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The contents of the Real and imaginary parts page were merged into Complex number on December 2011. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 10 sections are present. |
The article currently uses the calligraphic mathcal
font for the Re and Im operators:
While not exactly wrong this strikes me as unusal and unconventional, and I think
is much more common. – Tea2min ( talk) 10:10, 16 April 2021 (UTC)
This article could use a secton on the history of the applications of complex numbers (which I myself am completely UNqualified to develop). For example, I came to this article, out of curiosity, to learn how and when complex numbers were first used to describe electrical impedance, and other electromagnetic phenomena. (I still do not know...) Acwilson9 ( talk) 05:56, 9 January 2022 (UTC)
An editor has identified a potential problem with the redirect
Complex getal and has thus listed it
for discussion. This discussion will occur at
Wikipedia:Redirects for discussion/Log/2022 February 15#Complex getal until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ~~~~
User:1234qwer1234qwer4 (
talk) 20:28, 15 February 2022 (UTC)
An editor has identified a potential problem with the redirect
Nombre complexe and has thus listed it
for discussion. This discussion will occur at
Wikipedia:Redirects for discussion/Log/2022 February 15#Nombre complexe until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ~~~~
User:1234qwer1234qwer4 (
talk) 20:29, 15 February 2022 (UTC)
I'm looking for a better source for the geometric interpretation of multiplication, preferably a textbook. This interpretation seems to be well known e.g. it featured in this desmos course by Luke Walsh on complex numbers: #REDIRECT [ [1]], Grant Sanders heavily uses in his video on complex numbers: #REDIRECT [ [2]]. Grant refers to #REDIRECT [ [3]] by Ben Sparks. But I did not find a text book that features it. The only source beside these I found was deemed not reliable enough.
Note: The illustration below for and by Luke Walsh distributes to , i.e. , while Grant Sanderson and Ben Sparks in their visualizations distribute to , i.e. , but due to commutativity this does really matter. (In other words their presentations of match mine and Luke Walsh's of .)
This is a bit (together with two pictures) I'd like to add in the section Complex_number#Multiplication_and_square.
Happy for any suggestion. Qerez ( talk) 11:31, 20 February 2022 (UTC)
Aside from a lazy reversion of several improvements in my edits, we're back to an opening sentence that's unclear:
It needs to be clearly expressed, even if experts can wade through it to extract the correct meaning.
Tony (talk) 08:08, 4 July 2022 (UTC)
For what it’s worth, I think this lead section is pretty mediocre, suffering from a common problem of math textbooks of elevating formal definitions ahead of meaning. The #1 key thing to know about the complex numbers is that while multiplication by a "real number" represents scaling, multiplication by a complex number represents both scaling and planar rotation (Needham calls this an "amplitwist"). Here, the words “rotate”, “rotation”, etc. don’t occur until 1200 words into the article, and even then the explanation is that multiplication is "adding the angles" (To lay readers, this does not obviously and immediately come across as meaning rotation.) A proper explanation doesn’t come until about 4000 words into the article. The lead section says that multiplication is "a similarity" with a hyperlink: this is not remotely lay-accessible; most readers are going to skip right past that without understanding what is meant.
The #2 key thing to know about the imaginary unit i (after #1 knowing that it squares to –1) is that it represents some kind of movement or position at a right angle from the "straight" real-number direction. But words like "perpendicular", "orthogonal", "right angle" don’t show up anywhere here, and again, multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin
doesn’t show up until 1200 words in.
The lead image here is also not sufficient for the article. Just showing that a complex number can be pictured as a 2-dimensional Euclidean point or an arrow from the origin (representing a Euclidean planar vector) in a diagram with axes labeled Re and Im instead of x and y doesn’t explain anything. The key geometrical features of complex numbers are that they add tip-to-tail like a geometric vector and that they multiply by “amplitwisting”. Both of these key features can be easily depicted in a diagram or two.
