in the following i will demonstrate a 'proof' that 1+1=0 1+1=√1 +1 =√(-1)(-1) +1 = (√(-1))(√(-1)) +1 = (i)(i) +1 = i^{2} +1 = -1 + 1 = 0 I know I'm not the first to come up with this 'proof', and i have been told that the problem lies with splitting the radical between the 2nd and 3rd lines of working. But in the complex number system there is no problem with going from, say, √(-4) = √(-1)(4) = 2i so why is there a problem with the above working, isn't it the same line of thought? thanks, Michael.
yes, which is why i asked the question. it just seemed mathematically sound to me so i wanted to see why we disregard it. But it seems i have a lot to catch up on with complex exponentiation!
Ah ... yeah, I see from rereading your question that you aren't quite saying that you believe it, just that you are puzzled by it.
The problem is that for complex numbers, √(a+bi) has two complex roots. With real numbers, one is positive and the other is negative. We can assign the positive one to equal the positive square root. However, with complex numbers it isn't so simple. A complex root is neither negative nor positive, sense it doesn't make any sense to say a complex number is positive. So, it doesn't make sense to have a "positive" square root. This is where the issue arises.
Here's the explanation: The fun thing is you can go 'crazy' with the rule if you know it. say: ##i^{-i} = e^{\frac{\pi}{2}}##
okay, why are we allowed to go from ((4)*(-1))^{0.5} to (4)^{0.5}*(-1)^{0.5} (which is equal to 2i) but we aren't allowed to go from ((-1)(-1))^{0.5} to (-1)^{0.5}*(-1)^{0.5}?
and on another note, in my most recent post, isn't the first example a negative number and the second a positive (ie (4)(-1)= -ve and (-1)(-1)= +ve)
It's not possible. If 1+1 = 0, then +1 would have to be equal to -1, but it can't be equal to -1 and +1 at the same time. Same way you could try and prove for any "A" that A+A = 0
Alright, here's the rub. This isn't actually a "valid" operation. It just looks like what's going on. The notation convention used by every mathematician and math educator that I know is that ##\sqrt{a}## denotes the non-negative solution of ##x^2=a## if ##a## is a non-negative real number and ##\sqrt{a}## denotes the root ##i\sqrt{-a}## of ##x^2=a## when ##a## is a negative real number (so we've already agreed on what ##\sqrt{-a}## means), where ##i## denotes either (a) some made-up thing with the property that ##i^2=-1## or (b) the principle complex root (we singled one of them out and said "you're it") of ##x^2+1=0## depending on where you are in your math studies. There are no rules of exponents or multiplicative identities of roots or anything of that sort going on. It's all just definitions of notation; these are the symbols that we use to talk about these abstract mathematical concepts. In other words, ##\sqrt{-4}=i\sqrt{-(-4)}=i\sqrt{4}=2i## because somebody a long time ago decided that's what those symbols mean, and everybody else went along with it. It only looks like we're using ##\sqrt{-4}=\sqrt{-1\cdot 4}=\sqrt{-1}\cdot\sqrt{4}=i\cdot2## because either (a) you were clever enough to notice that might be the case based on your experience using rules that you learned in precalc or (b) you had a teacher (or tutor or helper on the interwebs) that told you a little white lie to convince you that the answer they gave was correct without realizing (or caring) that it might later cause serious strife for you and other teachers (or tutors or helpers on the interwebs) or (c) you had a teacher (or tutor or helper on the interwebs) who was incompetent tell you this is what was going on without understanding that it really wasn't.
if the same exponential laws applied to negative numbers as they do positive. a+a=√(a^2 )+a =√((-a)(-a) )+a =√(-a) √(-a)+a =√((-1)(a) ) √((-1)(a) )+a =(√(-1))(√a)(√(-1))(√a)+a =(i)(√a)(i)(√a)+a =i^2 a+a =-a+a =0
Isn't that pretty much equivalent to what your doing when you say √(-4) = i√(-(-4)) because your just taking √(i)^{2} out of the radical, which is the same as √(-1). so really your doing the exact same thing as me?
No, he is not doing the same thing. He is saying that we use the sequence of symbols "##\sqrt{a}##" as a convenient shorthand for "the non-negative solution of ##x^2=a## if ##a## is a non-negative real number; and the root ##i\sqrt{-a}## of ##x^2=a## when ##a## is a negative real number, where ##i## denotes some made-up thing with the property that ##i^2=-1##" That's a definition, and we can use that definition to show that if ##a## and ##b## are both non-negative reals, then ##\sqrt{ab} = \sqrt{a}\sqrt{b}##. However, we can also use this definition to show that, in general, this equality does not hold for negative numbers.
No it's not. Read again what I wrote regarding the definition of the meaning of the collection of symbols ##\sqrt{a}## when ##a## is a negative real number: If ##a## is a negative real number, the ##\sqrt{a}=i\sqrt{-a}##. There is nothing in between the left and right hand side that explains why it is true. There is not proof of this identity. It is not a matter of mathematics, but a matter of mathematical notation. It is simply true by definition of the notation being used. I'm not "taking √(i)^{2} out of the radical". The ##i## essentially materializes out of thin air, just like the ##\times## does in the definition of the exponential notation ##x^2=x\times x##.
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 √(a^{2}) = |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 = 4i^{2} + 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?
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!!