Transcription
Take a look at this equation, 1/2 factorial. We're taught that factorial of a number is a simple sequence of multiplication. 3! is 3 * 2 * 1, which is 6. 4! is 4 * 3 * 2 * 1, which equals 24. It's a very discrete operation. You can only calculate the factorial of a whole positive integer. You can't multiply a number by itself half a time. But what if we did?
What happens if you try to evaluate the factorial of exactly 1/2? Well, we have to stop thinking of factorials as basic multiplication and start visualizing them as geometry. If we plot the standard factorials on a graph, we're left with a series of disconnected points curving violently upwards into infinity. 1 is 1, 2 is 2, 3 is 6, 4 is 24, but math hates broken patterns. There must be a continuous equation that threads perfectly through every single one of these points, connecting the dots into a single curve.
That equation exists. It's called the gamma function. It's defined by an infinite integral that extends across all positive space. When we plot this integral, it sweeps perfectly through our integer points. But because the function is continuous, it does not only exist at whole numbers. It gives us a precise mathematical output for every single fraction in between.
If we travel along this curve to exactly 1/2 on the horizontal axis, we see something pretty strange. Evaluating the gamma function at 1/2 requires integrating the area of a standard Gaussian distribution, and that area is intrinsically tied to the geometry of a perfect circle. The factorial of 1/2 evaluates exactly to the square root of pi divided by 2.