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Number Systems: Irrational and Imaginary Number Emphasis

This number system is an extension of the real number system. Any polynomial equation has n complex roots, in general, in the complex number system. For example, the equation x2 + 1 = 0 has no real roots, but it has two complex roots given by +I and -I, where I is the "imaginary unit" given by the "square root" of -1. The complex number system is an extension of the real number system, just as the real number system is an extension of the rational numbers (numbers of the form of a ratio of two integers). Would you say that imaginary numbers really "exist"? What about, for that matter, the irrational numbers, which complete the real number system? (The ancient Greeks were surprised to find out that the square root of two is an irrational number, and that not all numbers can be expressed as the ratio of two integers.)

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These questions are to get you thinking about different types of number systems. There is actually nothing to solve, but is aimed at opening your mind to the idea about the structure of various number systems. I have made a schematic so you can visually see the different types of number systems. It is the statements listed above in a graphic format.

Absolutely, imaginary numbers exist! The name originates from the idea that when these imaginary numbers were first introduced, people would "imagine" what it would be like if there was a number equivalent to the square root of -1. This name stayed, but it doesn't mean that these numbers don't exist. Imaginary numbers show up all over the place in actual ...

Solution Summary

This solution provides an explanation of the different types of number systems. A diagram depicting the subsets that comprise each number system along with some examples is provided. The imaginary number system is defined with actual applications given to verify existence. The complex number system is defined and supplemented with a description of how to graphically interpret this number system. The existence of irrational numbers is explained and its existence proven.