Roots of complex equation zn = a

Lehman - Pesonen 2007
1. What are the solutions of zn = 1 (for n = 2, 3, 4) ?
Let z = x + i y = |z| eiC, with x, y and C real.
Play with the dynamic figure (called sketch) by pushing the buttons and moving z in the plane.
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Problems 1

Reset by pushing the computer keyboard 'R' button.
1a) Give the two solutions of the equation z2 = 1.
Answer 1a1): z1 = (      ) + i(      )
Answer 1a2): z2 = (      ) + i(      )
1b) Give the four solutions of the equation z4 = 1.
Answer 1b1): z1 = (      ) + i(      )
Answer 1b2): z2 = (      ) + i(      )
Answer 1b3): z3 = (      ) + i(      )
Answer 1b4): z4 = (      ) + i(      )
1c) The equation z3 = 1 has three solutions: 1, j and j2. Give approximative values of j and j2:
Answer 1c1): j = (      ) + i(      )
Answer 1c2): j2 = (      ) + i(      )
1d) For which values of z do the points z and z4 lay on top of each other?
Answer 1d1): z1 = (      ) + i(      )
Answer 1d2): z2 = (      ) + i(      )
Answer 1d3): z3 = (      ) + i(      )
Answer 1d4): z4 = (      ) + i(      )
1e) For which values of z do the points z2 and z4 lay on top of each other?
Answer 1e1): z1 = (      ) + i(      )
Answer 1e2): z2 = (      ) + i(      )
Answer 1e3): z3 = (      ) + i(      )
Answer 1e4): z4 = (      ) + i(      )
1f) Explain why there are less than 4 solutions.
Answer 1f):
2. What are the solutions of zn = a (for n = 2, 3, 4) ?
Let z = x + i y = |z| eiC, with x, y and C real.
Play with the dynamic figure (called sketch) by pushing the buttons and moving z in the plane.
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Problems 2

2a) Reset by pushing the computer keyboard 'R' button and do not move a. Use the variable point z to find the cartesian coordinates of a.
Answer 2a1):  Re a = (      ) + i(      )
Answer 2a2):  Im a = (      ) + i(      )
2b) Use the variable point z to find the polar coordinates of a.
Answer 2b1): arga = (      ) + i(      )
Answer 2b2): |a| = (      ) + i(      )
2c) Reset again. Click the buttons Show z^2 and |z| = |a|^1/2.
Find the two solutions of z2 = a.
Answer 2c1): z1 = (      ) + i(      )
Answer 2c2): z2 = (      ) + i(      )
2d) Reset again. Click the button Show z^3 and |z| = |a|^1/3.
Find the three solutions of z3 = a.
Answer 2d1): z1 = (      ) + i(      )
Answer 2d2): z2 = (      ) + i(      )
Answer 2d3): z3 = (      ) + i(      )
2e) What can you say about the (algebraic) sum of the arguments of the solutions?
Answer 2e):    
2f) Reset again. Click the button Show z^4 and |z| = |a|^1/4.
Find the four solutions of z4 = a.
Answer 2f1): z1 = (      ) + i(      )
Answer 2f2): z2 = (      ) + i(      )
Answer 2f3): z3 = (      ) + i(      )
Answer 2f4): z4 = (      ) + i(      )
2g) What can you say about z2/z1, z3/z1, z4/z1, z2/z4, z1/z3, zp/zq ?
Answer 2g):    
2h) What is the shape of the quadrangle z1 z2 z3 z4 ?
Answer 2h):    
3. About the solutions of a complex number equation zx = a, x real.
See the locus of x ⟼ zx
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Introduction to Problems 3

For a given complex number z there are q complex numbers that could be denoted zp/q, namely the q solutions of the equation in w: wq = zp.
If x is an integer, there is only one complex number zx, but if x is general real number, zx may not mean anything.
Here the complex number denoted by zx is the number |z|x eixC, where C is the real number such that arg(z) = C + k2π and -π < C < π.
The locus shown on the screen is the curve [0, x] C, t ⟼ |z|t eitC.

Problems 3

2a) Moving the point a, check that if r = |z| then |zx| = rx.
Answer 3a):    
3b) Choose x = 0.5; where are the values of a such that there is a complex number z for which zx is on top of a ?
Answer 3b):    
3c1) Choose x = 1.75; where are the points a for which two positions of z allows zx is on top of a for some x ?
Answer 3c1):    
3c2) Where are the points a for which there is only one position of z?
Answer 3c2):