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What is the mistake here? Is it matter of the unit?

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    $\begingroup$ Yes, the units don’t match across the 2nd equals sign $\endgroup$ – Alex Mar 12 at 0:20
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    $\begingroup$ Yes, you have to square the unit. Conversion of squared units is different: if 100 cents is a dollar, then $100^2$ cents squared is a dollar squred. $\endgroup$ – Dean Young Mar 12 at 0:20
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$\$0.01=(\sqrt{\$}0.1)^2$, not $(\$0.1)^2$.

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You can clearly see the fallacy if you keep track of the units:

  • In the second equality, $\$0.01 = \$0.1\times \$0.1$ is not true, if you are doing units.

  • Even if the second equality were true, the third one gives problems: since $c=\$/100$, you have $$ (\$0.1)^2=\left(\frac c{100}\,0.1\right)^2=\frac{c^2}{100}\times\frac1{10}=\frac{c^2}{1000}. $$ This is not $(10c)^2=100c^2$.

In conclusion, two equalities are bogus, and so is the argument.

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