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It is known that the number of imaginary components that could be added to the real numbers to define new arithmetics with operations of addition, multiplication, and division can be only 1, 3, and 7. The first two cases relate to the complex and quaternion numbers, respectively. In this chapter, we consider the arithmetic of numbers with a seven-component imaginary part, which generalizes the quaternion numbers and allows for adding, multiplying, and dividing the numbers. The properties, examples, and functions on such numbers are described. These numbers are called octonions or Cayley numbers and were discovered independently by the Irish mathematician John Graves (1806–1870) and the English mathematician Arthur Cayley (1821–1895).
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