Complex numbers, plane and geometry

In summary, the conversation discusses three complex numbers and their relationship to a set of propositions. The propositions include an equilateral triangle, a quadratic equation, and the sum of squares. The conversation also mentions a hint that involves constructing another equilateral triangle. The conversation does not come to a definitive conclusion but raises questions about the assumptions and approach being used.
  • #1
90nizam
2
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Homework Statement


There are three complex numbers a, b and c. Show that these propositions are equals.
1. ABC (triangle from the three points in complex plane) is equilateral (T1).
2. j or j2 is the solution for az2 + bz + c = 0.
3. a2 + b2 + c2 = ab + bc + ca


Homework Equations


There is a hint. Equilateral triangles made from the bases AB, BC, and CA have centres of gravity from which we can construct another equilateral triangle (T2).


The Attempt at a Solution


T1 and T2 are equal triangles. They have the same heights and sides. I've tried to use the equation to solve quadratic equation (quadratic formula) and assumed j is a solution, hence j2 is compliment of j or [tex]\bar{j}[/tex]. I found ac-3b=1. I have no idea how to use the equation. Is my assumption correct? Or my approach to the question is wrong?
 
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  • #2
Is this an engineering class? That is, is "j" the imaginary unit, j2= -1?
 
  • #3
This is a mathematic class and we use i as the imaginary unit. I don't think that the teacher mistyped it.
 

What are complex numbers?

Complex numbers are numbers that have both a real and an imaginary component. They are expressed in the form a + bi, where a is the real part and bi is the imaginary part with the imaginary number i being equal to the square root of -1.

How are complex numbers represented on a plane?

Complex numbers are represented on a plane called the complex plane. The horizontal axis represents the real numbers and the vertical axis represents the imaginary numbers. The point where these two axes intersect is the origin, and each complex number can be plotted as a point on this plane.

How do you add and subtract complex numbers?

To add or subtract complex numbers, you simply add or subtract the real parts and the imaginary parts separately. For example, (3 + 2i) + (1 + 4i) = (3 + 1) + (2i + 4i) = 4 + 6i.

What is the modulus of a complex number?

The modulus of a complex number is its distance from the origin on the complex plane. It is calculated using the Pythagorean theorem as the square root of the sum of the squares of the real and imaginary parts. For example, the modulus of 3 + 4i would be √(3^2 + 4^2) = √25 = 5.

How are complex numbers used in geometry?

Complex numbers are used in geometry to represent points in the complex plane. They can also be used to describe transformations, such as rotations and translations. Additionally, the complex plane can be used to visualize and solve geometric problems, such as finding the midpoint or distance between two points.

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