2=1? Is that true?

Often we see the evidence mathematically that 2=1. They show very logically. Some people who cannot know this mistake then they blindly assume that math is flawed.

I believe that mathematics is one of the perfect sciences. Because until now I have not found a defect in this science. Which I love, this science is connected to each other. Some people even consider that mathematics is the language of Deity.


Back to the topic of conversation. Ever seen evidence like this?

    \begin{eqnarray*} a & = & b\\ a^{2} & = & ab\\ a^{2}-b^{2} & = & ab-b^{2}\\ \left(a+b\right)\left(a-b\right) & = & b\left(a-b\right)\\ a+b & = & b\\ 2b & = & b\\ 2 & = & 1\end{eqnarray*}

Or something like this?

    \begin{eqnarray*} a & = & b\\ a^{2} & = & ab\\ a^{2}+a^{2} & = & a^{2}+ab\\ 2a^{2} & = & a^{2}+ab\\ 2a^{2}-2ab & = & a^{2}+ab-2ab\\ 2a^{2}-2ab & = & a^{2}-ab\\ 2\left(a^{2}-ab\right) & = & \left(a^{2}-ab\right)\\ 2 & = & 1\end{eqnarray*}

Lots of evidence like this is scattered on the internet. If we do not look closely, we will be caught in logical traps and as if they are true.

So, where is the error of evidence above?

It’s simple. The error of the above evidence is in the step that makes \frac{0}{0}=1. And we know that \frac{0}{0} is undefined.

I do not see the \frac{0}{0} . Where is it?

Yap, they deliberately wrap up \frac{0}{0} beautifully. In the first example, step 4 to step 5.They eliminate \left(a-b\right) on both sides. Though from the original definition we know that a=b. So \left(a-b\right) is 0. As a result, \frac{\left(a-b\right)}{\left(a-b\right)} should not be done.

Equivalent in the second example. They do the same to trick readers. They present \frac{0}{0} in the form of \left(a^{2}-ab\right).

So if there is something that looks logical but has a wrong conclusion or a contradiction, we should be suspicious. There is something wrong that may be deliberately hidden to make a false conclusion to be right.

Thanks for reading. May be useful. See another simply article in this link.

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