Answers and Explanations
Roughly speaking, any collection of objects that satisfies these properties is, by definition, a number system.
Note: this is only a rough statement. If you wanted to say things more accurately, you'd have to take into account the fact that
(a, b) times (c, d) = (ac-bd, ad+bc)which means that the pair (0,1), when squared, gives (-1,0) which in this context is considered to be the same as the real number -1 (since a complex number of the form (x,0) is indistinguishable by its arithmetic properties from the real number x, we can consider it to be describing the same underlying concept, much as we consider a fraction of the form x/1 to be describing the same underlying concept as the integer x).
Thus, the complex numbers are a number system in which there is an object whose square is -1. This object is denoted by i, and the pair (a,b) is written as "a + bi". When a=0, such a number is called an "imaginary number".
Complex numbers are extremely important mathematically. They have less direct relevance to real-world situations, since they aren't measurements of single quantities. However, they relate directly to a few real-world situations (such as the strength of an electromagnetic field), and they relate indirectly to many more since many properties of real numbers can be more easily understood in the larger context of complex numbers.
More information on this is available.
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