Download presentation

Presentation is loading. Please wait.

Published byBritney Christopher Modified over 3 years ago

1
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.

2
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

3
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.

4
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.

5
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.

6
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.

Similar presentations

OK

7.1, 7.2 & 7.3 Roots and Radicals and Rational Exponents Square Roots, Cube Roots & Nth Roots Converting Roots/Radicals to Rational Exponents Properties.

7.1, 7.2 & 7.3 Roots and Radicals and Rational Exponents Square Roots, Cube Roots & Nth Roots Converting Roots/Radicals to Rational Exponents Properties.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on 3d printing technology Ppt on non renewable energy sources Ppt on multiplexers and demultiplexers Ppt on society and culture Ppt on diffusion taking place in our daily life Ppt on indian railway reservation system Ppt on vegetarian and non vegetarian restaurant Free download ppt on alternative sources of energy Ppt on waxes Ppt on economic reforms in india 1991 economic reforms