Mathematicians are a clever lot. Just because a concept may not make sense at an intuitive level doesn’t mean that it can’t be used to help understand nature. Take imaginary numbers, for example. If you start with any “real” number and multiply it by itself, you get a positive number. For instance, 2 times 2 equals 4 but so does -2 times -2. That means the square root of 4 equals both 2 and -2. But what would the square root of -4 be? Mathematicians invented imaginary numbers to answer this question, defining the number i to equal the square root of -1 (making the square root of -4 equal to 2i).

       Imaginary numbers can be used to help explain tunnelling, a quantum mechanical process in which, for instance, a particle can spontaneously pass through a barrier. In trying to unify general relativity with quantum  mechanics, physicists used a related idea in which they would measure time with imaginary numbers instead of real numbers. By using this so-called imaginary time, physicists Stephen Hawking and Jim Hartle showed that the universe could have been born without a singularity.