In 1927, German physicist Werner Heisenberg was uncertain about something. Namely, the exact position and velocity of an electron. In one of the most fantastic discoveries of the 20th century, Heisenberg showed that certain pairs of physical properties cannot be simultaneously measured with arbitrary precision. In this talk, we will see a proof of Heisenberg's uncertainty principle for one spatial dimension via elementary Fourier analysis. We will then discuss the uncertainty principle in the context of quantum mechanics, revealing one of the many splendid connections between mathematics and physics.