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In the complex number system, why can't 1+1 = 0 ?

by pondzo
Tags: complex, number
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pondzo
#19
Feb19-14, 10:34 PM
P: 71
okay i think i understand what you said, sort of.

Are you saying that 'the square root' symbol is used to refer to only the principle (positive) root: ie √(a2) = |a| where a can be +ve or -ve but the radical implies the absolute (positive) value of it?

what i was getting most confused about was that we can go from ##√(-4) = √(-1)(4) = √(-1)√(4) ## but we cant go from ##√16 = √(-4)(-4) = √(-4)√(-4)## I realise now that its because the property ##√ab = √a√b## holds only if a and b are positive or one of them is negative. It does not hold if BOTH a and b are negative as this leads to inconsistencies.

But what if we weren't to classify these as inconsistencies and hence didn't ban the property √ab = √a√b for a negative a and negative b. could this mean that there would then be solutions to the system of equations (for example): (sorry this may seem off track, but this is why i started considering my original question on this thread)
a - 2b + 3c = -2
-a + b - 2c = 3
2a -b + 3c = 1
putting into a co efficient matrix:
[tex]
\begin{pmatrix}
1 & -2 & 3 & -2\\
-1 & 1 & -2 & 3\\
2 & -1 & 3 & 1
\end{pmatrix}
[/tex]
putting into row echelon form:
[tex]
\begin{pmatrix}
1 & -2 & 3 & -2\\
0 & -1 & 1 & 1\\
0 & 0 & 0 & 8
\end{pmatrix}
[/tex]
the last row says 0=8 which is not possible and leads this system to have no solutions.
what if we said that 0=8 is an identity that is necessary for there to be a solution to this system of equations and provable by (im now assuming that ##√ab = √a√b## for negative a and negative b is not treated as an inconsistency):
0 = -4 + 4
0 = 4i2 + 4
0 = (2i)(2i) + 4
0 = √(4)(-1)√(4)(-1) + 4
0 = √(-4)(-4) + 4
0 = 4 + 4
0 = 8
similarly it can be shown the 0=-8 which yields -8=0=8
now, going back to the matrix in row echelon form, treating 'c' as a free variable the solution space is ##{(-4-c,c-1,c)}##
lets say c = 2, then a=-6 and b=1 and plugging these values into the three equations we get:
a - 2b +3c = -2 <-------- this is true for these values
-a + b - 2c = 3 <-------- this is true for these values
2a - b + 3c = 1
2(-6) - 1 + 3(2) = 1
-7 = 1 <----- since we showed that -8=0=8 this is true also (under the assumptions that i stated)

so what im getting at is; if √ab = √a√b for negative a and b was not treated as an inconsistency and was allowed then could this mean that certain systems of equations that as it stands in modern mathematics have 'No solution' could actually have a solution?
pondzo
#20
Feb19-14, 10:46 PM
P: 71
ehhhh scratch all that i just realised that if -8=0=8 then by the same logic every single number is equal to every other number, which is definitely an inconsistency!!
7777777
#21
Feb20-14, 08:30 AM
P: 18
It makes me wonder, this same problem appears again and again, so I try to put my 2
cents in. I think that the problem lies already at the OP's question: "why can't 1+1=0?"

To me this kind of question means that the OP already knows that the result of his
calculation will be 0, because he wrote it that way. Had he instead asked: "what is 1+1= ?",
his problem might be avoided. It would mean that he had to admit that he does know the
answer. To me these kind of problems have always been very hard to understand, why
in mathematics we are many times calculating although the answer is known? Perhaps
we should use mathematics and calculate only those things whose answers are not known.
pondzo
#22
Feb20-14, 09:17 AM
P: 71
Quote Quote by 7777777 View Post
It makes me wonder, this same problem appears again and again, so I try to put my 2
cents in. I think that the problem lies already at the OP's question: "why can't 1+1=0?"

To me this kind of question means that the OP already knows that the result of his
calculation will be 0, because he wrote it that way. Had he instead asked: "what is 1+1= ?",
his problem might be avoided. It would mean that he had to admit that he does know the
answer. To me these kind of problems have always been very hard to understand, why
in mathematics we are many times calculating although the answer is known? Perhaps
we should use mathematics and calculate only those things whose answers are not known.
if you read my post #19 in this thread, i explicitly state that i was exploring a system of equations
(i then go on to give this system and tell you the problem) which yielded no solutions and this is what spurred me to investigate and ask the original question. so in fact i did not know the answer before i asked the question. and in any case i was wrong, so i didn't know the answer at any point.
7777777
#23
Feb20-14, 10:24 AM
P: 18
Quote Quote by pondzo View Post
if you read my post #19 in this thread, i explicitly state that i was exploring a system of equations
(i then go on to give this system and tell you the problem) which yielded no solutions and this is what spurred me to investigate and ask the original question. so in fact i did not know the answer before i asked the question. and in any case i was wrong, so i didn't know the answer at any point.
Ok, I did not know what you knew, that's why I asked. In your post #15, you arrive
at result a+a= -a+a. Lets not take the last step yet (the last step is that a+a=0).
Instead, lets assume that we meet some aliens who have no knowledge how to calculate with our rules. So that they don't know for example what are a+a and -a+a equal to. We give them two equations and ask them to calculate an answer, the question
is : What is a+a= ? What is a-a= ?
The alien might think that a+a=a-a, is this a correct conclusion? And even if it is, does
it automatically mean that ?=0, that you can introduce the number 0 and does it mean
anything to our alien. I don't know if I am very good at explaining, I hope you understand
at least something.
pondzo
#24
Feb20-14, 10:36 AM
P: 71
we have already established that i was wrong, and i accept that. i don't see what the problem is? and an alien wouldn't know what the operations +, - or even = meant so we would have to teach it our symbols and arithmetic before it could start to answer the question.
pondzo
#25
Feb20-14, 10:36 AM
P: 71
and sorry i missed the part where you asked me what i knew.
Integral
#26
Feb20-14, 12:09 PM
Mentor
Integral's Avatar
P: 7,320
Looks like this thread has run its course, I'll close it with this comment.

Pondzo, it appears that you have not studied the complex number system. I would recommend that either you take a course in Complex Analysis or perhaps find a good book. Complex Variables by Churchill is a good book.


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