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#4: Post edited
by
whybecause
·
2022-02-18T16:35:37Z (almost 3 years ago)
-1
- The mistake here is taking $i=\sqrt{-1}$. This is not correct, even though $i^2=-1$.
- How is that possible?!
Because $i$ in this page is NOT meant as the complex number, even though it is similar. $i$ is meant here as a purely symbolic object satisfying an algebraic property. Put another way, all you "get to know" about $i$ is just $i^2=1$ together with some of the other explicitly stated algebraic properties like $ij=k$ and so on. We do not define $\sqrt{-1}$.- If you want you can say that $\sqrt{-1}$ is any number such that its square is $-1$. But in this setting, therefore, there are many such numbers and then $\sqrt{-1}$ is not well-defined.
- ---
- Ok, that's that. Then to directly answer the question
- $$ ijk = (ij)k = kk = -1 $$
- The mistake here is taking $i=\sqrt{-1}$. This is not correct, even though $i^2=-1$.
- How is that possible?!
- Because $i$ in this page is NOT meant as the complex number, even though it is similar. $i$ is meant here as a purely symbolic object satisfying an algebraic property. Put another way, all you "get to know" about $i$ is just $i^2=-1$ together with some of the other explicitly stated algebraic properties like $ij=k$ and so on. We do not define $\sqrt{-1}$.
- If you want you can say that $\sqrt{-1}$ is any number such that its square is $-1$. But in this setting, therefore, there are many such numbers and then $\sqrt{-1}$ is not well-defined.
- ---
- Ok, that's that. Then to directly answer the question
- $$ ijk = (ij)k = kk = -1 $$
#3: Post edited
by
whybecause
·
2022-02-18T16:35:11Z (almost 3 years ago)
corrected -1
The mistake here is taking $i=\sqrt{-1}$. This is not correct, even though $i^2=1$.- How is that possible?!
- Because $i$ in this page is NOT meant as the complex number, even though it is similar. $i$ is meant here as a purely symbolic object satisfying an algebraic property. Put another way, all you "get to know" about $i$ is just $i^2=1$ together with some of the other explicitly stated algebraic properties like $ij=k$ and so on. We do not define $\sqrt{-1}$.
- If you want you can say that $\sqrt{-1}$ is any number such that its square is $-1$. But in this setting, therefore, there are many such numbers and then $\sqrt{-1}$ is not well-defined.
- ---
- Ok, that's that. Then to directly answer the question
- $$ ijk = (ij)k = kk = -1 $$
- The mistake here is taking $i=\sqrt{-1}$. This is not correct, even though $i^2=-1$.
- How is that possible?!
- Because $i$ in this page is NOT meant as the complex number, even though it is similar. $i$ is meant here as a purely symbolic object satisfying an algebraic property. Put another way, all you "get to know" about $i$ is just $i^2=1$ together with some of the other explicitly stated algebraic properties like $ij=k$ and so on. We do not define $\sqrt{-1}$.
- If you want you can say that $\sqrt{-1}$ is any number such that its square is $-1$. But in this setting, therefore, there are many such numbers and then $\sqrt{-1}$ is not well-defined.
- ---
- Ok, that's that. Then to directly answer the question
- $$ ijk = (ij)k = kk = -1 $$
#2: Post edited
by
whybecause
·
2022-02-18T16:29:55Z (almost 3 years ago)
elaboration of i
- The mistake here is taking $i=\sqrt{-1}$. This is not correct, even though $i^2=1$.
- How is that possible?!
Because $i$ is meant here as a purely symbolic object satisfying an algebraic property. Put another way, all you "get to know" about $i$ is just $i^2=1$ together with some of the other explicitly stated algebraic properties like $ij=k$ and so on. We do not define $\sqrt{-1}$.- If you want you can say that $\sqrt{-1}$ is any number such that its square is $-1$. But in this setting, therefore, there are many such numbers and then $\sqrt{-1}$ is not well-defined.
- ---
- Ok, that's that. Then to directly answer the question
- $$ ijk = (ij)k = kk = -1 $$
- The mistake here is taking $i=\sqrt{-1}$. This is not correct, even though $i^2=1$.
- How is that possible?!
- Because $i$ in this page is NOT meant as the complex number, even though it is similar. $i$ is meant here as a purely symbolic object satisfying an algebraic property. Put another way, all you "get to know" about $i$ is just $i^2=1$ together with some of the other explicitly stated algebraic properties like $ij=k$ and so on. We do not define $\sqrt{-1}$.
- If you want you can say that $\sqrt{-1}$ is any number such that its square is $-1$. But in this setting, therefore, there are many such numbers and then $\sqrt{-1}$ is not well-defined.
- ---
- Ok, that's that. Then to directly answer the question
- $$ ijk = (ij)k = kk = -1 $$
#1: Initial revision
by
whybecause
·
2022-02-18T16:28:52Z (almost 3 years ago)
The mistake here is taking $i=\sqrt{-1}$. This is not correct, even though $i^2=1$.
How is that possible?!
Because $i$ is meant here as a purely symbolic object satisfying an algebraic property. Put another way, all you "get to know" about $i$ is just $i^2=1$ together with some of the other explicitly stated algebraic properties like $ij=k$ and so on. We do not define $\sqrt{-1}$.
If you want you can say that $\sqrt{-1}$ is any number such that its square is $-1$. But in this setting, therefore, there are many such numbers and then $\sqrt{-1}$ is not well-defined.
---
Ok, that's that. Then to directly answer the question
$$ ijk = (ij)k = kk = -1 $$