De Moivre's theorem question (1 Viewer)

[marxman]

Currently working through Terry Lee's text, I noticed the worked solutions provided a swift workaround for using De Moivre's theorem, having ASSUMED that it is true for all rational numbers.

As far as I have seen from the ext 2 course, De Moivre's theorem is only proved for the integer set, due to the restrictions on not being allowed to use the e^i(theta) form of a complex number.

Essentially, I asked my teacher if we could simply just assume it is true for all rational numbers, as solving complex equations would become far easier - but he was fairly certain we should simply stick with the integer set. So on that note, in all school assessments I'm just going to use the longer, more unnecessarily tedious ways to solve these equations.

My question is, can we use the Terry Lee method in the HSC exams and assume it is true for all rationals?

 

[InteGrand]

Currently working through Terry Lee's text, I noticed the worked solutions provided a swift workaround for using De Moivre's theorem, having ASSUMED that it is true for all rational numbers.

As far as I have seen from the ext 2 course, De Moivre's theorem is only proved for the integer set, due to the restrictions on not being allowed to use the e^i(theta) form of a complex number.

Essentially, I asked my teacher if we could simply just assume it is true for all rational numbers, as solving complex equations would become far easier - but he was fairly certain we should simply stick with the integer set. So on that note, in all school assessments I'm just going to use the longer, more unnecessarily tedious ways to solve these equations.

My question is, can we use the Terry Lee method in the HSC exams and assume it is true for all rationals?

With non-integer powers, we actually end up with multi-valued things, which require specification of a Principal Value. But it is the case that one of the values out of these multiple values of a rational power will be what is specified by what De Moivre's Theorem would say.

E.g. something like (e^i(pi/4))^(1/2) would have two values, namely e^(i*pi/8) and e^(i*(-7pi/8)) (ignoring any Principal Value specifications for now). One of these is indeed what De Moivre's Theorem would say, i.e. e^(i*pi/8).

 

[Paradoxica]

Currently working through Terry Lee's text, I noticed the worked solutions provided a swift workaround for using De Moivre's theorem, having ASSUMED that it is true for all rational numbers.

As far as I have seen from the ext 2 course, De Moivre's theorem is only proved for the integer set, due to the restrictions on not being allowed to use the e^i(theta) form of a complex number.

Essentially, I asked my teacher if we could simply just assume it is true for all rational numbers, as solving complex equations would become far easier - but he was fairly certain we should simply stick with the integer set. So on that note, in all school assessments I'm just going to use the longer, more unnecessarily tedious ways to solve these equations.

My question is, can we use the Terry Lee method in the HSC exams and assume it is true for all rationals?

No. Once you start working over rationals, you have a multi-valued function.

i.e., you need to consider every possible rotation of the complex number that produces the number you want.

Which is no different from directly solving for the value as roots of a polynomial equation.

 

[uncertainty]

As far as I have seen from the ext 2 course, De Moivre's theorem is only proved for the integer set, due to the restrictions on not being allowed to use the e^i(theta) form of a complex number.

Why don't they let us use that form?

 

[dan964]

Why don't they let us use that form?

Because understanding it would require the understanding of all sorts of other things.