Mathematics took a leap forward in sophistication with the introduction of the COMPLEX NUMBERS [I.3 §1.5]. To define these, one suspends one’s disbelief, introduces a new number i, and declares that i^{2} = -1. A typical complex number is of the form *a* + i*b*, and the arithmetic of complex numbers is easy to deduce from the normal rules of arithmetic for real numbers. For example, to calculate the product of 1 + 2i and 2 + i one simply expands some brackets:

(1 + 2i)(2 + i) 2 + 5i + 2i^{2} = 5i,

the last equality following from the fact that i^{2} = -1. One of the great advantages of the complex numbers is that, if complex roots are allowed, every polynomial can be factorized into linear factors: ...

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