# Carl Friedrich Gauss

**Episode #2 of the course “Greatest Mathematicians”**

German-born Johann Carl Friedrich Gauss created one of the most well-known algebraic formulas at the age of 10. As the story goes, Gauss’ teacher told the class to find the sum of adding all the numbers 1-100. Two minutes later, Gauss turned in his paper. Instead of adding, he found a pattern: 1+100=101, 2+99=101, 3+98=101, and so on and so forth. Therefore, all he needed to do was figure out how many pairs there were (50) and multiply that number by 101.

Gauss was born in Brunswick, Germany in 1777. Sometimes referred to as the *Princeps mathematicorum*, Latin for “the Prince of Mathematicians,” Gauss is considered to be the greatest German mathematician of the 19th century. His discoveries and writings left a lasting mark in the areas of number theory, astronomy, geodesy, and physics, particularly the study of electromagnetism. He was 77 years old when he died in Göttingen, Germany in 1855.

Although he had already been dubbed a child prodigy probably long before, Gauss’ major breakthrough came when he was 19. In 1796, he showed that any regular polygon with a number of sides that is also a Fermat prime could be constructed by compass and straightedge. This helped solve construction problems, a major field of mathematics and one that had plagued builders since the days of the ancient Greeks. In that same year, he also became the first to prove the quadratic reciprocity law, which allows mathematicians to determine whether a quadratic equation in modular arithmetic is solvable or not.

Gauss’ most enduring contribution came in the form of his 1799 doctorate in absentia titled “A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree.” In it, Gauss proved that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This is known as the fundamental theorem of algebra. Renowned mathematicians before him had produced false proofs that were now proven to be inaccurate.

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