Alg1131W6T3 - Complex conjugates

Saturday, 14 April 2018

12:14 PM

Machine generated alternative text: Recall that if z = a + bi, then the real part of z is Re(z) both of these are actually real numbers! —8 —8i and So for u i) Re(u) = —8 and the imaginary part of z is Im(z) O Note carefully that ii) Im@) 8 iii) Re(8v) 24 iv) 1m u2 + v 120 Also recall that if z = a + bi then the complex conjugate is Note: remember to use I (capital i) for the imaginary unit It is reasonably obvious that a-bl Re(z) O and Im(E) -Im(z) More subtle is that a similar rule holds for both multiplication and division: that So for u On the other hand 8i and v = 3 8i, and so (uv) and + and and ( z -88+401 -88-401 _8+8*1 3+81 so (u) @) -88-401

 

Machine generated alternative text: Similarly while and so (40+881)/73 Sometimes it is useful to express the real and imaginary parts of a complex number z (40-881)/73 (40+881)/73 a + bi algebraically. This is done by combining z and its complex conjugate E. In particular Re(z) — 2 O Im(z) — O Im(z) — @ Im(z) — O Im(z) — . Similarly 2 2

 

Machine generated alternative text: Given that 3z — 2w = —10 — 17i and — = —3 + 4i, we can find that z Note: remember to use I (capital i) as the complex unit in Maple notation.

So therefore,

And

Machine generated alternative text: If —2i then i) Re z 2 ii)Re z2 iii)Re z3 iv)Re z4 while i) 1m z -2 ii)lm z2 iii)lm z3 iv)lm z4

 

Machine generated alternative text: In physics and engineering applications; complex numbers are a useful device to represent vectors. For example, the vector v z = 2 + 3i. We may visualize this vector as emanating from the origin 0 to the point [2, 3] which we may also label as z. In this case i) the complex number E represents the reflection of v in tne x-axis 2 in the plane can be represented by the complex number 3 ii) the complex number —z represents the iii) the complex number —E represents the iv) the complex number iz represents the reflection of v in the origin reflection ot v in the y-axis rotation of v by 90 degrees v) the complex number —iz represents the rotation of v by -90 degrees Note: the conventional direction for a rotation in mathematics is counter-clockwise, e.g. a rotation by —90 degrees would be a quarter turn clockwise.

 

 

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