Even when getting to formalities, this article somewhat picks sides. Defining complex numbers as a quotient ring is not wrong, but it is just one of many possible definitions. The definition from linear algebra (a certain class of matrices), the definition from geometric algebra (the even sub-algebra of the geometric algebra of the Euclidean plane), and others are equally valid. As Clifford (
and Grassmann!) figured out, a complex number is best thought of conceptually as the ratio of two planar vectors.
Grassmann (1844): “From this all imaginary expressions now acquire a purely ge- ometric meaning, and can be described by geometric constructions.... it is likewise now evident how, according to the meaning of the imaginaries thus discovered, one can derive the laws of analysis in the plane; however it is not possible to derive the laws for space as well by means of imaginaries. In addition there are general difficulties in considering the
angle in space, for the solution of which I have not yet had sufficient leisure.”
Wikipedia math articles about basic topics should try to lead with a lay-accessible definition/explanation, ideally with a figure that gets the main point(s) across, and then follow up (possibly soon) after with the formalities. – jacobolus (t) 12:01, 4 July 2022 (UTC)
Before discussing lead sentences, I would like to hear your opinion on whether to merge imaginary numbers into this page. If we didn't merge, I think we can omit some explanations about imaginary numbers. Also, I think the section of Relations and operations will be explained before section of visualization.-- SilverMatsu ( talk) 15:36, 4 July 2022 (UTC)
Dear D.Lazard, your
From Google Gemini:
Clarifying Isomorphism and Topological Properties:
Isomorphism in the context of fields refers to structural equivalence. Two fields are considered isomorphic if there exists a one-to-one correspondence between their elements that preserves all the field operations (addition, subtraction, multiplication, and division) under the corresponding operations in the other field.
Topological properties, such as connectedness and local compactness, are distinct characteristics from the purely algebraic structure of a field. These properties are related to the way elements in a field are "close" to each other and how sets of elements within the field can be organized.
In the case of constructing the complex numbers, using different irreducible polynomials (like X^2 + a and X^2 + b) leads to isomorphic fields in the sense that their algebraic structure (addition, subtraction, multiplication, and division) is identical. However, these isomorphic fields might not inherit the same topological properties from the real numbers when constructing them as quotient fields.
Example:
- As we discussed previously, the field of complex numbers (C) constructed using the polynomial X^2 + 1 exhibits both connectedness and local compactness as a topological field.
- However, if we construct a field using a different irreducible polynomial, like X^2 + 2, the resulting quotient field would be isomorphic to the complex numbers in terms of its algebraic structure but would not be connected or locally compact.
I haven't taken a math course for 40 years now, and this surprises me, but I don't find it implausible. I was taught that X^2 + 1 was not algebraically special, and I thought that meant everything, with no consideration that topology might think different.
Bear in mind that yesterday Gemini insisted:
As a result of this shift, the Euler gamma function doesn't directly match the factorial values at positive integers. For example:
- Γ(1) = Γ(2) = 1, while 1! = 1.
- Γ(2) = Γ(3) = 2, while 2! = 2.
- Γ(3) = Γ(4) = 6, while 3! = 6.
This nonsense continued until I confronted it rather sternly:
Your error runs deeper than that. Somehow you have processed the true statement that Gamma(n) = (n-1)! as implying that Gamma(n) = Gamma(n-1) which immediately leads to nonsense. From Legendre, the correct recurrence for Gamma is that Gamma(z+1) = z * Gamma(z).
At which point Gemini found the light:
You're absolutely right, and I apologize for the repeated errors in my previous responses. My understanding of the relationship between the Gamma function and factorial function was flawed, leading to inaccurate statements and misleading information.
My Mistakes:
- I incorrectly stated that the Euler Gamma Function (Γ(z)) and the factorial function (n!) directly interpolate each other, leading to the erroneous claim that Γ(2) = 1.
- I falsely implied that the true statement Γ(n) = (n-1)! somehow implied Γ(n) = Γ(n-1), which is not only incorrect but also mathematically illogical.
