4 ms·

You miss a key part of complex numbers if you think of them as just vectors: they are a field.

by nathan_compton 4y ago

adammarples 4y ago
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deleted 4y ago
[deleted]
dr_dshiv 4y ago
Because they are separate but interacting with real numbers? I don’t understand.
- tgv 4y ago
  Multiplying vectors differs from multiplying complex numbers.
- justin_ 4y ago
  He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0] You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity. You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1]. [0] https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra https://ncatlab.org/nlab/show/normed+division+algebra
  - nathan_compton 4y ago
    Exactly. But I want to point out that the field character of the complex numbers is not some incidental quality which happens to distinguish them from 2-vectors. Its absolutely essential to their mathematical character and usage and it also distinguishes them from other complexes we might want to form that behave in a real number like fashion. For instance, there is no way to form a field over the three vectors. In general, one has to give up more and more structure as the dimensions go up. I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).
mort96 4y ago
A vector field?
- C-x_C-f 4y ago
  It's a different concept [0], regrettably the word "field" is vastly overworked in math (and physics) [0] https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics)
  - deleted 4y ago
    [deleted]
0xBABAD00C 4y ago
And not just a field, but the algebraic closure of real numbers.

You miss a key part of complex numbers if you think of them as just vectors: they are a field.

adammarples 4y ago

deleted 4y ago

[deleted]

dr_dshiv 4y ago

Because they are separate but interacting with real numbers? I don’t understand.

tgv 4y ago
Multiplying vectors differs from multiplying complex numbers.
justin_ 4y ago
He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0] You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity. You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1]. [0] https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra https://ncatlab.org/nlab/show/normed+division+algebra
- nathan_compton 4y ago
  Exactly. But I want to point out that the field character of the complex numbers is not some incidental quality which happens to distinguish them from 2-vectors. Its absolutely essential to their mathematical character and usage and it also distinguishes them from other complexes we might want to form that behave in a real number like fashion. For instance, there is no way to form a field over the three vectors. In general, one has to give up more and more structure as the dimensions go up. I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).

mort96 4y ago

A vector field?

C-x_C-f 4y ago
It's a different concept [0], regrettably the word "field" is vastly overworked in math (and physics) [0] https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics)
- deleted 4y ago
  [deleted]

0xBABAD00C 4y ago

And not just a field, but the algebraic closure of real numbers.