3 ms·

I or j don’t need to originate from Sqrt(-1) ?

gus_massa 11mo ago
In math, officially i is the "root" of x^2+1=0 or to be more precise, C is R[x]/x^2+1, i.e. you take all the polynomials in x and pretend that the polynomials A and B they are equivalent when A-B is a multiple of x^2+1. There is also a construction with matices instead of polynomials. And perhaps others. Each of them are useful in some cases.
- anon291 11mo ago
  X*X + 1 = 0 is a fundamental statement on an algebraic rings behavior with the additive and multiplicative identities and the additive and multiplicative group operations. Namely, it says that the ring contains an element that when multiplied by itself is equal to the additive inverse of the multiplicative identity . Plenty of rings have such an element. You can complete any ring with such an element and call it whatever you want. The use of the term imaginary for it is incredibly unfortunate. There's nothing strange or mystical about it. It's very real. In fact the rational complex numbers are more real than the non complex real numbers
  - itishappy 11mo ago
    > In fact the rational complex numbers are more real than the non complex real numbers Fascinating. Can you say more about this or point me to where I may learn?
    - chunky1994 11mo ago
      Dummit and Foote is the classic abstract Algebra textbook to learn about how to precisely define these. Its treatment of ring theory is very well motivated and easy to grasp
    - anon291 11mo ago
      In general, determining if two arbitrary reals are the same is impossible per the halting problem. People claim to measure 'real' numbers. This is a lie. People can only measure rational numbers. A real number is either a rational or the supremum of some arbitrary set of rationals (perhaps an infinite one). A set is described by whether or not a number is in it. To be able to determine what number is in your set you need to have some sort of decision procedure (a program). However, more real numbers exist than there are possible written programs. Thus, the full set of reals is inexpressible On the other hand, it's very easy to see and measure rational complex numbers with a protractor.
anon291 11mo ago
No not at all. I is just something that behaves as if it is equivalent to negative one (that is, the additive inverse of the multiplicative identity) after combining it with itself in some way. We commonly call this multiplication. If such a thing comes with another operation called addition that behaves similarly to addition and multiplication (i.e. form a ring), then they will behave like i. Geometrically, multiplication by I can be seen as a 90deg rotation of a 2d vector. Complex numbers are simply 2-d coordinates (or rather, they are isomorphic to 2-d coordinates). Nothing special really. Easy to measure with a protractor and ruler. In general there are many algebraic rings with an element that, when multiplied by itself, produces the additive inverse of the multiplicative identity.
itishappy 11mo ago
Easier to see without the square root: i^2 = -1 What action when applied twice results in a sign change? "A 90 turn" is one answer. There are probably others.
- kgwgk 11mo ago
  “A -90 turn” would be another.
  - applied_heat 11mo ago
    Very interesting way of thinking about it, I don’t recall it ever being presented that way before. Thanks
    - anon291 11mo ago
      Everything makes sense when you see I for what it is -- an escape from the number line rotated by ninety degrees. Even the roots of a parabola that doesn't hit the z axis are actually the roots of the ninety degree rotated inverse analogue hitting the imaginary plane. Since the apex of such a parabola is always centered at 0i, the imaginary places it hits are symmetric, explaining why if a + bi is one imaginary root, then a - bi is as well. https://teaching-math.com/unlock-the-secrets-of-complex-roots-with-this-mind-blowing-math-trick/ https://teaching-math.com/unlock-the-secrets-of-complex-root... Again... There is nothing weird about imaginary numbers. They actually make a lot of sense. It's actually insane to only do math in one dimension when our world has three.

In math, officially i is the "root" of x^2+1=0 or to be more precise, C is R[x]/x^2+1, i.e. you take all the polynomials in x and pretend that the polynomials A and B they are equivalent when A-B is a multiple of x^2+1. There is also a construction with matices instead of polynomials. And perhaps others. Each of them are useful in some cases.

anon291 11mo ago
X*X + 1 = 0 is a fundamental statement on an algebraic rings behavior with the additive and multiplicative identities and the additive and multiplicative group operations. Namely, it says that the ring contains an element that when multiplied by itself is equal to the additive inverse of the multiplicative identity . Plenty of rings have such an element. You can complete any ring with such an element and call it whatever you want. The use of the term imaginary for it is incredibly unfortunate. There's nothing strange or mystical about it. It's very real. In fact the rational complex numbers are more real than the non complex real numbers
- itishappy 11mo ago
  > In fact the rational complex numbers are more real than the non complex real numbers Fascinating. Can you say more about this or point me to where I may learn?
  - chunky1994 11mo ago
    Dummit and Foote is the classic abstract Algebra textbook to learn about how to precisely define these. Its treatment of ring theory is very well motivated and easy to grasp
  - anon291 11mo ago
    In general, determining if two arbitrary reals are the same is impossible per the halting problem. People claim to measure 'real' numbers. This is a lie. People can only measure rational numbers. A real number is either a rational or the supremum of some arbitrary set of rationals (perhaps an infinite one). A set is described by whether or not a number is in it. To be able to determine what number is in your set you need to have some sort of decision procedure (a program). However, more real numbers exist than there are possible written programs. Thus, the full set of reals is inexpressible On the other hand, it's very easy to see and measure rational complex numbers with a protractor.

anon291 11mo ago

No not at all. I is just something that behaves as if it is equivalent to negative one (that is, the additive inverse of the multiplicative identity) after combining it with itself in some way. We commonly call this multiplication. If such a thing comes with another operation called addition that behaves similarly to addition and multiplication (i.e. form a ring), then they will behave like i. Geometrically, multiplication by I can be seen as a 90deg rotation of a 2d vector. Complex numbers are simply 2-d coordinates (or rather, they are isomorphic to 2-d coordinates). Nothing special really. Easy to measure with a protractor and ruler. In general there are many algebraic rings with an element that, when multiplied by itself, produces the additive inverse of the multiplicative identity.

itishappy 11mo ago

Easier to see without the square root: i^2 = -1 What action when applied twice results in a sign change? "A 90 turn" is one answer. There are probably others.

kgwgk 11mo ago
“A -90 turn” would be another.
- applied_heat 11mo ago
  Very interesting way of thinking about it, I don’t recall it ever being presented that way before. Thanks
  - anon291 11mo ago
    Everything makes sense when you see I for what it is -- an escape from the number line rotated by ninety degrees. Even the roots of a parabola that doesn't hit the z axis are actually the roots of the ninety degree rotated inverse analogue hitting the imaginary plane. Since the apex of such a parabola is always centered at 0i, the imaginary places it hits are symmetric, explaining why if a + bi is one imaginary root, then a - bi is as well. https://teaching-math.com/unlock-the-secrets-of-complex-roots-with-this-mind-blowing-math-trick/ https://teaching-math.com/unlock-the-secrets-of-complex-root... Again... There is nothing weird about imaginary numbers. They actually make a lot of sense. It's actually insane to only do math in one dimension when our world has three.