Why were complex numbers introduced in physics?

[axmls]

To me, it helps that one of the first things one learns in complex analysis is that a complex number is nothing more than an ordered pair (which can also be represented by a + bi). They're just ordered pairs that we know how to do operations with. Or from a physics viewpoint, it's almost like a unit vector that points upwards (with the exception of that little $i^2 = -1$ thing) if we assume that a number not multiplied by i points horizontally.

 

[UncertaintyAjay]

Complex numbers weren't invented with any specific application in mind. They were introduced to solve certain equations which had no real solutions. The fact that they are useful in physics was , I reckon, a coincidence. They weren't designed to be that way.

 

[anorlunda]

I recommend a very entertaining book An Imaginary Tale: The Story of i [the square root of minus one] By Paul J. Nahin

The science and math communities resisted the need for complex numbers for centuries. They had the same question as the OP. The book tells the story of how they were eventually persuaded.

Another book, QED: The Strange Theory of Light and Matter By Richard Feynman explains quantum mechanics using very clever graphics instead of complex numbers. Indeed instead of any number or equations. Studying Feynman's graphics makes it obvious that quantum mechanics can never be correctly described by just real arithmetic. Both the magnitude and phase are mandatory to get the correct answers.