4 ms·

It seems to me that i, j, and k are three things that have the same property, not three identical things.

handrous 5y ago
TIL that not every imaginary number is just "i" with optional scaling applied. Was taught "the square root of negative one is i".
- fault1 5y ago
  i wish i was taught in terms of euler' formula, which is probably the most useful: https://en.wikipedia.org/wiki/Euler%27s_formula https://en.wikipedia.org/wiki/Euler%27s_formula
- Twisol 5y ago
  That's a heck of a great observation. In the complex numbers, every imaginary number is, indeed, just "i" scaled. But the quaternions are like three copies of the complex numbers glued together; you have multiple kinds of "imaginary", each with their own unit -- like "i", but now also copies of it, "j" and "k". To bring this somewhere more familiar, you can probably imagine the real number line as a physical line stretching off to infinity. This line has a point called "1". Now if we take two more copies of this line, they each have their own point called "1" -- a different one for each line. And if you stick all three of these lines together as a three-dimensional set of axes, you get a world in which you have three 1s coexisting. We just say "in the X direction" to be clear about which we're talking about, or group them together in (x, y, z) triples -- in which case X's 1 is called (1, 0, 0). The situation is the same for quaternions -- we glue one real line together with three copies of the imaginary line. So we get three different i's, and we give them different names to distinguish them.
- mkl 5y ago
  Unfortunately that's not even correct for complex numbers: i is a square root of -1, and -i is the other. Complex numbers are really nice in that every non-zero complex number has two square roots, three cube roots, four fourth roots, etc. A better definitional statement is the other way around: i has the property that i^2 = -1.

TIL that not every imaginary number is just "i" with optional scaling applied. Was taught "the square root of negative one is i".

fault1 5y ago
i wish i was taught in terms of euler' formula, which is probably the most useful: https://en.wikipedia.org/wiki/Euler%27s_formula https://en.wikipedia.org/wiki/Euler%27s_formula
Twisol 5y ago
That's a heck of a great observation. In the complex numbers, every imaginary number is, indeed, just "i" scaled. But the quaternions are like three copies of the complex numbers glued together; you have multiple kinds of "imaginary", each with their own unit -- like "i", but now also copies of it, "j" and "k". To bring this somewhere more familiar, you can probably imagine the real number line as a physical line stretching off to infinity. This line has a point called "1". Now if we take two more copies of this line, they each have their own point called "1" -- a different one for each line. And if you stick all three of these lines together as a three-dimensional set of axes, you get a world in which you have three 1s coexisting. We just say "in the X direction" to be clear about which we're talking about, or group them together in (x, y, z) triples -- in which case X's 1 is called (1, 0, 0). The situation is the same for quaternions -- we glue one real line together with three copies of the imaginary line. So we get three different i's, and we give them different names to distinguish them.
mkl 5y ago
Unfortunately that's not even correct for complex numbers: i is a square root of -1, and -i is the other. Complex numbers are really nice in that every non-zero complex number has two square roots, three cube roots, four fourth roots, etc. A better definitional statement is the other way around: i has the property that i^2 = -1.