players keep track of the current beat, and if the pieces are dropped while the meter is in the scoring zone, it will allow the blocks to be destroyed. Doing so for a successive chain of blocks increases the score multiplier, while also making the scoring zone on the meter smaller.


0-D Beat Drop, Gameplay
0-D Beat Drop consists of 6 game modes: Planet Quest, Co-Op, Survival Four, Time Attack, Task and Multiplayer. One of the features of the game is the Beat-O-Matic, which can analyze music from a USB drive or the 360 hard drive, determine its BPM, and sync the game with it.


0-D Beat Drop, Reception
O-D Beat Drop has received generally positive reviews from critics and holds a rating of 80 on Metacritic based on 11 reviews.


0.0.0.0
In the Internet Protocol Version 4, the address 0.0.0.0 is a non-routable meta-address used to designate an invalid, unknown or non-applicable target. This address is assigned specific meanings in a number of contexts, such as on clients or on servers.


0.0.0.0, As a host address
In the context of servers, 0.0.0.0 can mean "all IPv4 addresses on the local machine". If a host has two IP addresses, 192.168.1.1 and 10.1.2.1, and a server running on the host is configured to listen on 0.0.0.0, it will be reachable at both of those IP addresses.


0.0.0.0, Routing
In the context of routing tables, a network destination of 0.0.0.0 is used with a network mask of 0 to depict the default route as a destination subnet. This destination is expressed as 0.0.0.0/0 in CIDR notation. It matches all addresses in the IPv4 address space and is present on most hosts, directed towards a local router.


0.0.0.0, Routing
In routing tables, 0.0.0.0 can also appear in the gateway column. This indicates that the gateway to reach the corresponding destination subnet is unspecified. This generally means that no intermediate routing hops are necessary because the system is directly connected to the destination.


0.0.0.0, In IPv6
In IPv6, the all-zeros address is typically represented by :: (two colons), which is the short notation of  0000:0000:0000:0000:0000:0000:0000:0000. The IPv6 variant serves the same purpose as its IPv4 counterpart.


0.01
0.01 is the debut studio album of H3llb3nt, released on February 20, 1996 by Fifth Colvmn Records. Sonic Boom called 0.01 an "exquisitely tuned finite element state machine which hums along with a groove all of its own" and only criticized the album for being too short.


0.5 mm
0.5 mm (0.5ミリ) is a 2014 Japanese drama film directed by Momoko Andō. It was released in Japan on November 8, 2014.


0.5 mm, Reception, Critical response
Maggie Lee of Variety called the film "a work of both cool precision and endearing eccentricity".


0.5 mm, Reception, Accolades
At the 36th Yokohama Film Festival, the film was chosen as the 3rd best Japanese film of the year and Momoko Andō won the award for Best Director.


0.5 mm, Reception, Accolades
At the 39th Hochi Film Awards, the film won the award for Best Picture and Masahiko Tsugawa won the award for Best Supporting Actor.


0.5 mm, Reception, Accolades
At the 69th Mainichi Film Awards, Momoko Andō won the award for Best Screenplay and Sakura Ando won the award for Best Actress.


0.8Syooogeki
0.8Syooogeki (0.8秒と衝撃。, Reitenhachi-byō to Shōgeki., "0.8 Seconds and a Crash.") were a two-member Japanese independent band, who debuted in 2009 with the single "Postman John".


0.8Syooogeki, Biography
The band was formed in 2008, with the members J.M. and Tadaomi Tōyama. J.M. was signed to an exclusive contract with Pinky fashion magazine, however the magazine disbanded in late 2009. Tōyama also works as a radio personality on Tokyo FM. Tadaomi Tōyama writes and produces the majority of the band's songs, with J.M. occasionally adding additional arrangement. Both members feature as vocalists, and Tōyama also performs guitar in the band.


0.8Syooogeki, Biography
Throughout 2008, the band entered music competitions, and applied to record company auditions. The band began performing lives in late August 2010, and released their debut single "Postman John" in September. The band's debut album, Zoo & Lennon was released a month later in October 2010, under independent label Evol Records' sub-label, Actwise.


