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# Precalculus MM: A New Day, Hall of Justice: Polar Coordinates and Complex Numbers 5: NWN 0148b

The Game feels compelled to force Zanzi to make progress by generating an unpleasant sound in her mind. The sound it generates is amplified exponentially according DeMoivre's theorem: z^n = r^n(cos(n.theta) + i.sin(n.theta)), where z is the complex number raised to the nth power, r^n is the modulus raised to the same nth power and theta is the argument multiplied by coefficient n. So with DeMoivre's theorem we raise the modulus by the same power to which the complex number is raised and we multiply the argument by the same value as the power. Note that the double angle identities for cosine (cos^2 theta - sin^2 theta) and sine (2 sin theta . cos theta) are required when deriving DeMoivre's theorem. Also note the DeMoivre's requires that the complex number be in trig form.
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All complex numbers can be written in the exponential form z = re^i.theta, which is called the Euler formula. So re^i.theta = r(cos theta + i.sin theta). This links exponential form with trigonometric form via base e and makes working with complex numbers relatively straightforward. Using the Euler formula it's quite easy to prove multiplication of complex numbers and DeMoivre's theorem and to derive the double angle identities. e^i.pi = -1 is an important Euler formula identity.
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The "infinity of silence" Zanzi experiences results from an almost instantaneous curtailment of sound caused when the nth power level of sound is cut to the nth root level of sound. As the amplification of sound is based on DeMoivre's theorem, the nth root will be obtained by setting z^n=x, where x is the current level of sound. E.g. if x=256i, then for z^2=256i means that the first of the two roots of 256i will be found using r^2(cos 2.theta+i.sin 2.theta); r=16 and 2.theta=pi/2+2n.pi. So theta=pi/4+n.pi. The first root will have n=0 giving 16(cos pi/4+i.sin pi/4)=16[(2^0.5)/2+i.(2^0.5)/2]=8.2^0.5+i.8.2^0.5. The second root will have n=1 giving 16(cos 5pi/4+i.sin 5pi/4)=16[(-2^0.5)/2-i.(2^0.5)/2]=-8.2^0.5-i.8.2.05.
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Note that when obtaining the roots of complex numbers: 1) If z^n=x then there will be n roots for x; 2); when roots are graphed there is always rotational symmetry because the angle between the roots remains constant, e.g. if n=3 then the angle between each root is 2pi/3 (or 120 deg); 3) the modulus can be any length but the arguments always lie on the unit circle; 4) knowing identities helps, e.g. for 256i the second root when n=1 could have been found without calculation by knowing the add pi identities: cos (theta+pi)=- cos theta and sin (theta+pi)=-sin theta.
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Further note that when obtaining the nth roots of a complex number, e.g. the fourth roots where n=4: 1) all roots have the same modulus and the modulus is obtained by taking nth root of the original modulus r, that is r^1/n; the fourth root is a root of degree 4, so r^1/4; 2) the primary (or first root) root has argument (1/n)(theta), so divide the original argument by the degree of the root, i.e. (1/4) (theta); 3) successive roots are found by adding 2pi/n; 4) there are exactly n distinct nth roots.