Key point: The invasion would have been so violent and costly that many on all both sides would have died.
One of the most controversial decisions in history was President Harry Truman's decision to drop atomic bombs on Hiroshima and Nagasaki in August 1945.
Some argue that Truman was haunted by estimates that Operation Downfall -- the proposed invasion of Japan in 1945 -- would cost a million American casualties. Others say that Japan was starving and exhausted, the casualty estimates were exaggerated, and that Truman had ulterior motives for dropping the Bomb, namely intimidating the Soviet Union with a display of America's technological might.
Like any counterfactual, there can never be any definitive proof of the outcome of a hypothetical invasion of Japan. But we can make a few reasonable assumptions.
First, we can take a good guess what an amphibious assault on Japan in November or December 1945 would have been like. Fresh in American minds would have been Operation Iceberg, the April 1945 assault on the island of Okinawa, 400 miles from the Japanese mainland and politically a part of Japan proper. Rather than suicidal banzai charges in the face of American firepower, the Japanese changed tactics: they retreated to fortified lines and caves in the Okinawan interior, where they fought for three months and almost to the last man. Meanwhile, wave after wave of kamikaze aircraft dove on U.S. and British Commonwealth ships (even the super-battleship Yamato made a suicide sortie). The result was more than 50,000 U.S. casualties, a quarter-million Japanese military and civilian dead, and more than 400 Allied ships sunk or damaged.