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System Of Complex Equations

1) Suppose that \(z_1, z_2, z_3, z_4 \) are complex numbers which satisfy

\[ \begin{cases} |z_1| = |z_2| = |z_3| = |z_4| = 1 , \\ z_1 + z_2 + z_3 + z_4 = 0. \\ \end{cases} \]

What can we conclude about \( z_1, z_2, z_3, z_4 \)? Must they lie on the corners of a square?


2) Suppose that \(z_1, z_2, z_3, z_4, z_5 \) are complex numbers which satisfy

\[ \begin{cases} |z_1| = |z_2| = |z_3| = |z_4| = |z_5| =1, \\ z_1 + z_2 + z_3 + z_4 + z_5 = 0. \\ \end{cases} \]

What can we conclude about \( z_1, z_2, z_3, z_4, z_5 \)? Must they lie on the corners of a regular pentagon?

You can refer to this similar note, which deals with the 3 variable case.

Note by Calvin Lin
2 years, 10 months ago

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For 1, I'm going to use the same argument as before.

Without loss of generality, let \(z_1\), \(z_2\), \(z_3\) and \(z_4\) be arranged counter-clockwise around the origin.

The resultant of two equal [in magnitude] vectors lies on the bisector of the angle between them.

So, \(z_1+z_2=a_1\) must lie on the bisector of the angle between \(z_1\) and \(z_2\).

Similarly, \(z_3+z_4=a_2\) must lie on the bisector of the angle between \(z_3\) and \(z_4\).

Since \(a_1+a_2=0\), we can conclude that they lie on the same line.

That means [you can see this visually, I don't know if it's rigorous] \(z_1\) and \(z_4\) lies on the same line and \(z_3\) and \(z_2\) lies on the same line.

And \(|z_1|+|z_4|\) and \(|z_2|+|z_3|\) are the diagonals of a quadrilateral and since they are equal and bisect one another, they are the diagonals of a rectangle.


You should add that the complex numbers are non-zero. I will work on 2 now. Mursalin Habib · 2 years, 10 months ago

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@Mursalin Habib For 2)

Must they lie on the corners of a regular pentagon?

Not necessarily. Here's a counter-example. If \(z_1\), \(z_3\) and \(z_5\) are the vertices of an equilateral triangle and \(z_2=-z_4\) and the angle between \(z_1\) and \(z_2\) is not equal to \(72\) degrees, then the complex numbers are not the vertices of a regular pentagon.

Notice that you can rotate the line that contains \(z_2\) and \(z_4\) along the origin. I'll add a picture later.

What can we conclude about \(z_1\), \(z_2\), \(z_3\), \(z_4\), \(z_5\)?

Well, we can say that the complex numbers are the vertices of a erm... a pentagon! :)


We can generalize this. Given any \(n\) complex numbers that satisfy the following conditions, it is not necessary that the numbers are the vertices of a regular \(n\) gon.

EDIT: Adding a picture.

Imgur

Imgur

\(BB'B'_1\) is an equilateral triangle and the complex numbers \(AB\), \(AC\), \(AB'\), \(AB'_1\) and \(AD\) are five complex numbers that satisfy the conditions but are not the vertices of a regular pentagon. Mursalin Habib · 2 years, 10 months ago

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@Mursalin Habib For 5 points, since we're looking to classify these sets, can we conclude that the set "must either be an equilateral triangle and a diameter, or a regular pentagon"? Or is there another set that is more 'random'? Calvin Lin Staff · 2 years, 10 months ago

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@Calvin Lin I think there exist various other sets. For example,

\(z_1 = a + i0\)

\(z_2 = -0.1a + ic\)

\(z_3 = -0.1a - ic\)

\(z_4 = -0.4a + id\)

\(z_5 = -0.4a - id\)

A rough figure of the shape formed by these points can be seen here.

