Do complex roots have a physical representation on a curve?

In summary: The roots are 2i and -2i, which exist on the curve in the complex plane.In summary, the conversation discusses the representation of the function y=x^2 -4 on a graph, and how the roots of the function can be found algebraically. The concept of complex numbers is introduced and it is explained that the roots of the function y=x^2+4 do not intersect with the x-axis in the real plane, but can be found in the complex plane. The idea of the curve having another dimension is also mentioned, but it is explained that it is difficult to draw in the real plane. Overall, the conversation provides a detailed explanation of the relationship between real and complex numbers in terms of graphing functions.
  • #1
mohammed El-Kady
32
2
If we have y=x^2 -4. This is represented by curve intersect x-axis at (-2, 0) and (2, 0) or if we wish to find it algebraically we set y =0 then we solve it. The roots must lie on the curve.
when y=x^2+4 the roots are 2i and -2i "complex" consequently there is no intersection with x-axis, so where the points (2i, 0) and (-2i, 0) on the curve?
Does it have another dimension or imagine dim.? if yes, is that mean the curve is solid?
 
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  • #2
mohammed El-Kady said:
If we have y=x^2 -4. This is represented by curve intersect x-axis at (-2, 0) and (2, 0) or if we wish to find it algebraically we set y =0 then we solve it. The roots must lie on the curve.
when y=x^2+4 the roots are 2i and -2i "complex" consequently there is no intersection with x-axis, so where the points (2i, 0) and (-2i, 0) on the curve?
Does it have another dimension or imagine dim.? if yes, is that mean the curve is solid?
You'll have to say what you are talking about. On the one hand you talked about the graphs ##\{\,(x,y)\in \mathbb{R}^2\,|\,y=x^2\pm 4\,\}##, which are parabolas in the Euclidean plane, one which crosses the ##x-##axis twice, and one which does not.

Now you brought in complex numbers. That is another problem, namely how real polynomials in one indeterminate split over the complex numbers. This is an algebraic theorem and pure abstract algebra. However, you took this now complex situation and asked about its real correspondence? There is none. The inclusion ##\mathbb{R} \subsetneq \mathbb{C} ## is a strict one. If you consider the complex numbers as a two dimensional real vector space, then the new function ##f\, : \, \mathbb{C} \longrightarrow \mathbb{C}## with ##f(z)=z^2+4## is a different one, and its graph has four real dimensions, making it difficult to draw.
 
  • #3
fresh_42 said:
You'll have to say what you are talking about. On the one hand you talked about the graphs ##\{\,(x,y)\in \mathbb{R}^2\,|\,y=x^2\pm 4\,\}##, which are parabolas in the Euclidean plane, one which crosses the ##x-##axis twice, and one which does not.

Now you brought in complex numbers. That is another problem, namely how real polynomials in one indeterminate split over the complex numbers. This is an algebraic theorem and pure abstract algebra. However, you took this now complex situation and asked about its real correspondence? There is none. The inclusion ##\mathbb{R} \subsetneq \mathbb{C} ## is a strict one. If you consider the complex numbers as a two dimensional real vector space, then the new function ##f\, : \, \mathbb{C} \longrightarrow \mathbb{C}## with ##f(z)=z^2+4## is a different one, and its graph has four real dimensions, making it difficult to draw.
thanks for your explanation i just need to know where the roots exists on the curve
 
  • #4
The imaginary roots do not exist on the real curve. They exist in the complex hyperplane, if the function is considered to be complex, in which case the "curve" is a there dimensional hyperspace of a four dimensional space.

The roots "exist" only in the complex extension of the real numbers, as algebraic entities in ##\mathbb{R}[x]/\langle x^2+1 \rangle##
 
  • #6
mfb said:
If you plot ##|x^4+4|## for x in the complex plane then you'll see it is 0 at +2i and -2i. This is not in the real line as the roots are not real. They are complex.
WolframAlpha can plot the real and imaginary part
Thank you for the valuable graphs, its explains too much information for me
 
  • #7
The roots exist in the curve ##f(z)=z^4+4 ## . Note, as someone said, ##x^4 \pm 4 ## is a subspace of ##z^4+4 ##. The map ##f(z)## goes from ##\mathbb R^2 \rightarrow \mathbb R^2 ##.
 

Related to Do complex roots have a physical representation on a curve?

1. What are complex roots and how do they differ from real roots?

Complex roots are solutions to a polynomial equation that involve the imaginary number i, which is equal to the square root of -1. They differ from real roots in that they cannot be graphed on a traditional x-y plane, as they involve both a real and imaginary component.

2. Can complex roots be represented on a curve?

No, complex roots cannot be represented on a curve in the traditional sense. This is because they involve both a real and imaginary component, making it impossible to plot them on a two-dimensional graph. However, they can be represented using other mathematical tools such as the Argand diagram.

3. Are complex roots important in real-world applications?

Yes, complex roots are important in many fields of science and engineering, including electrical engineering, quantum mechanics, and signal processing. They are used to model and understand systems that involve oscillations, such as electrical circuits and quantum mechanical systems.

4. How do complex roots affect the behavior of a curve?

Complex roots can affect the behavior of a curve in several ways. For instance, they can cause the curve to oscillate or spiral, depending on the nature of the complex roots. In some cases, complex roots can also result in a curve having multiple branches or loops.

5. Are there any physical representations of complex roots on a curve?

No, there are no physical representations of complex roots on a curve in the traditional sense. However, as mentioned earlier, they can be represented using mathematical tools such as the Argand diagram, which can help visualize and understand the behavior of the curve. Additionally, complex roots can also have physical implications in real-world systems, such as the resonance of an electrical circuit.

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