History of mathematics

:See Timeline of mathematics for a timeline of events in mathematics. See list of mathematicians for a list of biographies of mathematicians.

Complex numbers

Mathematics did not start with the concept of the complex numbers. It took many years and much discussion to get this far. Roughly speaking over time mathematicians have broadened the definition of number. Opinions differ as to how to treat the complex numbers philosophically.

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Many people argued that it was just an imaginary construct to solve the cubic and shouldn't be considered 'real'. This is the origin of the terms imaginary and real. However it was found that a whole new beautiful world of complex numbers opened up if you did allow them. To represent a solution to the equation shown above (i.e., X*X+1=0) mathematicians chose the letter i. Even with all of these extensions of the naturals we are still not finished.

Related Topics:
Cubic - Complex number

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In order to construct the complex numbers we need only one more assumption: Any set of real numbers with an upper bound has a least upper bound. This fills in the real line with all of the irrational numbers that cannot be derived merely from the equations above. It is worth noting that this is an entirely different type of extension. This is because of the cardinality of complex numbers is greater than that of the rationals. Once this is done all polynomial equations can be solved (although this can be done in smaller fields than the complex numbers).

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Mathematicians today rarely view the development of the complex numbers in this way (the preferred teaching method does not emphasize this stepwise development) but it demonstrates the tension in mathematics between the rigorous and the creative which is the main power behind much of modern mathematics.

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