Integers
Integers are whole numbers, including positive numbers, negative numbers, and zero.
...-3, -2, -1, 0, 1, 2, 3...
Integers have unique properties that have always interested mathematicians. For example, anyone who adds, subtracts, or multiplies two integers will always have an integer as the result. However, dividing two integers can result in a non-integer.
Throughout history, integers have been the building blocks for many advances in mathematics. The group-or set-of positive numbers, negative numbers and zero can also be called rational integers. Many think of positive integers as the first system humans developed for counting possessions. Ancient people probably used their fingers or small stones to count whole numbers. Positive integers are also called the counting numbers.
Records 4,000 years old show that Babylonians looked at the properties of positive integers (now thought of as 'pure' arithmetic). Diophantus of Alexandria, Greece, revised the system of algebra first developed by the Babylonians. Under the Diophantine method of analysis that evolved around A.D. 250, equations were solved with integers. The problems could involve single equations or systems of equations containing two or more unknowns, but the solutions were always whole numbers.
Many advances in the European study of mathematics began with the study of integers. Carl Friedrich Gauss (1777-1855), one of the most influential scholars in mathematics, enrolled at Göttingen University in Germany in 1795. While Gauss was an undergraduate, he spent three years writing a classic book on the mathematics of integers, Arithmetical Disquisitions. Albert Einstein later praised the importance of Gauss's many contributions to the field of mathematics and showed how Gauss's work helped Einstein develop the theory of relativity.
Other European mathematicians were intrigued with the challenge of defining integers. This was a problem that had to be resolved as part of the process of developing definitions of rational arithmetic. The mathematician Leopold Kronecker (1823-1891) tried to devise a system of math that depended on positive whole numbers and was so convinced of the key role of integers that he declared, only partly in jest, "God made the integers, all the rest is the work of man."
The importance of mathematics to other fields of study was aptly illustrated by the research of scientists John Dalton (1766-1844) and Dmitry Ivanovitch Mendeleev (1848-1907). Dalton built his work on the study of the atom in ancient Greece and noted that the atomic weights for all of the compounds he measured were ratios of simple integers, such as 4:1. The ratios were always whole numbers, never fractions. This observation led Dalton to conclude that elements could only combine in certain ways to form compounds. Building on Dalton's work, Mendeleev observed the relationships of valences and chemical elements and helped to develop the periodic law for the properties of chemical elements. The curiosity about whole numbers and measurements of elements and compounds led directly to the periodic table used by contemporary chemists. While the discoveries of Dalton and Mendeleev were not mathematically complex, they were of great scientific significance. Mathematics became more than a tool for analyzing numbers and was used in new ways to advance physics and chemistry.
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