Question

Huon has been learning that square roots of complex numbers are well... complicated. If a, b are both positive real numbers then we have the fact that ??a??b = ??ab. For a complex number this is no longer true. If w is a complex number with r = |w| and ? = Arg(w) then we define the complex squareroot of w to be ??r exp(i?) = ??r exp(i?/2). Note very carefully that ? is the principal argument of w. (a) Using this definition, enter the following complex squareroots in Cartesian form: (i) ??-1 = (ii) ??(-1) × (-1) = (iii) ??-1 × ??-1 = Please use Maple syntax for complex number. If your response is -5 + 19i then enter -5 + 19I Note the capital letter I. (b) Help Huon understand this definition by using it find the modulus and principal argument of ??z^2 where z = 25 exp(i 8?/13). |??z| = Arg(??z^2) = (c) For part (c) we are again using z = 25 exp(i 8?/13). Consider a complex number w of the form w = ? exp(i?) where ? is a positive real number and ? ? ?. What is the largest interval containing zero, so that ??zw = ??z??w. Enter your answer in interval notation. We need ? ? Enter your response in the box above using standard interval notation, for example, (-6,11] . The numbers you enter should use Maple syntax with a for multiplication, for example, 40*sqrt(Pi) for 40??? and infinity for ?. If your answer to part (c) above is incorrect, partial marks may be awarded for any working provided below:

          Huon has been learning that square roots of complex numbers are well... complicated.

If a, b are both positive real numbers then we have the fact that ??a??b = ??ab. For a complex number this is no longer true. If w is a complex number with
r = |w| and ? = Arg(w)
then we define the complex squareroot of w to be
??r exp(i?) = ??r exp(i?/2).

Note very carefully that ? is the principal argument of w.

(a) Using this definition, enter the following complex squareroots in Cartesian form:
(i) ??-1 =
(ii) ??(-1) × (-1) =
(iii) ??-1 × ??-1 =

Please use Maple syntax for complex number. If your response is -5 + 19i then enter
-5 + 19*I
Note the capital letter I.

(b)
Help Huon understand this definition by using it find the modulus and principal argument of ??z^2 where
z = 25 exp(i 8?/13).
|??z| =
Arg(??z^2) =

(c)
For part (c) we are again using z = 25 exp(i 8?/13). Consider a complex number w of the form
w = ? exp(i?)
where ? is a positive real number and ? ? ?. What is the largest interval containing zero, so that ??zw = ??z??w. Enter your answer in interval notation.

We need ? ?

Enter your response in the box above using standard interval notation, for example, (-6,11] . The numbers you enter should use Maple syntax with a * for multiplication, for example, 40*sqrt(Pi) for 40??? and infinity for ?.

If your answer to part (c) above is incorrect, partial marks may be awarded for any working provided below:

Huon has been learning that square roots of complex numbers are well... complicated.

If a, b are both positive real numbers then we have the fact that ??a??b = ??ab. For a complex number this is no longer true. If w is a complex number with
r = |w| and ? = Arg(w)
then we define the complex squareroot of w to be
??r exp(i?) = ??r exp(i?/2).

Note very carefully that ? is the principal argument of w.

(a) Using this definition, enter the following complex squareroots in Cartesian form:
(i) ??-1 =
(ii) ??(-1) × (-1) =
(iii) ??-1 × ??-1 =

Please use Maple syntax for complex number. If your response is -5 + 19i then enter
-5 + 19*I
Note the capital letter I.

(b)
Help Huon understand this definition by using it find the modulus and principal argument of ??z^2 where
z = 25 exp(i 8?/13).
|??z| =
Arg(??z^2) =

(c)
For part (c) we are again using z = 25 exp(i 8?/13). Consider a complex number w of the form
w = ? exp(i?)
where ? is a positive real number and ? ? ?. What is the largest interval containing zero, so that ??zw = ??z??w. Enter your answer in interval notation.

We need ? ?

Enter your response in the box above using standard interval notation, for example, (-6,11] . The numbers you enter should use Maple syntax with a * for multiplication, for example, 40*sqrt(Pi) for 40??? and infinity for ?.

If your answer to part (c) above is incorrect, partial marks may be awarded for any working provided below:

Added by Dawn H.

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