0.8Syooogeki, Biography
The band released an extended play, Ethnofunky Dostoyevsky Comecome Club EP, in August 2010, and at the same time Tōyama began writing a column for alternative magazine Skream!. The band's Ichi-bō Ni-bō San-bō Yon-bō Go-bō Roku-bō, Tōyō no Techno. was released in May 2011, preceded by the single "Machizō, Machiko, Hakai" in March 2011. These two releases were the first to chart on Oricon's singles and albums chart, reaching numbers 72 and 80 respectively.


0.8Syooogeki, Biography
In 2017, they announced that they were to end their activities together after a series of live shows.


0.9
0.9 is the fourth album by French rapper Booba and released on November 24, 2008, on Tallac Records via the major Barclay Records / Universal Music Group.


0.999...
In mathematics, 0.999... (also written as 0.9, in repeating decimal notation) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...). This number is equal to 1. In other words, "0.999..." and "1" represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. (In other systems, 0.999... can have the same meaning, a different definition, or be undefined.)


0.999...
More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all positional numeral system representations regardless of base. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, or disciplines—some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.


0.999..., Elementary proof
There is an elementary proof of the equation 0.999... = 1, which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. The proof, an exercise given by Stillwell (1994, p. 42), is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line there is no room left for placing a number between them and 1.


0.999..., Elementary proof
The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so 0.999... = 1.


0.999..., Elementary proof, Intuitive explanation
If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1.


0.999..., Elementary proof, Intuitive explanation
More precisely, the distance from 0.9 to 1 is 0.1 = 1/10, the distance from 0.99 to 1 is 0.01 = 1/102, and so on. The distance to 1 from the nth point (the one with n 9s after the decimal point) is 1/10n.


0.999..., Elementary proof, Intuitive explanation
Therefore, if 1 were not the smallest number greater than 0.9, 0.99, 0.999, etc., then there would be a point on the number line that lies between 1 and all these points. This point would be at a positive distance from 1 that is less than 1/10n for every integer n.    In the standard number systems (the rational numbers and the real numbers), there is no positive number that is less than 1/10n for all n.   This is (one version of) the Archimedean property, which can be proven to hold in the system of rational numbers. Therefore, 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc., and so 1 = 0.999....


0.999..., Elementary proof, Discussion on completeness
Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc. : a smallest number that is greater than all of the terms of the sequence. One of the axioms of the real number system is the completeness axiom, which states that every bounded sequence has a least upper bound. This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations. The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers in fact has a least upper bound, and that this least upper bound is equal to one.


0.999..., Elementary proof, Rigorous proof
The previous explanation is not a proof, as one cannot define properly the relationship between a number and its representation as a point on the number line. For the accuracy of the proof, the number 0.999...9, with n nines after the decimal point, is denoted 0.(9)n. Thus 0. (9)1 = 0.9, 0. (9)2 = 0.99, 0. (9)3 = 0.999, and so on. As 1/10n = 0.0...01, with n digits after the decimal point, the addition rule for decimal numbers implies


0.999..., Elementary proof, Rigorous proof
One has to show that 1 is the smallest number that is no less than all 0.(9)n. For this, it suffices to prove that, if a number x is not larger than 1 and no less than all 0. (9)n, then x = 1. So let x such that


0.999..., Elementary proof, Rigorous proof
which, using basic arithmetic and the first equality established above, simplifies to


0.999..., Elementary proof, Rigorous proof
This implies that the difference between 1 and x is less than the inverse of any positive integer. Thus this difference must be zero, and, thus x = 1; that is


0.999..., Elementary proof, Rigorous proof
This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This is the Archimedean property, that is verified for rational numbers and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers (infinitesimals) and infinitely large numbers (infinite numbers). When using such systems, notation 0.999... is generally not used, as there is no smallest number that is no less than all 0.(9)n. (This  is implied by the fact that 0. (9)n ≤ x < 1 implies 0. (9)n–1 ≤ 2x – 1 < x < 1).


