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1 4.4 Euler’s Form Appendix B does not include this work. See notes in Study Book 4.4 (& some of 4.3). 4.4 Euler’s Form Appendix B does not include this work. See notes in Study Book 4.4 (& some of 4.3). Objectives: Know how to convert any z = a + ib to Euler form r e i how to convert any z = a + ib to Euler form r e i & vice versa how to use Euler form to simplify calculation of powers, multiplication & division how to use Euler form to simplify calculation of powers, multiplication & division how to find nth roots using Euler form how to find nth roots using Euler form the algebraic & geometric properties of nth roots. the algebraic & geometric properties of nth roots.

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2 When complex no’s are multiplied their angles add. When they are divided, their angles subtract. Hence in the polar form r (cos + i sin ) the argument behaves like an exponent but the modulus r does not. To remind us of this, we write as an exponent: We define e i to be cos + i sin Then r (cos + i sin ) = r e i . And de Moivre’s Theorem, ( cos + i sin ) n = cos n + i sin n becomes much more intuitive: (e i ) n = e i n and z n = r n e i n

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3 There are many reasons why we use base e, not another base. (See Study Book and Taylor Series, Alg & Calc II.) Convert the following to Euler Form: (P Convert the following to Euler Form: (Plot first: easier!) 5 = 5 e i0 - 2 = 2 e iπ 5 = 5 e i0 - 2 = 2 e iπ 3i = 3 e iπ/2 -1 - i = sqrt(2) e i5π/4 3i = 3 e iπ/2 -1 - i = sqrt(2) e i5π/4 In reverse: e iπ = cos(π) + i sin(π) = -1 + i 0 = -1. Confirm by plotting e iπ which is 1 e iπ That is the point distance 1 on angle π So it gives the -1 on the x-axis.) Similarly 2e - iπ/2 is the point 2 units down the y axis, ie 0 – 2i or simply -2i. ie 0 – 2i or simply -2i. Also see Examples 4.1, 4.2.

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4 Note: multiplying any complex number by r e i causes an increase of in the angle, ie a rotation, and distance to change by the factor r. Eg: Multiplying any z by i (which is e i /2 ) Eg: Multiplying any z by i (which is 1 e i /2 ) causes anticlockwise rotation through angle /2. causes anticlockwise rotation through angle /2. Multiplying z by 2i (which is 2 e iπ/2 ) iz causes rotation through pi/2, z and doubling of distance (modulus). Also see Ex 4.3. Positive angles cause an anticlockwise rotation through angle . Negative angles cause a clockwise rotation.

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5 Finding nth roots of z. First write z in Euler form r e i . Then First write z in Euler form r e i . Then generalise its angle by adding revolutions 2k . take the (1/n)th power: r 1/n e (i + 2k ) /n. find n different roots using n successive values of k, eg k = 0, 1, 2, … Geometrically, the nth roots of a e ib Geometrically, the nth roots of a e ib are evenly spaced on a circle of radius a 1/n. Examples: Find & plot 1) the cube roots of 4 3 + 4 i 8 π/6 2) the 4th roots of - 8. Note: If we know one nth root, we can plot it Note: If we know one nth root, we can plot it and deduce the positions of the others. Eg: One cube root of -8 is -2. Plot it & deduce the others.

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6 Homework Re-visit Section 8.3, App B, p 438 Re-visit Section 8.3, App B, p 438 Working in Euler Form, write solutions to Working in Euler Form, write solutions to Q 27, 29, 31, 33, 39, 43, 45, 47, 51, 53. Q 27, 29, 31, 33, 39, 43, 45, 47, 51, 53.

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IMAGINARY NUMBERS AND DEMOIVRE’S THEOREM Dual 8.3.

IMAGINARY NUMBERS AND DEMOIVRE’S THEOREM Dual 8.3.

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