1.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

2.
1770
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As of the start of 1770, the Gregorian calendar was 11 days ahead of the Julian calendar, which remained in localized use until 1923. January 1 – Foundation of Fort George, Bombay laid by Colonel Keating, principal engineer, march 5 – Boston Massacre, Eleven American men are shot, five fatally, by British troops in an event that helps starting the American Revolutionary War five years later. March 26 – First voyage of James Cook, English explorer Captain James Cook, april 1818,00 – First voyage of James Cook, English explorer Captain James Cook and his crew become the first recorded Europeans to encounter the eastern coastline of the Australian continent. April 20 – Georgian king Erekle II defeats the Ottoman forces in the battle of Aspindza, despite being abandoned by an ally, Russian General Totleben. April 29 – First voyage of James Cook, Captain Cook drops anchor on HMS Endeavour in a bay about 16 km south of the present city of Sydney. Because the young botanist on board the ship, Joseph Banks, discovers 30,000 specimens of plant life in the area,1,600 of them unknown to European science, may 7 – Fourteen-year-old Marie Antoinette arrives at the French court. May 16 –Marie Antoinette marries Louis-Auguste, may 20 – A stampede at a celebration of the newly wedded Marie Antoinette and Louis-Auguste in Paris kills more than a hundred people. June 9 – Falklands Crisis, Some 1600 Spanish marines, sent by the Spanish governor of Buenos Aires in five frigates, the small British force present promptly surrenders. June 11 – First voyage of James Cook, HMS Endeavour grounds on the Great Barrier Reef, july 1 – Lexells Comet passes the Earth at a distance of 2184129 km, the closest approach by a comet in recorded history. July 5 – Battle of Chesma and Battle of Larga, The Russian Empire defeats the Ottoman Empire in both battles, august 1 – Russo-Turkish War – Battle of Kagul, Russian commander Pyotr Rumyantsev routs 150,000 Turks. Johann Gottfried Herder meets Johann Wolfgang von Goethe in Strasbourg, Joseph Priestley, British chemist, recommends the use of a rubber to remove pencil marks. The Baron dHolbachs materialist work Le Système de la Nature ou Des Loix du Monde Physique et du Monde Moral is produced in Neuchâtel, london, H. G. Bohn – via Hathi Trust

3.
1770s
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January 1 – Foundation of Fort George, Bombay laid by Colonel Keating, principal engineer, on the site of the former Dongri Fort. March 5 – Boston Massacre, Eleven American men are shot, five fatally, march 26 – First voyage of James Cook, English explorer Captain James Cook and his crew aboard HMS Endeavour complete the circumnavigation of New Zealand. April 1818,00 – First voyage of James Cook, English explorer Captain James Cook, April 20 – Georgian king Erekle II defeats the Ottoman forces in the battle of Aspindza, despite being abandoned by an ally, Russian General Totleben. April 29 – First voyage of James Cook, Captain Cook drops anchor on HMS Endeavour in a bay about 16 km south of the present city of Sydney. Because the young botanist on board the ship, Joseph Banks, discovers 30,000 specimens of plant life in the area,1,600 of them unknown to European science, may 7 – Fourteen-year-old Marie Antoinette arrives at the French court. May 16 –Marie Antoinette marries Louis-Auguste, may 20 – A stampede at a celebration of the newly wedded Marie Antoinette and Louis-Auguste in Paris kills more than a hundred people. June 9 – Falklands Crisis, Some 1600 Spanish marines, sent by the Spanish governor of Buenos Aires in five frigates, the small British force present promptly surrenders. June 11 – First voyage of James Cook, HMS Endeavour grounds on the Great Barrier Reef, july 1 – Lexells Comet passes the Earth at a distance of 2184129 km, the closest approach by a comet in recorded history. July 5 – Battle of Chesma and Battle of Larga, The Russian Empire defeats the Ottoman Empire in both battles, august 1 – Russo-Turkish War – Battle of Kagul, Russian commander Pyotr Rumyantsev routs 150,000 Turks. Johann Gottfried Herder meets Johann Wolfgang von Goethe in Strasbourg, Joseph Priestley, British chemist, recommends the use of a rubber to remove pencil marks. The Baron dHolbachs materialist work Le Système de la Nature ou Des Loix du Monde Physique et du Monde Moral is produced in Neuchâtel, January 9 – Emperor Go-Momozono accedes to the throne of Japan, following his aunts abdication. February 12 – Upon the death of Adolf Frederick, he is succeeded as King of Sweden by his son Gustav III, at the time, however, Gustav is unaware of this, since he is abroad in Paris. The news of his fathers death reaches him about a month later, march – War of the Regulation, North Carolina Governor William Tryon raises a militia to put down the long-running uprising of backcountry militias against North Carolinas colonial government. March 12 – The North Carolina General Assembly establishes Wake County from portions of Cumberland, Johnston, bloomsbury is made the informal county seat. March 15 – Society of Civil Engineers first meets, the worlds oldest engineering society, may 11 – War of the Regulation, North Carolina Governor William Tryon marches his military out of Hillsborough to come to the aid of General Hugh Waddells beleaguered forces. Tryons army stops at Alamance Creek,5 miles away from the Regulator army, may 16 – War of the Regulation, The Battle of Alamance commences after Regulators reject an appeal by Governor Tryon to peacefully disperse. Governor Tryons forces crush the rebellion, causing many Regulators to move to areas outside of North Carolina. May 23 – Battle of Lanckorona, A force of 4,000 Russians under Alexander Suvorov defeat a Polish formation of 1,300 men, july 13 – Russo-Turkish War, Russian forces occupy the Crimea under Prince Vasily Dolgorukov

4.
Seventeen Seventy, Queensland
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Originally known as Round Hill – after the creek it sits on – the name was changed in 1970 to commemorate the bicentenary of Cooks visit. The community of Seventeen Seventy hold the re-enactment of this historic landing each year as part of the 1770 Festival held in May. Although the town is referred to locally as 1770, the name of the town is Seventeen Seventy. The village is a tourist destination on Queenslands Discovery Coast and it is situated on a peninsula, with the Coral Sea and Bustard Bay on three sides. Agnes Water is eight kilometres to the south, Seventeen Seventy can be reached by a sealed road from Bundaberg,120 kilometres to the south, going through Agnes Water. The town sustains a small permanent population, a significant holiday population makes it to the area to take advantage of fishing, Great Barrier Reef trips, the area also has four national parks. These are Deepwater, Eurimbula, Mount Colosseum, and Round Hill and these all offer wilderness camping and hiking. It also offers day cruises and flights to the outer Great Barrier Reef, to nearby Lady Musgrave Island and Pancake Creek, and the historic Bustard Head lighthouse