Correct Information:
- The Euler Gamma Function extends the concept of factorial to non-integer and complex numbers. It does not directly interpolate the factorial function at positive integers due to a specific definition involving a shift in the argument (Γ(n) ≠ n! for most positive integers n).
- The correct recurrence relation for the Gamma function is Γ(z + 1) = z * Γ(z), as you accurately pointed out, referencing Legendre's work.
If Gemini's topological claims rise above the hallucination floor, it seems like something this article might profit from pointing out as a footnote-ish addition to the bottom of the relevant section concerning the extra topological specialness of X2 + 1. — MaxEnt 01:46, 28 February 2024 (UTC)
This article and the articles for real number, rational number, integer, and possibly others, have a large image of the symbol (in this article, ) even though the symbol is defined in the lead. This is redundant. I think these large images should be removed.— Anita5192 ( talk) 14:18, 26 March 2024 (UTC)
This article is very good from a mathematical perspective. Definitions, algebraic rules and so on are well explained. But I miss more emphasis on WHY. It would be much more inspiring for a reader to go through all those algebraic rules if the reader had a clue about what complex numbers can be used for. There isn't even a mention about a pendulum here!?
Complex numbers are often useful when a phenomenon can change between manifesting itself in two different ways.
One example is a pendulum where energy can be kinetic energy when the pendulum is moving fast at the bottom of its trajectory and positional energy at the highest points of a its trajectory. As the pendulum swings back and forth, the same energy changes in revealing itself in two different dimensions. One of the dimensions can be called "real", and the other then becomes "imaginary".
Another example is in an oscillating electrical circuit, where the same energy can change between revealing itself as a current flowing through a conductor or as a charge in a capacitor.
Another example is sound waves, where a sound measured at one point can change between revealing itself as a air pressure deviation and as a motion in the air.
Thus, two different manifestations or aspects of something can be modelled using just one complex number.
(If something like this had been told me when I started learning about complex number, this would have made my learning easier and more fun) Joreberg ( talk) 21:05, 14 April 2024 (UTC)
This
level-3 vital article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
The contents of the Real and imaginary parts page were merged into Complex number on December 2011. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 10 sections are present. |
The article currently uses the calligraphic mathcal
font for the Re and Im operators:
While not exactly wrong this strikes me as unusal and unconventional, and I think
is much more common. – Tea2min ( talk) 10:10, 16 April 2021 (UTC)
This article could use a secton on the history of the applications of complex numbers (which I myself am completely UNqualified to develop). For example, I came to this article, out of curiosity, to learn how and when complex numbers were first used to describe electrical impedance, and other electromagnetic phenomena. (I still do not know...) Acwilson9 ( talk) 05:56, 9 January 2022 (UTC)
An editor has identified a potential problem with the redirect
Complex getal and has thus listed it
for discussion. This discussion will occur at
Wikipedia:Redirects for discussion/Log/2022 February 15#Complex getal until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ~~~~
User:1234qwer1234qwer4 (
talk) 20:28, 15 February 2022 (UTC)
An editor has identified a potential problem with the redirect
Nombre complexe and has thus listed it
for discussion. This discussion will occur at
Wikipedia:Redirects for discussion/Log/2022 February 15#Nombre complexe until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ~~~~
User:1234qwer1234qwer4 (
talk) 20:29, 15 February 2022 (UTC)
I'm looking for a better source for the geometric interpretation of multiplication, preferably a textbook. This interpretation seems to be well known e.g. it featured in this desmos course by Luke Walsh on complex numbers: #REDIRECT [ [1]], Grant Sanders heavily uses in his video on complex numbers: #REDIRECT [ [2]]. Grant refers to #REDIRECT [ [3]] by Ben Sparks. But I did not find a text book that features it. The only source beside these I found was deemed not reliable enough.
Note: The illustration below for and by Luke Walsh distributes to , i.e. , while Grant Sanderson and Ben Sparks in their visualizations distribute to , i.e. , but due to commutativity this does really matter. (In other words their presentations of match mine and Luke Walsh's of .)
