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Index of Subjects --Apple-Mail-68--627903395 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=WINDOWS-1252; delsp=yes; format=flowed That reminds me of something I remember reading once: 2 is not equal to 3 =96 not even for large values of 2. Pat On Dec 20, 2006, at 7:22 PM, David & Alison Webster wrote: > Hi Chris & All, > But notice that one doesn't quite get to divide by zero because = the > previous line (below) says that a*zero =3D b*zero and this condition =20= > alone voids any conclusions that one might attempt to make about a and = =20 > b. > > Yt, DW > > > Factor : > a ( a - b -c) =3D b ( a - b -c) > > > > c.majka@ns.sympatico.ca wrote: > >> Hi Dave, >> >> Nice riddle: made me scratch my head for a few minutes, However, as =20= >> Paul points out (a - b - c) =3D 0 and dividing by 0 (at the last = step) =20 >> allows one to prove whatever one wishes. ;-> >> >> Cheers, >> >> Chris >> >> On 20-Dec-06, at 9:17 AM, David & Alison Webster wrote: >> >>> Dear All, Dec 20, 2006 >>> I recently came across an old article about Raymond Smullyan, a >>> highschool dropout who e.g. satisfied the course requirements for a = =20 >>> PhD >>> in Math and Philosophy by teaching the required courses. He also =20= >>> created >>> many logic puzzles. One thing leads to another, so I blew the dust =20= >>> off >>> of an old copy of Martin Gardner's 1959 Mathematical Puzzles and >>> Diversions. >>> >>> One of his fallacies follows. I hope you have not seen this =20 >>> previously. >>> Proof that unequal numbers are equal >>> >>> Given two numbers a & b such that b is smaller than a by an = amount >>> c; thus >>> >>> a =3D b + c >>> >>> prove that a =3D b >>> >>> multiply both sides by (a - b) to obtain-- >>> >>> a^2 - ab =3D ab +ac - b^2 - bc >>> >>> subtract ac from both sides to obtain >>> >>> a^2 - ab - ac =3D ab - b^2 - bc >>> >>> Factor : >>> a ( a - b -c) =3D b ( a - b -c) >>> >>> Divide both sides by ( a - b - c) to obtain >>> >>> a =3D b >>> >>> QED >>> >>> "The road to correct conclusions is full of pitfalls" (DHW, Dec =20= >>> 20, 2006). >>> >>> Merry Christmas, Dave Webster, Kentville >>> >>> >> >> > > > > =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20 =3D=3D Patrick Kelly Director of Computer Facilities =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20 =3D=3D Faculty of Architecture and Planning Dalhousie University =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20 =3D=3D PO Box 1000 Stn Central 5410 Spring Garden Road Halifax, Nova Scotia B3J 2X4 Halifax, Nova Scotia B3J 2X4 Canada Canada =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20 =3D=3D Phone:(902) 494-3294 FAX:(902) 423-6672 E-mail:patrick.kelly@dal.ca =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=20 =3D=3D --Apple-Mail-68--627903395 Content-Transfer-Encoding: quoted-printable Content-Type: text/enriched; charset=WINDOWS-1252 That reminds me of something I remember reading once: 2 is not equal to 3 =96 not even for large values of = 2.<fontfamily><param>Bookman</param> Pat </fontfamily> On Dec 20, 2006, at 7:22 PM, David & Alison Webster wrote: <excerpt>Hi Chris & All, But notice that one doesn't quite get to divide by zero because the previous line (below) says that a*zero =3D b*zero and this condition alone voids any conclusions that one might attempt to make about a and b. Yt, DW Factor : a ( a - b -c) =3D b ( a - b -c) c.majka@ns.sympatico.ca wrote: <excerpt>Hi Dave, Nice riddle: made me scratch my head for a few minutes, However, as=20 Paul points out (a - b - c) =3D 0 and dividing by 0 (at the last step)=20= allows one to prove whatever one wishes. ;-> Cheers, Chris On 20-Dec-06, at 9:17 AM, David & Alison Webster wrote: <excerpt>Dear All, Dec 20, 2006 I recently came across an old article about Raymond Smullyan, a highschool dropout who e.g. satisfied the course requirements for a=20 PhD in Math and Philosophy by teaching the required courses. He also=20 created many logic puzzles. One thing leads to another, so I blew the dust off of an old copy of Martin Gardner's 1959 Mathematical Puzzles and Diversions. One of his fallacies follows. I hope you have not seen this=20 previously. Proof that unequal numbers are equal Given two numbers a & b such that b is smaller than a by an amount c; thus a =3D b + c prove that a =3D b multiply both sides by (a - b) to obtain-- a^2 - ab =3D ab +ac - b^2 - bc subtract ac from both sides to obtain a^2 - ab - ac =3D ab - b^2 - bc Factor : a ( a - b -c) =3D b ( a - b -c) Divide both sides by ( a - b - c) to obtain a =3D b QED "The road to correct conclusions is full of pitfalls" (DHW, Dec=20 20, 2006). Merry Christmas, Dave Webster, Kentville </excerpt> </excerpt> </excerpt><fontfamily><param>Courier</param> = =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D Patrick Kelly Director of Computer Facilities = =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3