Hopper was born in New York City. She was the eldest of three children. Her parents, Walter Fletcher Murray and Mary Campbell Van Horne, were of Scottish and Dutch descent, and attended West End Collegiate Church.[10] Her great-grandfather, Alexander Wilson Russell, an admiral in the US Navy, fought in the Battle of Mobile Bay during the Civil War.
Grace was very curious as a child; this was a lifelong trait. At the age of seven, she decided to determine how an alarm clock worked and dismantled seven alarm clocks before her mother realized what she was doing (she was then limited to one clock).[11] For her preparatory school education, she attended the Hartridge School in Plainfield, New Jersey. Hopper was initially rejected for early admission to Vassar College at age 16 (her test scores in Latin were too low), but she was admitted the following year. She graduated Phi Beta Kappa from Vassar in 1928 with a bachelor's degree in mathematics and physics and earned her master's degree at Yale University in 1930.
In 1934, she earned a Ph.D. in mathematics from Yale[12] under the direction of Øystein Ore.[13][14] Her dissertation, New Types of Irreducibility Criteria, was published that same year.[15] Hopper began teaching mathematics at Vassar in 1931, and was promoted to associate professor in 1941.[16]
She was married to New York University professor Vincent Foster Hopper (1906–76) from 1930 until their divorce in 1945.[13][17] She did not marry again, but chose to retain his surname