In fact, my boyfriend told me a story he had read about the history of the 1st Division's patch that got me thinking about patches.
According to the 1st Infantry Division Society there was a tale that said a general with the 1st Division thought they needed a shoulder insignia and a lieutenant brought him a prototype using a piece of cloth from a captured German uniform on which he placed a red "1". Kind of an interesting history, if it's true. Unfortunately it may be more of a telltale then the truth, as many other sources have said the 1st Division received their big red "1" from being the first American division to land in France during World War I.
Either way it is an interesting thing to look into since uniform shoulder patches really did not become a way for identification until the very late stages of World War I. (To see some of the first shoulder patches from American uniforms click here!)