This is a bit (together with two pictures) I'd like to add in the section Complex_number#Multiplication_and_square.
Happy for any suggestion. Qerez ( talk) 11:31, 20 February 2022 (UTC)
Aside from a lazy reversion of several improvements in my edits, we're back to an opening sentence that's unclear:
It needs to be clearly expressed, even if experts can wade through it to extract the correct meaning.
Tony (talk) 08:08, 4 July 2022 (UTC)
For what it’s worth, I think this lead section is pretty mediocre, suffering from a common problem of math textbooks of elevating formal definitions ahead of meaning. The #1 key thing to know about the complex numbers is that while multiplication by a "real number" represents scaling, multiplication by a complex number represents both scaling and planar rotation (Needham calls this an "amplitwist"). Here, the words “rotate”, “rotation”, etc. don’t occur until 1200 words into the article, and even then the explanation is that multiplication is "adding the angles" (To lay readers, this does not obviously and immediately come across as meaning rotation.) A proper explanation doesn’t come until about 4000 words into the article. The lead section says that multiplication is "a similarity" with a hyperlink: this is not remotely lay-accessible; most readers are going to skip right past that without understanding what is meant.
The #2 key thing to know about the imaginary unit i (after #1 knowing that it squares to –1) is that it represents some kind of movement or position at a right angle from the "straight" real-number direction. But words like "perpendicular", "orthogonal", "right angle" don’t show up anywhere here, and again, multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin
doesn’t show up until 1200 words in.
The lead image here is also not sufficient for the article. Just showing that a complex number can be pictured as a 2-dimensional Euclidean point or an arrow from the origin (representing a Euclidean planar vector) in a diagram with axes labeled Re and Im instead of x and y doesn’t explain anything. The key geometrical features of complex numbers are that they add tip-to-tail like a geometric vector and that they multiply by “amplitwisting”. Both of these key features can be easily depicted in a diagram or two.
Even when getting to formalities, this article somewhat picks sides. Defining complex numbers as a quotient ring is not wrong, but it is just one of many possible definitions. The definition from linear algebra (a certain class of matrices), the definition from geometric algebra (the even sub-algebra of the geometric algebra of the Euclidean plane), and others are equally valid. As Clifford (
and Grassmann!) figured out, a complex number is best thought of conceptually as the ratio of two planar vectors.
Grassmann (1844): “From this all imaginary expressions now acquire a purely ge- ometric meaning, and can be described by geometric constructions.... it is likewise now evident how, according to the meaning of the imaginaries thus discovered, one can derive the laws of analysis in the plane; however it is not possible to derive the laws for space as well by means of imaginaries. In addition there are general difficulties in considering the
angle in space, for the solution of which I have not yet had sufficient leisure.”
Wikipedia math articles about basic topics should try to lead with a lay-accessible definition/explanation, ideally with a figure that gets the main point(s) across, and then follow up (possibly soon) after with the formalities. – jacobolus (t) 12:01, 4 July 2022 (UTC)
Before discussing lead sentences, I would like to hear your opinion on whether to merge imaginary numbers into this page. If we didn't merge, I think we can omit some explanations about imaginary numbers. Also, I think the section of Relations and operations will be explained before section of visualization.-- SilverMatsu ( talk) 15:36, 4 July 2022 (UTC)
Dear D.Lazard, your
From Google Gemini:
Clarifying Isomorphism and Topological Properties:
Isomorphism in the context of fields refers to structural equivalence. Two fields are considered isomorphic if there exists a one-to-one correspondence between their elements that preserves all the field operations (addition, subtraction, multiplication, and division) under the corresponding operations in the other field.
Topological properties, such as connectedness and local compactness, are distinct characteristics from the purely algebraic structure of a field. These properties are related to the way elements in a field are "close" to each other and how sets of elements within the field can be organized.