These complex numbers doesn't form a nice geometrical shape. We can just call it a pentagon. Prakhar Jaiswal · 2 years, 10 months ago

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I think we cannot conclude about both (1) and (2) . If we take z1,z3 as diametrical end points of unit circle and z2 , z4 as diametrical end points of same circle but a different diameter. It will satisfy both the conditions. And it is not necessary that both diameters will be perpendicular to each other. hence the 2 conditions are not sufficient to conclude that z1,z2,z3,z4 will be vertices of square Harsh Depal · 2 years, 10 months ago

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1) If you take any two numbers as diametrical points on a circle of radius 1. You get two diameters and if you rotate these two diameters about the origin independently you will get all the possible values of z1 z2 z3 z4 and they will not necessarily form a square 2) Similarly, If you take any three points as vertices of an equilateral triangle circumscribed by the circle of radius 1 and the remaining 2 points as diametrical points of the same circle you can rotate these two figures independently about the origin to get all the 5 points.

In general, for a system with n points, you can form subsets of regular polygon(s) and diameter(s) and rotate them to get the values. Ajinkya Jadhav · 2 years, 10 months ago

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THE ARE THE ROOTS OF THE EQUATION Z^5=1 Sai Krishna Chary · 2 years, 10 months ago

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well, since \(\left| { z }_{ 1 } \right| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =\left| { z }_{ 4 } \right| =1\) is true, \({ z }_{ 1 },{ z }_{ 2 },{ z }_{ 3 },{ z }_{ 4 }\) all lie on the unit circle with radius one and center as origin which is one obvious conclusion you make...so they make a cyclic quadrilateral when joined..and one of the properties of a cyclic quadrilateral states that the sum of the opposite angles is always \( \quad \pi \) radians... from the statement 2 given \({ z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 }+{ z }_{ 4 }=0\) using \({ z }_{ i }=\cos { { \theta }_{ i } } +i\sin { { \theta }_{ i } } \) i get \(\cos { { \theta }_{ 1 } } +\cos { { \theta }_{ 2 } } +\cos { { \theta }_{ 3 } } +\cos { { \theta }_{ 4 } } =0\) and similarly \(\sin { { \theta }_{ 1 } } +\sin { { \theta }_{ 2 } } +\sin { { \theta }_{ 3 } } +\sin { { \theta }_{ 4 } } =0\) and i know \({ \theta }_{ 1 }+{ \theta }_{ 2 }+{ \theta }_{ 3 }+{ \theta }_{ 4 }=2\pi \) if that cyclic quadrilateral were to be a square,,then i would have the sum of any two angles as \(\quad \pi \)..but from the two conditions we just got.. \(2\cos { \frac { { \theta }_{ 1 }+{ \theta }_{ 2 } }{ 2 } } \cos { \frac { { \theta }_{ 1 }-{ \theta }_{ 2 } }{ 2 } } =(-2)\cos { \frac { { \theta }_{ 3 }+{ \theta }_{ 4 } }{ 2 } } \cos { \frac { { \theta }_{ 3 }-{ \theta }_{ 4 } }{ 2 } } \)

\(2\sin { \frac { { \theta }_{ 1 }+{ \theta }_{ 2 } }{ 2 } } \cos { \frac { { \theta }_{ 1 }-{ \theta }_{ 2 } }{ 2 } } =(-2)\sin { \frac { { \theta }_{ 1 }+{ \theta }_{ 2 } }{ 2 } } \cos { \frac { { \theta }_{ 3 }-{ \theta }_{ 4 } }{ 2 } } \) we get \({ \tan { \frac { { \theta }_{ 1 }+{ \theta }_{ 2 } }{ 2 } } }^{ 2 }=1\) if it were a square i should get the sum of any two angles as \(\pi \) but that doesn't satisfy the condition we just ended on..so..i conclude the points don't have to make a square...

this is my thinking....i might be wrong..if so,then i would be glad to know the flaw in my approach... Shriya Mandarapu · 2 years, 10 months ago

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THEY ARE THE ROOTS OF THE EQUATION Z^5=0 Sai Krishna Chary · 2 years, 10 months ago

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No, It is not necessary that they should lie on corners of square. Second condition gives points are collinear and first gives they lie on a circle with radius 1. example (1,0) (-1,0) (1/2,root3/2) (-1/2,-root3/2) Manish Mehta · 2 years, 10 months ago

