Why did President Harry Truman decide to drop the atomic bomb on Japan?

There are
several important reasons. First, Japan had rebuffed all talks of surrender of any kind. They
had turned down the proposed Potsdam Declaration, taking the stance of
mokusatsu, "to kill by silence." It appeared that the Japanese
would defend their islands to the last man, as they had previously at Okinawa and Iwo Jima.
Secondly, Allied estimates concerning an invasion of the islands calculated that losses would
probably be at about one million killed, wounded and/or captured. The Allied command decided
that dropping the bombs would save hundreds of thousands of American, English and other Allied
forces' lives. They also figured that the atomic bombs would show Japan the depth of the Allies'
new power and force a surrender.