In the case of constructing the complex numbers, using different irreducible polynomials (like X^2 + a and X^2 + b) leads to isomorphic fields in the sense that their algebraic structure (addition, subtraction, multiplication, and division) is identical. However, these isomorphic fields might not inherit the same topological properties from the real numbers when constructing them as quotient fields.
Example:
- As we discussed previously, the field of complex numbers (C) constructed using the polynomial X^2 + 1 exhibits both connectedness and local compactness as a topological field.
- However, if we construct a field using a different irreducible polynomial, like X^2 + 2, the resulting quotient field would be isomorphic to the complex numbers in terms of its algebraic structure but would not be connected or locally compact.
I haven't taken a math course for 40 years now, and this surprises me, but I don't find it implausible. I was taught that X^2 + 1 was not algebraically special, and I thought that meant everything, with no consideration that topology might think different.
Bear in mind that yesterday Gemini insisted:
As a result of this shift, the Euler gamma function doesn't directly match the factorial values at positive integers. For example:
- Γ(1) = Γ(2) = 1, while 1! = 1.
- Γ(2) = Γ(3) = 2, while 2! = 2.
- Γ(3) = Γ(4) = 6, while 3! = 6.
This nonsense continued until I confronted it rather sternly:
Your error runs deeper than that. Somehow you have processed the true statement that Gamma(n) = (n-1)! as implying that Gamma(n) = Gamma(n-1) which immediately leads to nonsense. From Legendre, the correct recurrence for Gamma is that Gamma(z+1) = z * Gamma(z).
At which point Gemini found the light:
You're absolutely right, and I apologize for the repeated errors in my previous responses. My understanding of the relationship between the Gamma function and factorial function was flawed, leading to inaccurate statements and misleading information.
My Mistakes:
- I incorrectly stated that the Euler Gamma Function (Γ(z)) and the factorial function (n!) directly interpolate each other, leading to the erroneous claim that Γ(2) = 1.
- I falsely implied that the true statement Γ(n) = (n-1)! somehow implied Γ(n) = Γ(n-1), which is not only incorrect but also mathematically illogical.
Correct Information:
- The Euler Gamma Function extends the concept of factorial to non-integer and complex numbers. It does not directly interpolate the factorial function at positive integers due to a specific definition involving a shift in the argument (Γ(n) ≠ n! for most positive integers n).
- The correct recurrence relation for the Gamma function is Γ(z + 1) = z * Γ(z), as you accurately pointed out, referencing Legendre's work.
If Gemini's topological claims rise above the hallucination floor, it seems like something this article might profit from pointing out as a footnote-ish addition to the bottom of the relevant section concerning the extra topological specialness of X2 + 1. — MaxEnt 01:46, 28 February 2024 (UTC)
This article and the articles for real number, rational number, integer, and possibly others, have a large image of the symbol (in this article, ) even though the symbol is defined in the lead. This is redundant. I think these large images should be removed.— Anita5192 ( talk) 14:18, 26 March 2024 (UTC)
This article is very good from a mathematical perspective. Definitions, algebraic rules and so on are well explained. But I miss more emphasis on WHY. It would be much more inspiring for a reader to go through all those algebraic rules if the reader had a clue about what complex numbers can be used for. There isn't even a mention about a pendulum here!?
Complex numbers are often useful when a phenomenon can change between manifesting itself in two different ways.
One example is a pendulum where energy can be kinetic energy when the pendulum is moving fast at the bottom of its trajectory and positional energy at the highest points of a its trajectory. As the pendulum swings back and forth, the same energy changes in revealing itself in two different dimensions. One of the dimensions can be called "real", and the other then becomes "imaginary".
Another example is in an oscillating electrical circuit, where the same energy can change between revealing itself as a current flowing through a conductor or as a charge in a capacitor.
Another example is sound waves, where a sound measured at one point can change between revealing itself as a air pressure deviation and as a motion in the air.
Thus, two different manifestations or aspects of something can be modelled using just one complex number.
(If something like this had been told me when I started learning about complex number, this would have made my learning easier and more fun) Joreberg ( talk) 21:05, 14 April 2024 (UTC)