SECTION 9.3 The Complex Plane; De Moivre’s Theorem 631 Now Work PROBLEM 37 Notice that the solution to Example 4(b) demonstrates that the argument of a complex number is periodic. (b) π π π π π π π π ( ) ( ) ( ) ( ) = + + = = = = ⎡ − + − ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = π π π π π π ( ) ( ) ⋅ ⋅ ⋅ − ⋅ − ⋅ z w i i e e e e i i e 3 cos 9 sin 9 5 cos 5 9 sin 5 9 3 5 3 5 3 5 3 5 cos 4 9 sin 4 9 3 5 cos 14 9 sin 14 9 3 5 i i i i i 9 5 9 9 5 9 4 9 14 9 Write the complex numbers in exponential form. Use formula (6). The argument must be between 0 and π2 . Polar form of the quotient z w Exponential form of the quotient z w THEOREM The argument of a complex number is periodic. θ π θ π ( ) ( ) [ ] = = + + + θ θ π ( ) + re re r k k k cos 2 sin 2 an integer i i k2 You are asked to prove this theorem in Problem 71. 4 Use De Moivre’s Theorem We have seen that the four fundamental operations of addition, subtraction, multiplication, and division can be performed with complex numbers. De Moivre’s Theorem, stated by Abraham De Moivre (1667–1754) in 1730, but already known to many people by 1710, is important because it allows the last two fundamental algebraic operations, raising to a power and extracting roots, to be used with complex numbers. De Moivre’s Theorem, in its most basic form, is a formula for raising a complex number z to the power n, where ≥ n 1 is an integer. Suppose = θ z rei is a complex number. Then ( ) = = = ⋅ = θ θ θ θ ( ) n z re re re r e 2: i i i i 2 2 2 2 Formula (5) ( ) = = = ⋅ = θ θ θ θ ( ) ( ) n z re r e re r e 3: i i i i 3 3 2 2 3 3 Formula (5) ( ) = = = ⋅ = θ θ θ θ ( ) ( ) n z re r e re r e 4: i i i i 4 4 3 3 4 4 Formula (5) Do you see the pattern? When written in exponential form, rules for exponents can be used to raise a complex number to a positive integer power. THEOREM De Moivre’s Theorem Exponential Form If = θ z rei is a complex number, then = θ ( ) z r e n n i n where ≥ n 1 is an integer. Polar Form If θ θ ( ) = + z r i cos sin is a complex number, then θ θ [ ] ( ) ( ) = + z r n i n cos sin n n where ≥ n 1 is an integer.
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