Maybe the most well-known anecdote about Gauss is one that happened when he was just a boy. When Gauss was in school at age 10, his teacher, perhaps needing a quiet half hour, gave his class the problem of adding all the integers from 1 to 100. Gauss immediately wrote down the answer on his slate (this would have been in the year 1787). He had quickly noticed that the sum (1 + 2 + 3 + ... + 100) could be rearranged to form the pairs (1 + 100) + (2 + 99) + (3 + 98) + ... + (50 + 51), and that there were 50 pairs each equaling 101. This reduced the problem to the simple product 50 x 101 = 5050.
The fact that Gauss isn't as well-known as scientific giants, such as Archimedes, Newton, Einstein, and Hawking, is curious, since his accomplishments equal (some might say surpass) even those great scientific minds. Consider the following accomplishments:
- In 1796, at the age of 19, Gauss proved the constructability of a heptadecagon (a regular polygon with 17 sides) using only a straghtedge and a compass. Greek philosophers had believed this construction was possible, but a proof escaped them. Note that Guass proved the construction was possible without actually providing the steps necessary. The first explicit construction of a heptadecagon wasn't given until 1800, by Johannes Erchinger. Gauss was so proud of the proof of the construction that he requested the shape be placed on his tombstone. The stonemason refused, stating that the shape would have been indistinguishable from a circle.
- In 1799, in his doctoral dissertation, Gauss provided a proof for what is now known as the Fundamental Theorem of Algebra, which states that every polynomial has at least one root that is a complex number. Gauss would go on to provide four separate proofs of this theorem throughout his career.
- In 1801, Gauss had an Annus Mirabilis of his own. In that year, proved the Fundamental Theorem of Arithmetic (that any positive integer greater than 1 can be represented as the product of prime numbers in exactly one way), systematized the study of number theory with the publication of Disquisitiones Arithmeticae, and proved that every number is the sum of at most three triangle numbers. In the same year, he developed the method of least squares fitting and used it to calculate an accurate orbit of the asteroid Ceres (now classified as a dwarf planet) from only three data points.
- In 1809, Gauss published Theoria Motus, his massive treatise on celestial mechanics, which is still in print today. In writing Theoria Motus, Gauss analyzed astronomical data using normal distributions, which are also referred to as Gaussian distributions after Gauss' definition of their probability density function.
- In 1818, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances. Gauss used the device to aid him in the land survey of the Kingdom of Hanover.
- In 1833, together with Wilhelm Weber, Gauss built the first electromagnetic telegraph used for regular communication, in Göttingen, Germany. This accomplishment is often attributed (by many Americans) to Samuel Morse, who independently developed and patented an electrical telegraph in the U.S. in 1837.
- Gauss lived and worked by the personal motto "pauca sed matura," or "few but ripe." Consequently, he published only a small fraction of his work. Years after his death, publication of his diary and letters confirmed that Gauss was the earliest pioneer of non-Euclidean Geometry, which is normally (and rightly) attributed to Bolyai and Lobachevsky (the two men developed non-Euclidean geometry independently from Gauss, and one another).