Complex numbers and their geometrical properties

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?

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Reply 1

username4407536

Original post by AronAdamski

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.

Reply 2

AronAdamski

Original post by 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.

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

Reply 3

username4407536

Original post by AronAdamski

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

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

Reply 4

Htx_x346

Original post by golgiapparatus31

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

Reply 5

mqb2766

Original post by Htx_x346

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

No. That is 4 equal rotations without any stepping.
The problem is a unit step, then rotate, then unit step, then rotate.

(edited 2 years ago)

Reply 6

Htx_x346

Original post by mqb2766

No. That is 4 equal rotations without any stepping.
The problem is a unit step, then rotate, then unit step, then rotate.

oh. hmmm.
but the magnitude is 1?

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

Reply 7

Itsmikeysfault

Original post by Htx_x346

oh. hmmm.
but the magnitude is 1?

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

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

Reply 8

mqb2766

Original post by Htx_x346

oh. hmmm.
but the magnitude is 1?

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

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?

Reply 9

Htx_x346

Original post by 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?

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?

Original post by 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

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

(edited 2 years ago)

Reply 10

mqb2766

Original post by 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.

They want you to solve the problem by using complex numbers to represent translations and rotations. That sounds about right and you should note it is a simple geometric series so you can use the standard result.

(edited 2 years ago)

Reply 11

Htx_x346

Original post by 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.

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?

Reply 12

mqb2766

Original post by 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?

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.

(edited 2 years ago)

Reply 13

tetrad4444

anyone know where the OP got the question from

Reply 14

Itsmikeysfault

Original post by tetrad4444

anyone know where the OP got the question from

It's a question from the edexcel further maths core pure 2 textbook

Reply 15

Htx_x346

Original post by 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.

thanks

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

Reply 16

mqb2766

Original post by Htx_x346

thanks

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

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

(edited 2 years ago)

Reply 17

Htx_x346

Original post by mqb2766

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

thank you

have a good evening

Reply 18

mqb2766

Original post by 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

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?