Why do physicists introduce fictitious forces when working within a non-inertial frame?

Answer

To make Newton's second law (F=ma) appear to work within that accelerating system

Fictitious or inertial forces must be introduced in accelerating (non-inertial) frames so that Newton's second law can mathematically describe observed motion within that system.

Why do physicists introduce fictitious forces when working within a non-inertial frame?

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