Real analysis begins with the natural numbers and develops methods for analyzing sequences and series of numbers. Integers and rational numbers are introduced so that subtraction and division, respectively, always produce an answer. For example, without negative numbers, five minus seven would make no sense. The real number line is also constructed by proving the existence of irrational numbers, which can not be written as the ratio of two integers. The theory of functions, measurement theory and calculus can also be developed once the foundations of the real numbers are in place.
Complex numbers are formed by combining real numbers and imaginary numbers, or the root of a negative number. Complex analysis deals with the same fundamental issues as real analysis, such as measurement theory and integrability, but the presence of imaginary numbers changes many of the conclusions reached. Although imaginary numbers are often thought to sound fanciful, they are used extensively in applied mathematics, from electrical engineering to acoustics.
Although the introduction of new operations required the introduction of new numbers, from the integers to the complex numbers, complex numbers have been proven to be algebraically closed. This means that any algebraic equation that can be expressed with complex numbers can also be solved using complex equations. This does not mean that complex numbers are the only algebraically closed system, but the inclusion of new types of numbers is not necessary.
In 2000 the Clay Mathematics Institute announced that it would pay 1 million dollars to anyone who could solve one of seven problems that were deemed to be particularly difficult and important. One of these Millennium Prize problems, the Riemann hypothesis, is closely related to complex analysis. Although on its face the problem has to do with the distribution of prime numbers, attempting to analyze the Riemann Zeta function requires a deep understanding of complex analysis. The first Millennium Prize was awarded in 2010 for the solution of the Poincare Conjecture.