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.