Why study mathematics?
For the things of this world cannot be made known without a knowledge of mathematics.— Roger Bacon (c. 1214-1294), English scholastic philosopher, scientist, and Franciscan, from Opus Majus part 4 Distinctia Prima cap 1, 1267
Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure.— Martin Gardner, from Gardner on Gardner: JPBM Communications Award Presentation. Focus: The Newsletter of the Mathematical Association of America, v. 14, no. 6, December 1994
In many disciplines, the history of the twentieth century tells of the introduction of more and more mathematical techniques. Mathematics has been established as a universal ingredient in the understanding of the world, and is the language used in conveying this understanding. Now that we have entered the twenty-first century, higher mathematics is not just a tool for physical scientists and engineers, but it is also of crucial importance in business, economics, and even the social sciences. Many mathematicians still delight in the aesthetic appeal of their subject; however, it is ultimately the application of mathematics that makes it a critical element in modern civilization.
Why declare a math major?
Undergraduate training in mathematics is an important step along many career paths. It is essential for those who intend to continue toward a graduate degree in Mathematics, Applied Mathematics, Statistics, or Computer Science. Admissions committees for graduate studies in Astronomy, Physics, Engineering, Economics, or Finance often prefer students with undergraduate mathematics majors (together with the proper minor) over students who have majored in their own subject. Some University of Arizona mathematics graduates have also used their degree as the first step toward professional degrees in law, medicine, and even divinity. 
The mathematics major is not just a path toward graduate education. Mathematical training is training in general problem solving. Many employers recognize this fact; they also recognize that a Bachelors degree in mathematics is proof of an ability and willingness to work hard. Many Mathematics majors have found success by completing more than one major. Double majors in Mathematics and Science, Mathematics and Business, or even Mathematics and a Social Science, prepare graduates with enhanced ability to compete well in the job market.
Mathematics majors are advised to develop both computing skills and communication skills during their time as a student. Computing skills are increasingly important in today's society. Experience using graphing calculators and standard mathematical software packages, programming experience, and experience using the Internet are extremely useful. And in virtually any type of position, the need for good communication skills is paramount.
More reasons to become a math major
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.— Paul A.M. Dirac (1902-1984), English physicist, professor of mathematics at Cambridge, worked in quantum mechanics, from P. J. Davis and R. Hersh, The Mathematical Experience
The Advantage is that mathematics is a field in which one's blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one's best moments that count and not one's worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician's reputation.— Norbert Wiener (1894-1964), mathematician, from Ex-Prodigy: My Childhood and Youth
Let us not forget one of the main reasons for becoming a mathematics major. It is fun! The mathematics program affords students plenty of time to link their mathematical interests to other intellectual endeavors. Students who combine mathematics, computer science, and another area will find themselves with many options when they graduate with an undergraduate degree.
The Department of Mathematics created the Math Center to ensure that our mathematics majors have a home base to turn to during their undergraduate studies. One of our goals is to integrate the undergraduate mathematics major into the intellectual life of this university and this nation. The working mathematician is involved in three areas: the creation of new mathematics, the application of mathematics to solve real-world problems, and the teaching of mathematics. Many of our undergraduate mathematics majors are able to perform these same activities during their undergraduate years through various research and teaching/tutoring opportunities and summer internship opportunities.