1 = -1, which part of this proof is wrong?

In summary, the conversation was about a proof that 1 is equal to -1 and the mistake lies in assuming that the square root of a negative number can be split into two square roots. This is only true for real numbers, not complex numbers. The conversation also included a link to an article discussing common mistakes with complex numbers.
  • #1
Byeonggon Lee
14
2
Of course 1 isn't same as -1.
This proof must be wrong but I can't find which part of this proof is wrong.
Could you help me with this problem?
(1)$$1 = \sqrt{1}$$
(2)$$= \sqrt{(-1)(-1)}$$
(3)$$= \sqrt{(-1)} \cdot i$$
(4)$$= i \cdot i$$
$$=-1$$
 
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  • #2
Would you like better to start with [itex]i^2+1=0[/itex] ?
 
  • #3
Byeonggon Lee said:
Of course 1 isn't same as -1.
This proof must be wrong but I can't find which part of this proof is wrong.
Could you help me with this problem?
(1)$$1 = \sqrt{1}$$
(2)$$= \sqrt{(-1)(-1)}$$
(3)$$= \sqrt{(-1)} \cdot i$$
(4)$$= i \cdot i$$
$$=-1$$
Read this insight article by @micromass : https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/
 
  • Like
Likes Demystifier
  • #4
In the transition between the second line, [itex]= \sqrt{(-1)(-1)}[/itex], and the third line, [itex]= \sqrt{-1}\cdot i[/itex] you are assuming that [itex]\sqrt{ab}= \sqrt{a}\cdot \sqrt{b}[/itex]. That is true for real numbers but not for general complex numbers.
 
  • #5
##1=\sqrt{1}##
##=\sqrt{(-1)(-1)}##
##=\sqrt{-1}\cdot\sqrt{-1}##
##=i^2##
##=-1##
 
  • #6
Deepak suwalka said:
##1=\sqrt{1}##
##=\sqrt{(-1)(-1)}##
##=\sqrt{-1}\cdot\sqrt{-1}##
##=i^2##
##=-1##
You are just repeating what others have pointed out to be wrong.
 
  • #7
In a thread that is 7 months old.
 
  • #8
And for an OP who has not been here since posting the question.
 
  • #9
LCKurtz said:
In a thread that is 7 months old.
sorry
 
  • #10
Deepak suwalka said:
sorry

It's ok.
 

1. Is it possible for 1 to equal -1?

No, it is not possible for 1 to equal -1. This goes against the fundamental rules of mathematics and logic.

2. What is the proof that 1 = -1?

There is no valid proof that 1 = -1. This statement is false and cannot be proven.

3. Can you explain the logic behind this statement?

The logic behind this statement is flawed. Any equation that results in 1 = -1 is mathematically incorrect and goes against the basic principles of equality.

4. What part of the proof is incorrect?

The entire proof is incorrect. The statement 1 = -1 is false and cannot be proven using valid mathematical reasoning.

5. Why is it important to understand that 1 does not equal -1?

Understanding that 1 does not equal -1 is crucial for maintaining logical, accurate thinking and problem-solving skills in mathematics. It is also important for avoiding errors and misunderstandings in more complex equations and proofs.

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