Perhaps one of the reasons that Carl Friedrich Gauss was able to create so much mathematics in his lifetime was that he got a very early start. Just consider the following two events: when he was 3 years old he corrected his father's arithmetic, and in school, when he was in the third grade, he developed a formula for finding the sum of any arithmetic progression.
Gauss was born in Brunswick, Germany as the only son of poor peasants living in miserable conditions. He exhibited such early genius that his family and neighbors called him the "wonder child". When he was two years old, he gradually got his parents to tell him how to pronounce all the letters of the alphabet. Then, by sounding out combinations of letters, he learned (on his own) to read aloud. He also picked up the meanings of the number symbols and learned to do arithmetical calculations. The story as told by Eric T. Bell:
"One Saturday Gerhardt Gauss was making out the weekly payroll for the laborers under his charge, unaware that his young son was following the proceedings with critical attention. Coming to the end of his long computations, Gerhardt was startled to hear the little boy pipe up, 'Father, the reckoning is wrong, it should be ..." A check of the account showed that the figure named by the young Gauss was correct."
Eric Temple Bell, Men Of Mathematics , Simon Schuster, Inc., New York, 1937
When Gauss was ten years old he was allowed to attend an arithmetic class taught by a man (Buttner) who had a reputation for being cynical and having little respect for the peasant children he was teaching. The teacher had given the class a difficult summation problem in order to keep them busy and so that they might appreciate the "shortcut" formula he was preparing to teach them. Gauss took one look at the problem, invented the shortcut formula on the spot, and immediately wrote down the correct answer. This act was apparently so astonishing that Herr Buttner was transformed into a champion for this young boy. "Out of his own pocket he paid for the best textbook on arithmetic obtainable and presented it to Gauss. The boy flashed through the book." (E. T. Bell). Buttner, realizing that he could teach this young genius no more, recommend him to the Duke of Brunswick, who granted him financial assistance to continue his education into secondary school and finally into the University of Gottingen.
In 1799, Gauss got his doctorate; his dissertation was a brilliant proof of the fundamental theorem of algebra. In 1801 when he was 24, he completed his work Disquisitiones Arithmeticae which became the most significant contribution to number theory up to that time. In that volume he led the way to many new areas of mathematics, including the use of imaginary numbers and his theory of congruent numbers.
Immediately following this abstract work in pure mathematics, Gauss plunged into the realm of applied mathematics -- in particular, astronomy. The newly discovered asteroid Ceres had been observed by many astronomers for 40 days, but none of them could get a correct computation for its orbit. Gauss was able to accurately compute the orbit after only three observations. This he did by inventing the method of least squares.
On another occasion, while interested in the abstract problem of geodesics, (shortest distance between two points on a surface such as the Earth) he invented the heliotrope, a surveying instrument that used the sun's rays to obtain accurate measurements. He also developed the mathematics of error analysis for measurements in general, giving rise to probability analysis and hypothesis testing. The normal probability curve is known as the Gaussian curve. His work with Wilhelm Weber resulted in an advancement of the theory of electromagnetism.
Because of his motto "few but ripe", there were certain ideas that Gauss had done work on but did not publish, since he felt that they were incomplete. Some of these unknown works included complex variables, non-euclidean geometry, and the mathematical foundations of physics. Every one of these ideas were discovered later by other mathematicians. Although he didn't get credit for these particular discoveries, his reward in pursuing such research was the pleasure of finding the truth for its own sake. It is rightfully said that Gauss was probably the greatest mathematician of all time.