0.999..., Algebraic arguments
The matter of overly simplified illustrations of the equality is a subject of pedagogical discussion and critique. Byers (2007, p. 39) discusses the argument that, in elementary school, one is taught that 1⁄3=0.333..., so, ignoring all essential subtleties, "multiplying" this identity by 3 gives 1=0.999.... He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation 0.999...."


0.999..., Algebraic arguments
Most undergraduate mathematics majors encountered by Byers feel that while 0.999... is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to 1. Richman (1999) discusses how "this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking", but also suggests that the argument may lead skeptics to question this assumption.


0.999..., Algebraic arguments
Students who did not accept the first argument sometimes accept the second argument, but, in Byers' opinion, still have not resolved the ambiguity, and therefore do not understand the representation for infinite decimals. Peressini & Peressini (2007), presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness. Baldwin & Norton (2012), citing Katz & Katz (2010a), also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.


0.999..., Algebraic arguments
The same argument is also given by Richman (1999), who notes that skeptics may question whether x is cancellable –  that is, whether it makes sense to subtract x from both sides.


0.999..., Analytic proofs
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of one or more digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as b0 and one can neglect negatives, so a decimal expansion has the form


0.999..., Analytic proofs
The fraction part, unlike the integer part, is not limited to finitely many digits. This is a positional notation, so for example the digit 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.


0.999..., Analytic proofs, Infinite series and sequences
A common development of decimal expansions is to define them as sums of infinite series. In general:


0.999..., Analytic proofs, Infinite series and sequences
For 0.999... one can apply the convergence theorem concerning geometric series:


0.999..., Analytic proofs, Infinite series and sequences
Since 0.999... is such a sum with a = 9 and common ratio r = 1⁄10, the theorem makes short work of the question:


0.999..., Analytic proofs, Infinite series and sequences
This proof appears as early as 1770 in Leonhard Euler's Elements of Algebra.


0.999..., Analytic proofs, Infinite series and sequences
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the algebraic proof given above, and as late as 1811, Bonnycastle's textbook An Introduction to Algebra uses such an argument for geometric series to justify the same maneuver on 0.999... A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.


0.999..., Analytic proofs, Infinite series and sequences
A sequence (x0, x1, x2, ...) has a limit x if the distance |x − xn| becomes arbitrarily small as n increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:


0.999..., Analytic proofs, Infinite series and sequences
The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that 1⁄10n → 0 as n → ∞, is often justified by the Archimedean property of the real numbers. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms.


0.999..., Analytic proofs, Infinite series and sequences
For example, the 1846 textbook The University Arithmetic explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 Arithmetic for Schools says, "when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small". Such heuristics are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.


0.999..., Analytic proofs, Nested intervals and least upper bounds
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.


0.999..., Analytic proofs, Nested intervals and least upper bounds
If a real number x is known to lie in the closed interval [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10].


0.999..., Analytic proofs, Nested intervals and least upper bounds
The number x must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits b0, b1, b2, b3, ..., and one writes


0.999..., Analytic proofs, Nested intervals and least upper bounds
In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.


0.999..., Analytic proofs, Nested intervals and least upper bounds
One straightforward choice is the nested intervals theorem, which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their intersection. So b0.b1b2b3... is defined to be the unique number contained within all the intervals [b0, b0 + 1], [b0.b1, b0.b1 + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.


0.999..., Analytic proofs, Nested intervals and least upper bounds
The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or suprema. To directly exploit these objects, one may define b0.b1b2b3... to be the least upper bound of the set of approximants {b0, b0.b1, b0.b1b2, ...}. One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,


0.999..., Analytic proofs, Nested intervals and least upper bounds
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.


0.999..., Proofs from the construction of the real numbers
Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ordering, so that one number can be compared to another and found to be less than, greater than, or equal to another number.


0.999..., Proofs from the construction of the real numbers
The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.