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that z1=z3=-1 and z2=z4=1 Nabil Ahmed · 2 years, 10 months ago

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@Finn Hulse @Rajsuryan Singh @Adrian Neacșu @Harsh Depal @Vishal Sharma @Jack D'Aurizio @Mursalin Habib @Michael Mendrin Thanks for working on the 3 variable case. Care to tackle the 4 variable and 5 variable case? Calvin Lin Staff · 2 years, 10 months ago

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@Calvin Lin I'd love to! I'll get right to work... tomorrow. :D Finn Hulse · 2 years, 10 months ago

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1) Let \(z_1 = a+bi\). Then, \(z_2 = \pm c \pm \sqrt{a^2+b^2-c^2}i\), \(z_3 = -a-bi\), \(z_4 = \mp c \mp \sqrt{a^2+b^2-c^2}i\), \(c \in \mathbb{R}\) (or some permutation of this; WLOG \(z_1\), \(z_2\), \(z_3\) and \(z_4\) are arranged clockwise or counter-clockwise around the origin), i.e. we have that \(z_1\) and \(z_3\) are reflections of each other over the origin, as are \(z_2\) and \(z_4\). Geometrically, the object \(z_1 z_2 z_3 z_4\) is a rectangle centered around the origin. I'll come back to you on 2)! Michael Lee · 2 years, 10 months ago

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@Michael Lee Can you substantiate why \(z_i\) must have that form? I see that \( z_1, z_2 \) are still unrestricted, but why must \( z_3, z_4 \) have that particular form? Calvin Lin Staff · 2 years, 10 months ago

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According to me I think it the four complex numbers can lie on any rectangle Milun Moghe · 2 years, 10 months ago

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@Milun Moghe I mean the vertices of a rectangle Milun Moghe · 2 years, 10 months ago

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@Milun Moghe As with the previous note, the challenge is to prove your claim. Calvin Lin Staff · 2 years, 10 months ago

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\((Z_1*Z_2) / (Z_3*Z_4) =1\) ////no El Kattan · 2 years, 10 months ago

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@El Kattan Hm, is that a sufficient and necessary condition? What does that tell us about the complex numbers? Is there any possible description? Calvin Lin Staff · 2 years, 10 months ago

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i think it's not possible .if condition 1 is true ,then 2 must be false and vice-verse Mamun Uddin · 2 years, 10 months ago

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\(z_{1}, z_{2}, z_{3},\) and \(z_{4}\) are vertices of a rectangle ABCD. Here is the proof(rigorous):

In general, we can prove it for any \(|z| = a(>0)\) rather than 1.

First, you should know that the value of \(|p+q|^2 = 2|p^2|(1 + \cos \theta)\), where \(\theta\) is theta is the angle that complex numbers \(p\) and \(q\) lying on a circle centered about origin subtend at origin.

Let \(\alpha\) be the angle between lines OA and OB, and \(\beta\) be the angle between lines OC and OD,

\(z_{1}+ z_{2} = -(z_{3}+z_{4})\)

Taking modulus and squaring, we get:

\(2|z^2|(1+ \cos \alpha|) = 2|z^2| (1 + \cos \beta)\)

\(\cos \alpha = \cos \beta\) , or \(\alpha = 2 \pi - \beta\) (not possible)

Similarly, if \(\gamma \) and \(\delta\) are angles between \(OA\) and \(OD\) , and \(OB\) and \(OC\) respectively, we can prove that

\(\gamma = \delta\)

Now, as \(\alpha, \beta\), and \(\gamma, \delta\) are pairs of vertically opposite angles, we get that

Lines AOC , BOD are straight, i.e. the diagonals intersect at O.

Thus, all angles of the quadrilateral are angle in semicircle, i.e. \(90^{\circ}\) , and the quadrilateral is rectangle.

Also, it is not necessiraly a square. As a counter example, \(z , \bar{z},-z , -\bar{z}\) always satisfy the given relations, but don't necessarily form a square. Jatin Yadav · 2 years, 10 months ago

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