Algebra with Complex Numbers & Imaginary Unit

[rbj]

1mile,

i think the wikipedia article is clear about this. i don't know what the problem they have with it.

$-1$ and $+1$ are not equivalent. one is the multiplicative identity and the other is not.

in contrast, $-i$ and $i$ are qualitatively equivalent. there is not one single property that one has that the other does not. but they are not zero, so being negatives of each other, they cannot be equal.

 

[lostcauses10x]

Stuff to catch up on.
A quick scan:

my objection is to teach complex numbers (which i have done in the context of electrical engineering classes) solely as ordered pairs.

I agree.

 

[lostcauses10x]

1MileCrash
I do not know what to say to you. I really don't.
I have encountered this before, and for some reason it just does not get across.
(0+1i) does not = (0-1i).
How do you come to the conclusion you have?

Edited to remove shorthand.

 

[rbj]

I have encountered this before, and for some reason it just does not get across.

there's always hope. he might be able to get it across to you eventually.

(0+1i) does not = (0-1i).
How do you come to the conclusion you have?

please list (using words) a single property that $0 + 1i$ has and that $0 - 1i$ does not have. or vise versa.

can you respond to the Wikipedia article pointed to several times?

 

[R136a1]

please list (using words) a single property that $0 + 1i$ has and that $0 - 1i$ does not have. or vise versa.

That they have the "same" property, doesn't mean that they are equal. That's all he said. Same as a category has the "same" properties as its dual, doesn't mean that they're equal.

 

[HallsofIvy]

they are qualitatively different. only one of those two numbers are the multiplicative identity.



it is no mistake to think of $-i$ and $i$ as qualitatively the same. every property $-i$ has, $+i$ also has.

No, -i has the property that it is equal to -1 times i. i does not have that property!

 

[rbj]

No, -i has the property that it is equal to -1 times i. i does not have that property!

sorry, Halls. epic fail.

replace every occurrence of $-i$ with $i$ (which has the consequence that every occurrence of $i$ is replaced by $-i$) and you will see your mistake.

 

[R136a1]

sorry, Halls. epic fail.

Can we please have a civil discussion without immature stuff like "epic fail". This is not a contest.

 

[rbj]

$-1$ and $+1$ are not equivalent. one is the multiplicative identity and the other is not.

in contrast, $-i$ and $i$ are qualitatively equivalent. there is not one single property that one has that the other does not. but they are not zero, so being negatives of each other, they cannot be equal.

That they have the "same" property,

all of their properties are the same.

doesn't mean that they are equal.

please respond to what i said, not to what i didn't say (in fact what i actually explicitly denied).

That's all he said. Same as a category has the "same" properties as its dual, doesn't mean that they're equal.

i didn't say "equal". i said being that "they are not zero [and] being negatives of each other, they cannot be equal."

but they are equivalent. $i$ and $-i$ are interchangeable (and every single theorem in every single textbook and journal article would continue to be just as valid). and that cannot be said of $1$ and $-1$.

 

[Mark44]

No, -i has the property that it is equal to -1 times i. i does not have that property!

sorry, Halls. epic fail.

What HallsOfIvy wrote is obviously true, so it's difficult to see why you think this is an "epic fail."

replace every occurrence of $-i$ with $i$ (which has the consequence that every occurrence of $i$ is replaced by $-i$) and you will see your mistake.

Since the OP hasn't been back for quite some time, it seems to me that this thread has run its course. I am closing it.