Nathan was born Nathan Bachrach in Germany. After immigrating to the United States in 1857, he and his brother Nathan changed their surname to Barnum and moved west to Kansas; Nathan to Newton and Simon to Topeka. The name change may have been to make it easier to do business in the 19th century, as Bachrach is a recognizably Jewish surname. However, even after the name change they were both active in the Jewish community in Kansas.
Bacharach, also Bachrach, is a German surname, mostly Jewish, referring to the medieval town of Bacharach in the Mainz-Bingen district of the Rhineland-Palatinate.