0.999..., Proofs from the construction of the real numbers, Dedekind cuts
In the Dedekind cut approach, each real number x is defined as the infinite set of all rational numbers less than x. In particular, the real number 1 is the set of all rational numbers that are less than 1. Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers r such that r < 0, or r < 0.9, or r < 0.99, or r is less than some other number of the form


0.999..., Proofs from the construction of the real numbers, Dedekind cuts
Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, all elements of 1 are rational numbers that can be written as


0.999..., Proofs from the construction of the real numbers, Dedekind cuts
with b > 0 and b > a. This implies


0.999..., Proofs from the construction of the real numbers, Dedekind cuts
by the definition above, every element of 1 is also an element of 0.999..., and, combined with the proof above that every element of 0.999... is also an element of 1, the sets 0.999... and 1 contain the same rational numbers, and are therefore the same set, that is, 0.999... = 1.


0.999..., Proofs from the construction of the real numbers, Dedekind cuts
The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872. The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in Mathematics Magazine, which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students. Richman notes that taking Dedekind cuts in any dense subset of the rational numbers yields the same results; in particular, he uses decimal fractions, for which the proof is more immediate.


0.999..., Proofs from the construction of the real numbers, Dedekind cuts
He also notes that typically the definitions allow{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [ ...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning." A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1⁄3 has no representation; see "Alternative number systems" below.


0.999..., Proofs from the construction of the real numbers, Cauchy sequences
Another approach is to define a real number as the limit of a Cauchy sequence of rational numbers. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between x and y is defined as the absolute value |x − y|, where the absolute value |z| is defined as the maximum of z and −z, thus never negative. Then the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance. That is, in the sequence (x0, x1, x2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |xm − xn| ≤ δ for all m, n > N. (The distance between terms becomes smaller than any positive rational.)


0.999..., Proofs from the construction of the real numbers, Cauchy sequences
If (xn) and (yn) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (xn − yn) has the limit 0. Truncations of the decimal number b0.b1b2b3... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number. Thus in this formalism the task is to show that the sequence of rational numbers


0.999..., Proofs from the construction of the real numbers, Cauchy sequences
has the limit 0. Considering the nth term of the sequence, for n ∈ ℕ, it must therefore be shown that


0.999..., Proofs from the construction of the real numbers, Cauchy sequences
This limit is plain if one understands the definition of limit. So again 0.999... = 1.


0.999..., Proofs from the construction of the real numbers, Cauchy sequences
The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872. The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation. The book is written specifically to offer a second look at familiar concepts in a contemporary light.


0.999..., Proofs from the construction of the real numbers, Infinite decimal representation
Commonly in secondary schools' mathematics education, the real numbers are constructed by defining a number using an integer followed by a radix point and an infinite sequence written out as a string to represent the fractional part of any given real number. In this construction, the set of any combination of an integer and digits after the decimal point (or radix point in non-base 10 systems) is the set of real numbers.


0.999..., Proofs from the construction of the real numbers, Infinite decimal representation
This construction can be rigorously shown to satisfy all of the real axioms after defining an equivalence relation over the set that defines 1 =eq 0.999... as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version. With this construction of the reals, all proofs of the statement "1 = 0.999..." can be viewed as implicitly assuming the equality when any operations are performed on the real numbers.


0.999..., Proofs from the construction of the real numbers, Dense order
One of the notions that can resolve the issue is the requirement that real numbers are densely ordered. Students are taking for granted that 0.99999...{\displaystyle 0.99999...} is before 1{\displaystyle 1} while this kind of intuitive ordering is better defined as purely lexicographical.


0.999..., Proofs from the construction of the real numbers, Dense order
"... the ordering of the real numbers is recognized as a dense order. However, depending on the context, students can reconcile this property with the existence of numbers just before or after a given number (0.999... is thus often seen as the predecessor of 1)."


0.999..., Proofs from the construction of the real numbers, Dense order
Dense order requires that there is a third real value strictly between 0.99999...{\displaystyle 0.99999...} and 1{\displaystyle 1}, but there is none: we cannot change a single digit in either of the two to obtain such a number. If 0.99999...{\displaystyle 0.99999...} and 1{\displaystyle 1} are to represent real numbers they have to be equal. Dense ordering implies that if there is no new element strictly between two elements of the set, the two elements must be considered equal.


0.999..., Generalizations
The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.


