Why did Harry Truman order the atomic bomb?

Answer 1

He was faced with the choice of using the Bomb or of ordering a conventional invasion of mainland Japan, which The Pentagon estimated would cost up to a million Allied lives. Because the number of casualties in the Far East had already been horrifically high, and because the Japanese military were known to be extremely cruel to their captors, he regarded the conventional option as unacceptable.

Answer 2

Truman was dealing with significant Russian aggression after the end of World War II. Cold War tensions were building and an arms race had ensued between Russia and the US the only logical thing Truman could think of was to order the building of the H-bomb.

Answer 3

He thought dropping the bomb would save American lives by making the Japanese surrender.