Thursday, February 22, 2018

algebra precalculus - Why does $2+2=5$ in this example?


I stumbled across the following computation proving $2+2=5$


calculation proving 2+2=5


Clearly it doesn't, but where is the mistake? I expect that it's a simple one, but I'm even simpler and don't really understand the application of the binomial form to this...


Answer



The error is in the step where the derivation goes from $$\left(4-\frac{9}{2}\right)^2 = \left(5-\frac{9}{2}\right)^2$$ to $$\left(4-\frac{9}{2}\right) = \left(5-\frac{9}{2}\right)$$



In general, if $a^2=b^2$ it is not necessarily true that $a=b$; all you can conclude is that either $a=b$ or $a=-b$. In this case, the latter is true, because $\left(4-\frac{9}{2}\right) = -\frac{1}{2}$ and $\left(5-\frac{9}{2}\right) = \frac{1}{2}$. Once you have written down the (false) equation $-\frac{1}{2} = \frac{1}{2}$ it is easy to derive any false conclusion you want.


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