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The Magic Cafe Forum Index » » Puzzle me this... » » -1 = 1 (0 Likes) Printer Friendly Version

Nir Dahan
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Inner circle
Munich, Germany
1390 Posts

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Since there were a few of those posted lately I thought why not post another - sorry if it was already posted...

-1 = -1

-1/1 = 1/-1

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

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

multiply in diagonals...

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

cancel the sqrt...

-1 = 1 !!!!!!

has the world of numbers gone bizzare ????!!! or could you help "captain math" find the error ???
stanalger
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Special user
St. Louis, MO
996 Posts

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Error occurs when going from
line 3 to line 4.

sqrt(a/b) = (sqrt(a)) / (sqrt(b))

only if all of the square roots are real numbers.
James F
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Atlanta
1097 Posts

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http://www.themagiccafe.com/forums/viewt......01&2

that's close to what I posted. At least the errors in the math is the same. That mistake is pretty tough to catch if you aren't up on your math.

James
Nir Dahan
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Munich, Germany
1390 Posts

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Stan,

we don't assume real numbers only.
complex numbers are allowed and the mistake is not there...
it is somewhere else.
there is a more beautiful version of this "falacy" in the Moscow puzzles book. look for the mosquito and elephant puzzle.
the one in the book is even more hard to catch.

Nir
TomasB
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Sweden
1143 Posts

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I guess it is from going to line 3 as you have missed writing all solutions when you take the root out of both sides of the equation. The solution you decided to keep is not the correct one.

/Tomas
Nir Dahan
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Munich, Germany
1390 Posts

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Right Tomas!
if you have the moscow puzzles book try to look up the similar problem there. that is by far more hidden in their problem...
Jonathan Townsend
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Eternal Order
Ossining, NY
27017 Posts

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I got lost when you suggested that i/1 == 1/i ie I = (i)**-1
...to all the coins I've dropped here
stanalger
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St. Louis, MO
996 Posts

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Jonathan,

Agreed.

There is no error on line 3.
The error certainly occurs when you go to line 4.

Earlier I wrote:
sqrt(a/b) = (sqrt(a)) / (sqrt(b))
only if all of the square roots are real numbers.

I shouldn't have included the word "only".

Perhaps I should have said:
sqrt(a/b) = (sqrt(a))/(sqrt(b))
whenever a is a non-negative real number
and b is a positive real number.

Although it is also true in some other
cases, it is not true for all real a, b.

Case in point: sqrt(1/-1) is not equal to
sqrt(1) / sqrt(-1).

(But sqrt(-1/1) is equal to sqrt(-1) / sqrt(1).)

Stan
Jonathan Townsend
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Eternal Order
Ossining, NY
27017 Posts

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There are usually two square roots of a number. We call one of them the principal root.
...to all the coins I've dropped here
James F
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Inner circle
Atlanta
1097 Posts

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A lot of people say the fallacy comes by not writting the "plus or minus" sign. (as stated by Nir Dahan) However, with this example the fallacy is simplifying sqrt(-1/1) = sqrt(1/-1) as sqrt-1/sqrt1 = sqrt1/sqrt-1 ...You cannot do that. Although in a lot of cases this will work it will not work all the time and it is the source of fallacy so strangler was correct. Read his explanation, it is right on.

James
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