In <a href="http://carma.newcastle.edu.au/jon/digits.pdf">100 Digits Challenge: an Extended Review (March 2005)</a> Jon Borwein lists "the 20th century's Top Ten" algorithms:

(1) 1946: *The Metropolis Algorithm for Monte Carlo*. Through the
use of random processes, this algorithm offers an effcient way to stumble
toward answers to problems that are too complicated to solve exactly.

(2) 1947: *Simplex Method for Linear Programming*. An elegant solution
to a common problem in planning and decision-making.

(3) 1950: *Krylov Subspace Iteration Method*. A technique for rapidly
solving the linear equations that abound in scientific computation.

(4) 1951: *The Decompositional Approach to Matrix Computations*. A
suite of techniques for numerical linear algebra.

(5) 1957: *The Fortran Optimizing Compiler*. Turns high-level code into
efficient computer-readable code.

(6) 1959: *QR Algorithm for Computing Eigenvalues*. Another crucial matrix
operation made swift and practical.

(7) 1962: *Quicksort Algorithms for Sorting*. For the efficient handling of
large databases.

(8) 1965: *Fast Fourier Transform*. Perhaps the most ubiquitous algorithm
in use today, it breaks down waveforms (like sound) into periodic components.

(9) 1977: *Integer Relation Detection*. A fast method for spotting simple
equations satisfied by collections of seemingly unrelated numbers.

(10) 1987: *Fast Multipole Method*. A breakthrough in dealing with the complexity of $n$-body calculations, applied in problems ranging from celestial
mechanics to protein folding.

Please refer to the original paper for many examples illustrating Jon's point.