What are Complex Numbers?Points in the plane with the operations of addition and multiplication just given are called the complex numbers. The plane with these two operations on its points is called the complex numbers plane, or more briefly the complex numbers. A Change of NotationWe will now change to a more standard notation for them. We may and often will write the rectangular coordinates z = [a,b] as z = a+ib, We will further call the abscissa a, the real part of the complex number z = a+ib. We will also call the ordinate b, the imaginary part of the complex number z = a+ib. Note the previous notations [a, b] and (r,q) will be continued to be used for points in the plane in the further discussion of the properties of real and complex numbers. Purely Imaginary: We will say that the complex number z = a+ib
is purely imaginary when and only when its real part a = 0. The angle of
a purely imaginary complex number z = a+ib = 0+ib =
(0,b) is 90 degrees or 270 degrees (modulo 360 degrees). Note: Two quantities x and y are equal modulo a third quantity c, if and only if their difference x-y = kc for some whole number or integer k.360 degrees), depending on the sign of the imaginary part b. When b > 0, the angle is 90 degrees (modulo 360 degrees). When b < 0, the angle is 270 degrees (modulo 360 degrees). Real: We will also say that z = a+ib is (purely) real when and only when its imaginary part b is zero. The angle of a (purely) real complex number z = a+ib = a+i0 = (a,0) is 0 degrees or 180 degrees (modulo 360 degrees), depending on the sign of the real part a. If a > 0, this angle is 0 degrees (modulo 360 degrees) while if a > 0, this angle is 180 degrees (modulo 360 degrees). VocabularyFor each point or complex number z = a+b i = (a,b) = [r,q] in this plane, we say that a is the real part of z; that b is the imaginary part of z; that r = |z| is the magnitude, modulus or absolute value of z (different texts prefer different terms); and that q is the angle or argument of z.Horizontal and Vertical ComponentsThe horizontal component of a complex number z = a + ib = [a, b] is the point a = [a, 0] = real part, while the vertical component of the point z = a + ib = [a, b] is the points ib= [0, b] = imaginary part times i. Show or observe that a horizontal component times an horizontal component has a horizontal result, that a horizontal component times a vertical component has vertical result; and that a vertical times a vertical has a horizontal result. y |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers. Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision . D1 to D6 after provide a review of vectors. More on Complex Numbers:
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