Of view, we can try all of the techniques we have used in To study such an object from a geometric point The domain and two for the range, giving a two-dimensional surface inįour-dimensional space. The total graph requires four real dimensions, two for Real and imaginary parts, so we have two real coordinates for z and A single complex number already requires two real numbers, its We can analyze the function algebraically all right, but howĬan we graph it? A problem with dimensions precludes a simple graph on What about complex functions of a complex number? We can still talkĪbout the equation w = z 2, where now z and w represent complex This representation gives a tremendous amount of insight into the symmetry of the function as well as the location of its minimum points. The geometric form of this graph is a parabola, symmetric with respect to the vertical axis and passing through the origin. X 2 can be graphed in the plane by plotting all pairs of the form ( x, x 2). In ordinary analytic geometry, the equation u = The graphing of functions of one real variable on a two-dimensional One of the most effective techniques in ordinary analytic geometry is The graph of the real parabola in real two-space. The interplay between the algebraic and the geometric accounts for the rich structure of the complex number system and for its surprisingly many applications in science and engineering. The most significant thing about this construction is that we can relate the algebraic properties of complex numbers to the geometric properties of the plane. Thus an element exists whose square is the negative of the unit element, the key property of the complex numbers. But in this new system, we also have an element (0, 1) such that (0, 1)(0, 1) = (-1, 0). For example, since ( x, y)(1, 0) = ( x, y), the number pair (1, 0) is a unit element for multiplication. The justification for this rule is that so many of the desirable algebraic properties of real numbers still hold. In a sense, we can say that the essence of complex numbers is that they are the collection of number pairs endowed with the usual rule for addition and an unusual rule for multiplication: ( x, y)( u, v) = ( xu - yv, yu + xv). As these examples illustrate, the complex numbers share very many of the properties of the real numbers. For any x + yi not equal to 0, there is another complex number u + vi such that ( x + yi)( u + vi) = 1, specifically u + vi = x/( x 2 + y 2) - yi/( x 2 + y 2). Thus the product of two complex numbers is another expression of the ( x + yi)( u + vi) = ( xu + yui + xvi + yvi 2) = ( xu - yv) + ( yu + xv) i Product of x + yi by the real number c is cx + ( cy) i.īut it is also possible to define a multiplication of one complex number with another, giving as the product of two complex numbers The sum of x + yi and u + vi is ( x + u) + ( u + v) i, and the To rules for addition and scalar multiplication for complex To add number pairs and multiply them by real scalars, and this leads Part and the imaginary part of the complex number. The two coordinates of the pair ( x, y) are called the real Plane, so we may say that the complex numbers form a two-dimensionalĬollection. They called this collection the complex numbers.Įach complex number x + yi corresponds to a number pair ( x, y) in the Their system all numbers of the form x + yi, with Numbers and multiply them by real numbers, they had to include in Mathematicians introduced a new symbol, i, with the property ToĬonstruct a larger number system where this equation can be solved, Square of any real number is never negative, it is impossible to findĪ real number that solves the equation x 2 = -1. Inadequate for solving some simple equations. Sufficient for very many purposes in algebra and geometry, they are N-tuples of real numbers, starting with the points on a Up to this point, we have considered pairs, triples, and Complex Numbers as Two-Dimensional Numbers Complex Numbers as Two-Dimensional Numbers
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |