Complex Analysis/Complex numbers

The field of the complex numbers Historically, it was observed that the equation has no solution for a real (since for ). Since mathematicians wanted to solve this equation, they just defined a number , called the imaginary unit, such that . Of course, there exists no such number. But if we write a two-tuple with as and calculate with these two-tuples using the calculation rule , that is, and (where we already wrote a two-tuple as , which we will continue to do throughout this book), then the set of all two-tuples with this addition and multiplication forms a field. Indeed, the required axioms for a commutative ring are easy to check, and an inverse of , and not both zero, is given by , as can be checked by a direct computation. 0 100% #FFFFFF Definition 1.1: A complex number is a two-tuple of real numbers , written , where is called the real part of the number and is called the imaginary part. The field of the complex numbers, denoted by , is the set of all such numbers, together with addition and multiplication . Absolute value, conjugation To each complex number, we can assign an absolute value as follows: A complex number () is actually a two-tuple , which is as such an element of . Now in , we have the Euclidean absolute value, namely , and thus we just define: 0 100% #FFFFFF Definition 1.2: The absolute value of a complex number is defined as . Note that the absolute value of the absolute value of a complex number is always a real number, since the root function maps everything in to (in fact to ). To each complex number (), we also assign a different quantity, which is obtained by reflecting along the first axis: 0 100% #FFFFFF Definition 1.3: Let a complex number be given. Then the conjugate of , denoted , is defined as . That is, the second component changed sign; if, in precise terms, , then . We observe: 0 100% #FFFFFF Theorem 1.4: Let two complex number be given. Then . Proof: and . With this notation, we can write the absolute value of a complex only in terms of without referring to or : 0 100% #FFFFFF Theorem 1.5: Let a complex number be given. Then . Here juxtaposition denotes multiplication, as usual (albeit complex multiplication in this case). Proof: . From this follows that the absolute value has the following crucial property: 0 100% #FFFFFF Corollary 1.6: Let be complex numbers. Then . Proof: by theorems 1.4 and 1.5. Note that the argument of the square roots was always real. The complex plane Since each complex number is in fact a two-tuple , , the set of all complex numbers can be visualized as the plane, where is the first coordinate and the second coordinate. The situation is indicated in the following picture: 400px The horizontal axis (or -axis) indicates the real part and the vertical (or -) axis indicates the imaginary part. Exercises Compute the absolute value of the following complex numbers: , , . Assume that and are natural numbers which can be written as the sum of two squares of natural numbers: and for some . Prove that the product can also be written as the sum of two squares. Hint: Plug in that (and similarly for ) and use the rules of computation for complex numbers. Prove the following relation connecting complex multiplication and the standard scalar product of : . This exercise introduces central concepts in algebra. Make yourself familiar with the concept of a field in algebra. If is a field, a subfield is defined to be a subset of which is closed under the addition, multiplication, subtraction and division inherited from and contains the elements and (ie. the neutral elements of addition and multiplication) of . Prove: Let be a family of subfields of a field . Prove that the intersection is also a subfield of . Make yourself familiar with the concept of a partially ordered set, and prove that the set of subfields of a given field is partially ordered by inclusion (ie. ). Prove that with regard to that order, any family of subfields has a greatest lower bound. Prove that a field has a smallest subfield, called the prime field, and identify the prime field of .