# Complex numbers and their geometrical properties

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AronAdamski

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#1

An ant walks forward one unit and then turns to the right by 2pi/9. It repeats this a further 3 times. Show that the distance of the ant from its initial position is sin(4pi/0)/sin(pi/9).

Can someone show me their approach?

Can someone show me their approach?

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golgiapparatus31

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#2

(Original post by

An ant walks forward one unit and then turns to the right by 2pi/9. It repeats this a further 3 times. Show that the distance of the ant from its initial position is sin(4pi/0)/sin(pi/9).

Can someone show me their approach?

**AronAdamski**)An ant walks forward one unit and then turns to the right by 2pi/9. It repeats this a further 3 times. Show that the distance of the ant from its initial position is sin(4pi/0)/sin(pi/9).

Can someone show me their approach?

The position is given by the sum of all this.

Take its magnitude and you have the distance.

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AronAdamski

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#3

(Original post by

It first walks 1 forward, then e^(2pi/9) then e^(4pi/9) then e^(6pi/9).

The position is given by the sum of all this.

Take its magnitude and you have the distance.

**golgiapparatus31**)It first walks 1 forward, then e^(2pi/9) then e^(4pi/9) then e^(6pi/9).

The position is given by the sum of all this.

Take its magnitude and you have the distance.

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golgiapparatus31

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#4

(Original post by

So you just add all of the complex numbers? I can't imagine how this would work.

**AronAdamski**)So you just add all of the complex numbers? I can't imagine how this would work.

The real part gives the x-displacement

The imaginary gives the y-displacement

Yes just add

the complex numbers behave like vectors

if you don't believe me then try it

say you go east by 2 then north by 1

your position is described by 2+i

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Htx_x346

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#5

(Original post by

It is vectors

The real part gives the x-displacement

The imaginary gives the y-displacement

Yes just add

the complex numbers behave like vectors

if you don't believe me then try it

say you go east by 2 then north by 1

your position is described by 2+i

**golgiapparatus31**)It is vectors

The real part gives the x-displacement

The imaginary gives the y-displacement

Yes just add

the complex numbers behave like vectors

if you don't believe me then try it

say you go east by 2 then north by 1

your position is described by 2+i

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mqb2766

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#6

(Original post by

i know this is really late lol but would that just be (e^2pi/9)^4?

**Htx_x346**)i know this is really late lol but would that just be (e^2pi/9)^4?

The problem is a unit step, then rotate, then unit step, then rotate.

Last edited by mqb2766; 1 week ago

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Htx_x346

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#7

(Original post by

No. That is 4 equal rotations without any stepping.

The problem is a unit step, then rotate, then unit step, then rotate.

**mqb2766**)No. That is 4 equal rotations without any stepping.

The problem is a unit step, then rotate, then unit step, then rotate.

but the magnitude is 1?

so would I have to add e^2pi/9 4 times?

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Itsmikeysfault

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#8

(Original post by

oh. hmmm.

but the magnitude is 1?

so would I have to add e^2pi/9 4 times?

**Htx_x346**)oh. hmmm.

but the magnitude is 1?

so would I have to add e^2pi/9 4 times?

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mqb2766

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#9

**Htx_x346**)

oh. hmmm.

but the magnitude is 1?

so would I have to add e^2pi/9 4 times?

Have you sketched what you need to do?

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Htx_x346

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#10

(Original post by

For the first point, you've just done demoivre. e^ix is a rotation by x. e^i4x is a rotation by 4x.

Have you sketched what you need to do?

**mqb2766**)For the first point, you've just done demoivre. e^ix is a rotation by x. e^i4x is a rotation by 4x.

Have you sketched what you need to do?

So i need to sum e^0 then e^(2pi/9) then e^(4pi/9) then e^(6pi/9)...and that's it?

(Original post by

Using complex numbers just makes the problem more complex that it needs to. You could get the answer by drawing a diagram and using trig

**Itsmikeysfault**)Using complex numbers just makes the problem more complex that it needs to. You could get the answer by drawing a diagram and using trig

Last edited by Htx_x346; 1 week ago

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mqb2766

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#11

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#11

(Original post by

ah right. yeah i've sketched it.

So i need to sum e^0 then e^(2pi/9) then e^(4pi/9) then e^(6pi/9)...and that's it?

this was from the complex numbers part of the textbook so im just trying to solve it using CN.

**Htx_x346**)ah right. yeah i've sketched it.

So i need to sum e^0 then e^(2pi/9) then e^(4pi/9) then e^(6pi/9)...and that's it?

this was from the complex numbers part of the textbook so im just trying to solve it using CN.

Last edited by mqb2766; 1 week ago

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#12

(Original post by

They want you to solve the problem by using complex numbers to represent displacements and rotations. That sounds about right and you should note it is a simple geometric sequence.

**mqb2766**)They want you to solve the problem by using complex numbers to represent displacements and rotations. That sounds about right and you should note it is a simple geometric sequence.

But would it not be slightly more tedious using the sum of geometric sequences equation in this case when you can just add them?

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#13

(Original post by

thanks.

But would it not be slightly more tedious using the sum of geometric sequences equation in this case when you can just add them?

**Htx_x346**)thanks.

But would it not be slightly more tedious using the sum of geometric sequences equation in this case when you can just add them?

Its not going to be that long either way.

Hint is to think about the form of the answer.

Last edited by mqb2766; 1 week ago

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tetrad4444

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Itsmikeysfault

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#15

(Original post by

anyone know where the OP got the question from

**tetrad4444**)anyone know where the OP got the question from

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#16

(Original post by

Id go for the series, but why not try both and see what you find out.

Its not going to be that long either way.

Hint is to think about the form of the answer.

**mqb2766**)Id go for the series, but why not try both and see what you find out.

Its not going to be that long either way.

Hint is to think about the form of the answer.

also, just to clarify, what exactly was the reason for (e^2pi/9)^4 being incorrect? thanks again

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#17

(Original post by

thanks

also, just to clarify, what exactly was the reason for (e^2pi/9)^4 being incorrect? thanks again

**Htx_x346**)thanks

also, just to clarify, what exactly was the reason for (e^2pi/9)^4 being incorrect? thanks again

Last edited by mqb2766; 1 week ago

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#18

(Original post by

Its e^(i8pi/9). So a distance of 1 from the origin.

**mqb2766**)Its e^(i8pi/9). So a distance of 1 from the origin.

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**Itsmikeysfault**)

Using complex numbers just makes the problem more complex that it needs to. You could get the answer by drawing a diagram and using trig

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Itsmikeysfault

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#20

(Original post by

Can't see why using complex numbers makes it more complex, though obviously you can do it with straight geometry/trig. With complex numbers, after writing down the geometric series expression, you can "centre" both the numerator and denominator and pretty much write down the answer?

**mqb2766**)Can't see why using complex numbers makes it more complex, though obviously you can do it with straight geometry/trig. With complex numbers, after writing down the geometric series expression, you can "centre" both the numerator and denominator and pretty much write down the answer?

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