0.999..., Generalizations
Second, a comparable theorem applies in each radix or base. For example, in base 2 (the binary numeral system) 0.111... equals 1, and in base 3 (the ternary numeral system) 0.222... equals 1. In general, any terminating base b expression has a counterpart with repeated trailing digits equal to b − 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.


0.999..., Generalizations
Alternative representations of 1 also occur in non-integer bases. For example, in the golden ratio base, the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for almost all q between 1 and 2, there are uncountably many base-q expansions of 1. On the other hand, there are still uncountably many q (including all natural numbers greater than 1) for which there is only one base-q expansion of 1, other than the trivial 1.000.... This result was first obtained by Paul Erdős, Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the Komornik–Loreti constant q = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the Thue–Morse sequence, which does not repeat.


0.999..., Generalizations
A more far-reaching generalization addresses the most general positional numeral systems. They too have multiple representations, and in some sense the difficulties are even worse. For example:


0.999..., Generalizations, Impossibility of unique representation
That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered lexicographically. Indeed, the following two properties account for the difficulty:


0.999..., Generalizations, Impossibility of unique representation
The first point follows from basic properties of the real numbers: L has a supremum and R has an infimum, which are easily seen to be equal; being a real number it either lies in R or in L, but not both since L and R are supposed to be disjoint. The second point generalizes the 0.999.../1.000... pair obtained for p1 = "0", p2 = "1".


0.999..., Generalizations, Impossibility of unique representation
In fact one need not use the same alphabet for all positions (so that for instance mixed radix systems can be included) or consider the full collection of possible strings; the only important points are that at each position a finite set of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an order preserving map from the collection of strings to an interval of the real numbers cannot be a bijection: either some numbers do not correspond to any string, or some of them correspond to more than one string.


0.999..., Generalizations, Impossibility of unique representation
Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary point-set topology"; it involves viewing sets of positional values as Stone spaces and noticing that their real representations are given by continuous functions.


0.999..., Applications
One application of 0.999... as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. Examples include:


0.999..., Applications
E. Midy proved a general result about such fractions, now called Midy's theorem, in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If it can be proved that if a decimal of the form 0.b1b2b3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem. Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.


0.999..., Applications
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest fractals, the middle-thirds Cantor set:


0.999..., Applications
The nth digit of the representation reflects the position of the point in the nth stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.


0.999..., Applications
Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying his 1891 diagonal argument to decimal expansions, of the uncountability of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines. A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.


0.999..., Skepticism in education
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:


0.999..., Skepticism in education
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999...


0.999..., Skepticism in education
Many of these explanations were found by David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven't specified how many places there are' or 'it is the nearest possible decimal below 1'".


0.999..., Skepticism in education
The elementary argument of multiplying 0.333... = 1⁄3 by 3 can convince reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated. Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a supremum definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division. Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the fractional proof, insist that "logic" supersedes the mathematical calculations.


0.999..., Skepticism in education
Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."


0.999..., Skepticism in education
As part of Ed Dubinsky's APOS theory of mathematical learning, he and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.


0.999..., Cultural phenomenon
With the rise of the Internet, debates about 0.999... have become commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup sci.math, arguing over 0.999... is described as a "popular sport", and it is one of the questions answered in its FAQ. The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.


0.999..., Cultural phenomenon
A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999... via 1⁄3 and limits, saying of misconceptions,


0.999..., Cultural phenomenon
The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart. Nonsense.


0.999..., Cultural phenomenon
A Slate article reports that the concept of 0.999... is "hotly disputed on websites ranging from World of Warcraft message boards to Ayn Rand forums". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of Blizzard Entertainment's Battle.net forums that the company issued a "press release" on April Fools' Day 2004 that it is 1:


0.999..., Cultural phenomenon
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.


0.999..., Cultural phenomenon
Two proofs are then offered, based on limits and multiplication by 10.


0.999..., Cultural phenomenon
Q: How many mathematicians does it take to screw in a lightbulb?


0.999..., In alternative number systems
Although the real numbers form an extremely useful number system, the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999... = 1 is a convention as well:


0.999..., In alternative number systems
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.


0.999..., In alternative number systems
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of the real number system (rather than independent alternatives to it), so 0.999... = 1 continues to hold.