What is the name given to that large piece of land that the President and America bought in 1803?

The name that is given to that large piece of land that the President and America bought in 1803 is the Louisiana Purchase.

Explanation:

The Louisiana Purchase encompassed 530,000,000 acres of territory in North America that the United States purchased from France in 1803 for $15 million.

The Louisiana Purchase was a land purchase made by United States president, Thomas Jefferson, in 1803. He bought the Louisiana territory from France, which was being led by Napoleon Bonaparte at the time, for 15,000,000 USD. First, the men sent to France were allowed to spend up to 10 million USD in order to buy New Orleans and, if possible, the west bank of the Mississippi River. But then the French government said that for 5 million more dollars they would sell all of the Louisiana territory. Thomas Jefferson approved the deal and used his constitutional power to sign treaties to buy the land.

Explanation:

classical republicanism, also known as civic humanism, is a form of republicanism  developed in the renaissance inspired by the governmental forms and writings of  classical  antiquity, especially such  classical  writers as aristotle, polybius, and cicero.