Born in Manchester, Great Britain, Green came to the United States by way of Canada as a child.
On December 6, 1941, Green and 4 others came together to buy a Dallas based company called Geophysical Service Inc., and in 1951, changed its name to Texas Instruments, which is responsible for the production of calculators, pocket transistor radios (which they were the first to produce) and other electronically controlled machines.
Green served as vice president of Texas Instruments from 1941 - 1951, and as president from 1951 - 1955. It is known that Green's philanthropic efforts total over 200 million dollars, and most of this money was given to charitable organizations invoving education and medical affairs. He was given an honorary knighthood in 1991 (at age 91) by Queen Elizabeth II.