March 2026
Intermediate
512 pages
14h 19m
English
Content preview from Mathematics in Cybersecurity
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8Introduction to Complex Numbers
8.1 Introduction to Complex Numbers
Real numbers (the vast collection of natural numbers, whole numbers, integers, rational numbers, and irrational numbers) have their limitations when it comes to the solutions to some types of equations. As an illustration, consider:
If we attempt to solve this equation, we find it is impossible to solve by the method of factoring, and so we turn to the quadratic formula. In this case, a = 1, b = 3, c = 11, so the formula yields:
Since the radicand of the square root is negative, it is impossible to simplify this result further by using any of the methods seen thus far. Nevertheless, it becomes important in applications for us to be able to establish a solution to the equation, and for this reason, a new set of values, the complex numbers, is introduced.
In other words, complex numbers are motivated by the need to simplify square (and higher even-index) roots of negative numbers. In mathematics, it is often necessary to introduce a new concept to resolve a particular problem; doing so spawns consequences unforeseeable when the definition is initially made, which (in this case) leads to a rich field of mathematics ...
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