Computers cannot represent every real number perfectly. They use a finite number of bits (like IEEE 754 double precision), which introduces tiny rounding errors ( ) in almost every operation. Conditioning (Problem Sensitivity): condition number of a matrix, denoted as
In the real world, data is noisy. Suppose you are trying to fit a line through a scatter plot of stock prices. There is likely no single line that passes through every point. You have an "overdetermined" system—more equations than unknowns. In linear algebra terms, there is no exact solution $x$ to $Ax = b$.