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# The Imaginary Number System

Imaginary numbers are just as real as real numbers. Presumably one would think that this number system does not exist at all. To the early mathematicians, it was puzzling to think that there existed less than zero of anything. Therefore, they could not fathom their solutions to equations when it came out either negative or a negative square root. These answers were seen as nonexistent and useless because how could one have negative apples or a negative amount of loaves of bread. However, in today's society, imaginary numbers are accepted as being real, not just a fictitious number. .
To the ancient Greeks, numbers were only thought of to be rational. "The ancient Greeks once believed that all numbers were rational numbers; that is, that every number could be expressed as the ratio of two integers" (Engel, 1997). Therefore, if a solution came out to be irrational or negative, it was deemed useless. However, as time went on, these numbers that were thought to not exist were given a name, that name being negative. Furthermore, for many centuries there were quadratic equations that remained unsolved. Equations such as x2 = -1 did not have solutions. "These numbers were based upon what seemed an extremely mysterious idea, that there must exist an entity which, when squared, would produce the number -1" (Bunch, 2000, p. 303). The squares of positive and negative numbers always come out positive; therefore there was seemingly no solution to quadratics that resulted in a negative integer. As Engel continues on to state, "Solutions for equations like these can be found, however, if we decide to invent a completely new number whose square is -1; of course, it is not a number that we have seen before. We name this number "i". The square of -i is also -1" (Engel, 1997). Mathematically, the definition of an imaginary number is a number with a negative square. Moreover, -1 has been named i and any multiple of -1 is called i.

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