TITLE: Which mathematical ideas have done most to change history?
QUESTION [68 upvotes]: I'm planning a course for the general public with the general theme of "Mathematical ideas that have changed history" and I would welcome people's opinions on this topic.  What do you think have been the most influential mathematical ideas in terms of what has influenced science/history or changed the way humans think, and why?
I won't expect my audience to have any mathematical background other than high-school.
My thoughts so far are: non-Euclidean geometry, Cantor's ideas on uncountability, undecidability, chaos theory and fractals, the invention of new number systems (i.e. negative numbers, zero, irrational, imaginary numbers), calculus, graphs and networks, probability theory, Bayesian statistics.
My apologies if this has already been discussed in another post.

REPLY [7 votes]: Probably it can be viewed as a variant on already posted answers (cryptography etc.), but the study of permutation groups and its application in cracking the Enigma code literally changed history (namely, the outcome of World War II). Here is an article by Marian Rejewski, one of the people involved in the code-cracking, explaining what was done and how:
http://www.impan.pl/Great/Rejewski/article.html
Rejewski and his achievements were also mentioned in answers to the following MO questions:
Real-world applications of mathematics, by arxiv subject area?
Notable mathematics during World War II