/sci/ - Science & Math

[Anonymous No.2173210]

Hey /sci/,

why is the following assumption wrong?

1 = sqr(1) = sqr( (-1)*(-1) ) = sqr(-1) * sqr(-1) = i*i = -1

with "i" being the imaginary unit. (i² = -1)

[dolan !Uq56cE.45s No.2173213]

It's a paradox scientists have been trying to solve for many centuries. Nothing new.

[Anonymous No.2173215]

sqr( (-1)*(-1) ) = sqr(-1) * sqr(-1)
nope
only valid if those variables are positive

[Anonymous No.2173217]

>>2173210
because the definition of imaginary unit is i²=(-1)

[Colin !!GWicPmfSgsD No.2173222]

>Rollover for detail
>6 KB

[Anonymous No.2173225]

>>2173215

we're talking complex dude

[Anonymous No.2173231]

>>2173217
What he said. sqrt(-1)= i or -i

[Anonymous No.2173242]

sqrt(x) = +-x

You ignore that a few times

Just like anywhere else, you have to throw out the bullshit solution since that is positive OR minus values of x (assuming x is real)

[Anonymous No.2173268]

>>2173242
why wouldn't this work if x is complex?
I'm pretty sure if Z=A*A , (-A)*(-A) is also equal to Z. (Z and A complex)

[Anonymous No.2173271]

>>2173242

sqrt(x) = y>0 with y² = x only if x>0 (in case of x being real, otherwise sqrt wouldn't be a valid function.)

In the case of x being complex, sqrt(x) with x<0 is valid.

In case if x being complex sqrt can be applied to x<0

[Anonymous No.2173294]

>>2173268
>>2173271
Correct

The same logic with real also applies to imaginary/complex

This can be confirmed by GOOGLE

[Anonymous No.2173507]

Op you ignored that which is you can't take number our from squ as negative. (Absolute Value)

[Anonymous No.2173766]

>>2173507

depends on wether x is real or complex!

[Anonymous No.2173779]

The fallacy is that the rule <span class="math">\displaystyle \sqrt{xy} = \sqrt{x}\sqrt{y}[/spoiler] is generally valid only if at least one of the two numbers x or y is positive, which is not the case here.

[Anonymous No.2173786]

<span class="math">
1 \neq \sqrt(1)
{-1, +1} = \sqrt(1)[/spoiler]

[Anonymous No.2174487]

>>2173210
Thou shalt not change branches of the complex logarithm

[Anonymous No.2176642]

Don't dissapear

[Anonymous No.2176861]

no use logic only

[Anonymous No.2176870]

because i is not defined as "i = sqrt(-1)" but as one of the solutions of X² = -1 in an extension of the field of the real numbers. sqrt(-1) has no fucking sense with your usual square root. And when you want to choose a determination of the square root in the complex plane, you have to choose a determination of the logarithm, which usually is not defined on the negative real numbers. You can actually give a sense to sqrt(-1) but not in the sense you mean at first with your trolling problem, and your algebraic manipulations would not be correct with this definition.

Now get back to highschool and try to understand complex numbers.

[Anonymous No.2176880]

sqrt(-1) = -1 * -1 = 1 = sqrt(1) * 1 = -1

[Anonymous No.2177146]

sqrt(a*b) = sqrt(a)*sqrt(b)

OP proved that the above is not true if a and b are negative by contradiction: he assumed it was true for negatives, and it resulted in a contradiction, therefore it is not true for negatives.

[Anonymous No.2177205]

While it is true that sqrt (ab) = sqrt (a) * sqrt (b) where a, b are positive real number, and sqrt means the positive square root. But the same does not apply in general for complex numbers, where sqrt means square root primary instance, and a single example to is what you get: sqrt (-1) * sqrt (-1) = i ^ 2 = - 1, while sqrt ((-1) * (-1)) = sqrt (1) = 1
Know that arg (z) is ambiguous, and the main square root sqrt (z) is defined by the main argument Arg (z) to z.

But the main argument Arg and the argument arg do not have the same properties: we have always arg (z * w) = arg (z) + arg (w) (when arg interpreted is ambiguous), but it is not true in general that

Arg (z * w) = Arg (z) + Arg (w),

- eg, we have that arg ((-1) * (-1)) = arg (1) = 0, but it is also true that arg (1) =

arg ((-1) * (-1)) = arg (-1) + arg (-1) = pi + pi = 2pi.

However, Arg (1) = 0 by definition, not 2pi.

[Anonymous No.2177242]

we need one of these for this faggotry too