What Is the Error in Spivak's 2=1 Proof?

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Hi, I've gotten Spivak's calculus and I have a question on the second proof in the first chapter

What is wrong with the following "proof"?

suppose x=y

1. x² = xy

2. x² - y²= xy - y²

3. (x + y)(x - y)=y(x - y)

4. x + y=y

5. 2y = y

6. 2 = 1


I just want to clarify that the error is in the transition from step three to step four as subtracting both sides by (x - y) is to subtract by zero as if x=y then x -y = 0.

Step two is also saying that 0 = 0.

 

[Mark44]

Hi, I've gotten Spivak's calculus and I have a question on the second proof in the first chapter

What is wrong with the following "proof"?

suppose x=y

1. x² = xy

2. x² - y²= xy - y²

3. (x + y)(x - y)=y(x - y)

4. x + y=y

5. 2y = y

6. 2 = 1


I just want to clarify that the error is in the transition from step three to step four as subtracting both sides by (x - y) is to subtract by zero as if x=y then x -y = 0.

No, that's not it at all. You have correctly identified the step that is incorrect, but not the reason. What they have done in going from step 3 to step 4 is to divide by x - y, not subtract x - y. There is never a problem subtracting the same amount from both sides of an equation, but you can run into problems by dividing both sides by a quantity that happens to be zero.

Step two is also saying that 0 = 0.

 

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Sorry I meant divide, misuse of language. Good to hear I got it, Spivak doesn't seem so difficult now




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