Carl Friedrich Gauss is a German mathematician and physicist who has made significant contributions to many fields of mathematics and natural sciences. Gauss is sometimes referred to as "the greatest mathematician" and "the greatest mathematician since ancient times". He has had an extraordinary influence on many fields of Mathematics and science and is one of the most influential mathematicians in history.

Early years

Carl Friedrich Gauss was born to a poor working class parents. His mother was illiterate and never recorded his date of birth, she only remembered that he was born on a Wednesday, eight days before the feast of Ascension (which is 39 days later of Easter). Gauss later solved this mystery about his date of birth in the context of finding the date of Easter and derived methods for calculating the date in years past and future. He was christened and confirmed in a church near the school he attended as a child. Gauss was a child prodigy. Wolfgang Sartorius von Waltershausen wrote in his memorial about Gauss that, when he was just three years old, he corrected a mathematical mistake made by his father; and that at the age of seven he was solving a series arithmetic problem faster than anyone in his class of 100 students. Gauss Summation A legend suggests that Gauss came up with a new method of summing sequences at a very young age. The legend says that his math teacher asked the class to add the numbers 1 to 100. In other words, the teacher wanted them to add 1 + 2 + 3 + 4 + 5... all the way up to 100! The teacher assumed that this would take the students a very long time. Think about how long it would take you to add up all the numbers from 1 to 100 one by one. However, Gauss answered 5050 almost immediately. This story may not be entirely true. But, it reminds us that the youngest students are sometimes the ones to discover new mathematical patterns. Now, let's think about the pattern that Gauss used to solve this problem quickly. The trick that Gauss used to solve this problem is that it doesn't matter what order we add the numbers. No matter what order we follow, we will get the same result. For example: 2 + 3 has the same answer as 3 + 2. We can reorder the numbers from 1 to 100 in a clever way. This can help us add them more quickly. Here is a simple example which will show you how this grouping strategy works. Say you wanted to add the numbers from 1 to 10. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ? Now, you might have noticed something strange. Each of these pairs adds up to 11. So, we can think about our problem like this (1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6) = ? (11) + (11) + (11) + (11) + (11) = ? Since we have 5 pairs, our answer is 11 + 11 + 11 + 11 + 11 = 11 x 5 = 55 We can write this as: $$ \begin{split} \text{Total Sum} &= {(\text{Number of Pairs}) \times (\text{Sum of each pair}) \over 2} \\ \text{Total Sum} &= {(10) \times (11) \over 2} \\ \text{Total Sum} &= 55 \\ \end{split}$$ Gauss's intellectual abilities caught the attention of the Duke of Braunschweig, who sent him to the Collegium Carolinum (now the Braunschweig University of technology ), which he attended from 1792 to 1795, and to the University of Gottingen from 1795 to 1798. During his studies Gauss independently rediscovered several important theorems. Gauss's first significant discovery, in 1792, was that a regular polygon of 17 sides can be constructed by ruler and compass alone. Its significance lies not in the result but in the proof, which rested on a profound analysis of the factorization of polynomial equations and opened the door to later ideas of Galois theory. His doctoral thesis of 1797 gave a proof of the fundamental theorem of algebra: every polynomial equation with real or complex coefficients has as many roots (solutions) as its degree (the highest power of the variable). Gauss's proof, though not wholly convincing, was remarkable for its critique of earlier attempts. Gauss later gave three more proofs of this major result, the last on the 50th anniversary of the first, which shows the importance he attached to the topic. Gauss's recognition as a truly remarkable talent, though, resulted from two major publications in 1801. Foremost was his publication of the first systematic textbook on algebraic number theory, Disquisitiones Arithmeticae. This book begins with the first account of modular arithmetic, gives a thorough account of the solutions of quadratic polynomials in two variables in integers, and ends with the theory of factorization mentioned above. This choice of topics and its natural generalizations set the agenda in number theory for much of the 19th century, and Gauss's continuing interest in the subject spurred much research, especially in German universities. The second publication was his rediscovery of the asteroid Ceres. Its original discovery, by the Italian astronomer Giuseppe Piazzi in 1800, had caused a sensation, but it vanished behind the Sun before enough observations could be taken to calculate its orbit with sufficient accuracy to know where it would reappear. Many astronomers competed for the honor of finding it again, but Gauss won. His success rested on a novel method for dealing with errors in observations, today called the method of least squares. Thereafter Gauss worked for many years as an astronomer and published a major work on the computation of orbits the numerical side of such work was much less onerous for him than for most people. As an intensely loyal subject of the duke of Brunswick and, after 1807 when he returned to Gottingen as an astronomer, of the duke of Hanover, Gauss felt that the work was socially valuable.