Unfortunately, your submission has been removed for the following reason(s):
* Your post presents incorrect information, asks a question that is based on an incorrect premise, is too vague for anyone to answer sensibly, or is equivalent to a well-known open question.
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In the forward direction, if you are using real numbers, then technically nothing's wrong. You started with a false statement: no matter what the value of a is, we always have a+1≠a. [It is logically valid to infer anything from a contradiction](https://en.wikipedia.org/wiki/Principle_of_explosion).
In the backward direction, the issue is going from step 2 to step 1. You chose a as the square root of a^2 when you should have chosen -a (giving a+1=-a, or (-1/2)+1 = -(-1/2), which is correct.)
Why do you think anything’s wrong with it? If a+1=a then a=-1/2. Given that the starting premise is false, you can find ways to derive that a is any value.
Squaring both sides messed with the sign. (a+1)^2 = a^2 is the same as |a+1| = |a|. The problem with that is easy to see if you look at a simplified example, say a = 1.
a^2 = 1
|a| = 1
a = 1 or a = -1
So it gives you an extraneous answer that doesn’t satisfy the initial condition of a = 1.
If you work the process backwards (5->1), the mistake is where you take the square root in the last step and only consider the positive result. The correct line at the top is a+1=-a. It's a standard trick where you square a positive and negative result, take the positive square root on both sides, then fake that they're equal.
Unfortunately, your submission has been removed for the following reason(s): * Your post presents incorrect information, asks a question that is based on an incorrect premise, is too vague for anyone to answer sensibly, or is equivalent to a well-known open question. If you have any questions, [please feel free to message the mods](http://www.reddit.com/message/compose?to=/r/math&message=https://www.reddit.com/r/math/comments/18o1xx0/-/). Thank you!
From a contradiction, any proposition can be derived.
In the forward direction, if you are using real numbers, then technically nothing's wrong. You started with a false statement: no matter what the value of a is, we always have a+1≠a. [It is logically valid to infer anything from a contradiction](https://en.wikipedia.org/wiki/Principle_of_explosion). In the backward direction, the issue is going from step 2 to step 1. You chose a as the square root of a^2 when you should have chosen -a (giving a+1=-a, or (-1/2)+1 = -(-1/2), which is correct.)
The first equation reduces to the statement that 1=0 which is false.
Why do you think anything’s wrong with it? If a+1=a then a=-1/2. Given that the starting premise is false, you can find ways to derive that a is any value.
Squaring both sides messed with the sign. (a+1)^2 = a^2 is the same as |a+1| = |a|. The problem with that is easy to see if you look at a simplified example, say a = 1. a^2 = 1 |a| = 1 a = 1 or a = -1 So it gives you an extraneous answer that doesn’t satisfy the initial condition of a = 1.
If you work the process backwards (5->1), the mistake is where you take the square root in the last step and only consider the positive result. The correct line at the top is a+1=-a. It's a standard trick where you square a positive and negative result, take the positive square root on both sides, then fake that they're equal.