To understand better the influence that Johann Carl Friedrich Gauss, Mathematician and Physicist, had on modern mathematics, you might need to understand the man himself. He was born the only child to a poor family in Brunswick, Germany in 1777. Gauss was often portrayed as a child prodigy when it came to math. The stories range from correcting his father on a math problem at the age of three, to being able to figure out large sums quickly in his head in Primary School. While in school he was awarded the chance to study at the Collegium Carolinum from 1792 to 1795 by the Duke of Brunswick. He later attended college at the University of Gottingen from 1795 to 1798. He continued to amaze those around him with his mathematical genius. His love and understanding of math set him up for the path that his life would take.
While still in college in 1796 at the age of 19, he proved that one could use a straight-edge and compass in order to construct certain types of polygons, using the heptadecagon in his example. He showed that you could use basic math and square roots to express trigonometric functions. This was an important discovery to mathematicians everywhere in that the actual construction of such objects had often been wondered about.
In that same year he also invented the method of modular arithmetic, which is used most notably in the 24 hour clock. This method gets it's name from the fact that numbers will "modulo" or "wrap around" once they reach a certain value. He was also the first one to show proof of the number theory law of quadratic reciprocity in that same year. Quadratic reciprocity is used to determine if certain quadratic equations can be solved. The importance of this is that it gives the mathematician the ability to know if they can even be solved without first having to solve them. Think of it like this, would you want to spend all your time trying to fit a square peg in a round hole; or would you rather know right off the bat that they don't go together? The fact that it can save time in knowing which quadratic equations go together can give mathematicians the opportunity to avoid using unsolvable equations when trying to reach a solution in complex problems.
There are many more things that Johann Carl Friedrich Gauss contributed to the world of mathematics. For instance the prime number theorem and the fundamental theorem of algebra. Throughout his lifetime he used math to solve problems in various fields from physics and astronomy to optics, geometry, and magnetism. He has been awarded several honors; in 1804 he was elected to the Fellows of the Royal Society of London; in 1820 he was elected to the Fellows of the Royal Society of Edinburgh; in 1838 he was awarded the Copley Medal, this medal is the highest award given by the Royal Society of London. Along with his numerous awards and honors, he also has several lunar features named after him. From the Crater Gauss on the moon to the asteroid 1001 Gaussia, which was discovered in 1923. The first expedition ship from Germany to explore Antarctica was named the Gauss, and that same ship discovered an extinct volcano which they named Gaussberg.
His proven theories are still in use today and his influence is everywhere in the world of mathematics and beyond. If you would like to do some more research on this noted genius, please visit the sites that I used for references. These are as follows: