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    Imaginary and irrational numbers

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    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 seem philosophical. We have to know what "exist" means. In general, I would say that all numbers are man made and are defined for our convenience. They are just concepts rather than physical entities although some of them can be useful to describe things in real world.

    Would you say that imaginary ...

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    This provides answers to the question of the existence of imaginary and irrational numbers.

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