Moreover, the latter is obtained from the former by rotation through 90 o in the positive ( counterclockwise) direction. If you compare two points (x, y) and (y, -x) on the plane (even if for hust a few specific values of a and y) and join the two to the origin, you'll be able to observe that the two segments are perpendicular to each other. Multiplication by i has a curious effect: Elsewhere, it is standard to denote it i: i = (0, 1).
In engineering sciences, the number (0, 1) is sometimes denoted as j. It has the remarkable feature of having a negative square. The third number of importance is (0, 1). Again, there is a good reason to say that the two are one and the same. This number plays an important role in multiplication that stems from the following property:Īmong complex numbers (1, 0) behaves like the real unit 1 among the real numbers. We shall see shortly that there is a good reason to think of the two zeros - real and complex - as one and the same number.Īnother complex number of consequence is (1, 0). The symbol is exactly the same as used to identify the "real" 0. It is therefore natural to identify it with 0. For example, the number (0, 0) has the properties of 0: Several complex numbers play exclusive roles. It only has weakened analogues in R 4 ( quaternions) and R 8 ( octonions).Īddition and multiplication of complex numbers inheret most of the properties of addition and multiplication of real numbers: Multiplication on the other hand is peculiar to complex numbers. (x 1, y 1)(x 2, y 2) = (x 1x 2 - y 1y 2, x 1y 2 + x 2y 1).Īddition is defined componentwise in a relatively standard way that extends to spaces of higher dimension. R 2 considered as a set of complex numbers C is called Argand diagram or Argand plane or Gauss plane. With two operations - addition and multiplication - defined below, the set R 2 becomes the set C of complex numbers.
Two such pairs are equal if their corresponding components coincide: We consider the set R 2 =, i.e., the set of ordered pairs of real numbers. When you come to quadratic equations you will be confronted with an entity (or non-entity) whose name is written this way - √ -1, and pronounced "square root of minus one."Ĭomplex numbers are points in the plane endowed with additional structure.
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