5 = -1 Proof

 

Hereīs another proof to pretzel your mind. See if you can figure out whatīs wrong with this one:

We start with:

x2 + 6x + 5

=

x2 - 1

Factor both sides of the equation

(x + 5)(x + 1)

=

(x - 1)(x + 1)

Divide both sides by (x + 1)

(x + 5)

=

(x - 1)

Subtract x from both sides

5

=

-1    

Q.E.D. smileyface

Canīt see whatīs wrong?

Figure out what the value of x is by doing the following:

We start with:

x2 + 6x + 5

=

x2 - 1

Subtract x2 from both sides

x2 + 6x + 5 - x2

=

x2 - 1 - x2

Simplify

6x + 5

=

 - 1

Subtract 5 from both sides

6x + 5 - 5

=

 - 1 - 5

Simplify

6x

=

 - 6

Divide both sides by 6

x

=

 - 1

Uh-oh. Now do you see it?

Whatīs the value of (x + 1) in our proof above? Ah ha! Zero. Just like the 2 = 1 proof. See now why we have to disallow division by zero in our system of mathematics? If we didnīt, our number system would lapse into chaos, and mathematics would no longer work. No more jets, bridges, stereos. Technology as we know it would collapse.

Also note how subtly in both proofs the division by zero occurs. In the 2 = 1 proof itīs fairly blatant. Here itīs more insidious. So always be careful with your algebra.

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Sunday, January 20, 2008