stevesu@bronze.UUCP (Steve Summit) (09/10/83)
The proof that a = b was pretty obvious. Here's one I discovered by accident and still can't quite figure out: 1 = 1 reflexive property of equality 2/2 = 2/2 another name for 1 2/2 2/2 -1 = -1 raise -1 to both sides 1 2 b/c th b -1 = sqrt( ( -1 ) ) a == c root of a 2 -1 = sqrt( 1 ) reduce ( -1 ) -1 = 1 reduce sqrt( 1 ) The mistake is probably in the fourth step, but no book I've seen places restrictions on a, b, or c in that identity. Steve Summit
laura@utcsstat.UUCP (Laura Creighton) (09/10/83)
you cannot take the square root of something in that way. You have to split your question instead. if k**2 is 16, thn you have to say that k is + or - the square root of 16. thus k is +4 or k is -4. In the same way with your question, you prove that 1 = 1 or 1 = -1. While the second statement is false, the union of them is true. Laura Creighton utzoo!utcsstat!laura
laura@utcsstat.UUCP (Laura Creighton) (09/10/83)
Actually, you proved that -1 is equal to 1, or -1 is equal to -1, but for the reasons I stated. laura
asente@decwrl.UUCP (Paul Asente) (09/11/83)
It's really quite straightforward: sqrt is a multiple valued relation in mathematics, although you don't usually think of it that way: sqrt(4) = {+2, -2} (I really mean a set here!) Transitivity does not hold in this situation, any more than it does in the situation "John is a boy; Tom is a boy; therefore John is Tom." -paul asente
debray@sbcs.UUCP (Saumya Debray) (09/12/83)
To say that 1 2 b/c th b -1 = sqrt( ( -1 ) ) a == c root of a is only partially correct. There are n values for the nth root of any number (not all of which might be real, e.g. cube roots of 1). Thus, we don't have a one-to-one mapping between (bth. root of a) and ((bth. root of a) to the bth. power). Saumya Debray SUNY at Stony Brook
ecn-ec:ecn-pc:ecn-ed:vu@pur-ee.UUCP (09/13/83)
O The mistake: a **(b/c) = c th root of ( a ** b ) is right, BUT it might be - c th root of ( a ** b ), in case there are two different real root, i.e. in case c is even. So -1 = - sqrt(1) and that's more than legal. Hao-Nhien Vu (pur-ee!norris)