Who cares about complex numbers??  Summary and Q&A
TL;DR
Complex numbers are essential in solving equations and formulas that involve both real and imaginary components, leading to a deeper understanding of mathematics.
Questions & Answers
Q: Why do complex numbers disappear in certain equations?
Complex numbers appear in equations, such as quadratic and cubic formulas, to ensure the correct solutions are obtained. However, the imaginary components often disappear in the final result, leaving only real numbers.
Q: What is the significance of irrational numbers in the Fibonacci sequence?
In the Fibonacci sequence, which is defined recursively, the presence of the square root of 5 (an irrational number) in the formula is necessary to generate the correct sequence, even though the resulting terms are all real numbers.
Q: Why were Italian mathematicians interested in complex numbers?
Italian mathematicians, such as Cardano, embraced complex numbers to gain an advantage in mathematical duels, where the ability to solve complex equations was highly esteemed. They were driven to discover formulas that would give them an edge over their rivals.
Q: Why are complex numbers considered "imaginary"?
Complex numbers are termed "imaginary" because they include an imaginary component, represented by the square root of a negative number, which is not part of the natural numbers, integers, or real numbers. These numbers were initially viewed as "fictitious" or nonreal elements in mathematical formulas.
Summary & Key Takeaways

Complex numbers play a crucial role in solving quadratic and cubic equations, allowing us to find both real and irrational roots.

The presence of imaginary numbers in certain formulas, even when the end result is purely real, highlights their significance in mathematics.

Italian mathematicians in the 16th century, like Cardano, were driven to explore complex numbers in order to gain an advantage in mathematical duels and secure prestigious positions.