[net.math] Whats wrong with this equation?

dave (12/02/82)

Recently I was shown the following equation, which is obviously incorrect, but
was unable to figure out the logic error.  Could someone enlighten me?

(Note: for lack of a character, I will use sqrt() for square root)

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

2)	sqrt(-1/1) = sqrt(1/-1)

3)	sqrt(-1)/sqrt(1) = sqrt(1)/sqrt(-1)

	cross-multiply
4)	sqrt(-1) sqrt(-1) = sqrt(1) sqrt(1)

5)	-1 = 1.

					Dave Cohrs
					...uwvax!dave

spence (12/02/82)

#R:uwvax:-69900:harpo:9100001:000:113
harpo!spence    Dec  2 10:33:00 1982

step three is incorrect


sqrt(-1)=j

then sqrt(-1)/sqrt(1)=j

while sqrt(1)/sqrt(-1)=1/j or -j



harpo!spence

gsp (12/03/82)

In the equations:

	sqrt(1) = 1
	sqrt(1) = -1
	therefore 1 = -1

The = sign does not mean equality, but more literally means
"has as part of its set" or "is part of the set."  So there
is no paradox at all.

	let X = the set of all sqrt(1)'s
	1 is in X
	-1 is in X
	that does not imply they are the same but share a property.

Okay, how about his one:
	Kepler thought that the number of planets equals six.
	The number of planets equals nine.
	Therefore, Kepler thought six equals nine.

	Gary Perlman	Bell Labs	MH 5D-105	ucbvax!mhb5b!gsp

dap1 (12/04/82)

#R:uwvax:-69900:ihlpb:6200003:  0:1488
ihlpb!dap1    Dec  3 20:15:00 1982

I think that there is a much more fundamental problem here which shows up
when you map out what is going on in the complex plane.  First, a few facts:

 1. Every complex number can be represented by an amplitude (angle the point
    makes with the real axis in the complex plane) and a modulus (absolute
    value).  I assume everyone is pretty familiar with this stuff.

2. The division of two complex numbers, a/b, has modulus mod(a)/mod(b) and
   amplitude amp(a)-amp(b).

3. Sqrt(a) has modulus Sqrt(mod(a)) and amplitude amp(a)/2.

Soooo...
-1/1 has modulus 1 and amplitude of 180 degrees while 1/-1 has modulus 1 and
amplitude of -180 degrees(!).  Therefore, when you take the square root of
-1/1 you get a modulus of 1 with amplitude of 90 degrees, that is, j.  On the
other hand, when you take the square root of 1/-1 you get a modulus of 1 and
an amplitude of -90 degrees or -j.  Thus, you are implicitly saying that
                             j=-j
in step 3 from which it is no trouble at all to go on to the rather odd
result.  That is, you are implicitly using the two distinct roots on opposite
sides of the equation.
This is all circumvented in the reals since the sqrt function is
always defined to be the positive root.  Otherwise, you could also "prove"
that:
          sqrt(4)=2=-2 -> 1=-1.

                                                Darrell Plank
                                                BTL-IH
                                                ihlpb!dap1

dap1 (12/04/82)

#R:mhb5c:-102200:ihlpb:6200004:  0:456
ihlpb!dap1    Dec  3 20:23:00 1982

Hey, you guys seem to forget that square roots in the real number domain are
by definition positive!  Sure, x*x=1 has two solutions but only one of them
is the square root of 1.  As far as '=' meaning that two sets are the same,
that all depends on whether you are talking about an equivalence relation
or the more standard definition of '=' (which is a special case of an
equivalence relation).  In standard usage, '=' is the identity relation for
reals.

mcewan (12/05/82)

#R:mhb5c:-102200:uiucdcs:28200001:000:52
uiucdcs!mcewan    Dec  5 13:36:00 1982

Yes, but sqrt(-1) is not in the real number domain.