These exercises are optional extra material for students who are done with the exercises of the day but have the time for and the interest in more.
Extra (Optional) Exercises – Short Day#
Extra Exercise 1: “Proofs” ending in Contradictions#
Question a#
The following “proof” of the claim \(1=0\) is proposed:
\[\begin{split}
\begin{array}{ll}
a=b & \text{The assumption. $a$ and $b$ are assumed real.}\\
& \text{Multiply by $a$}\\
a^2=ab & \\
& \text{Subtract $b^2$}\\
a^2-b^2=ab-b^2 & \\
& \text{Rewrite}\\
(a+b)(a-b)=(a-b)b & \\
& \text{Divide by $a-b$}\\
a+b=b & \\
& \text{Use the assumption that $a=b$}\\
2a=a & \\
& \text{Divide by $a$}\\
2=1 & \\
& \text{Subtract $1$}\\
1=0 & \\
\end{array}
\end{split}\]
It’s clear that something is wrong - where is the mistake?
Question b#
Again, a “proof” is presented for the claim \(1=0\):
\[\begin{split}
\begin{array}{ll}
-20=-20 & \\
& \text{Rewrite}\\
25-45=16-36 & \\
& \text{Rewrite}\\
5^2-9\cdot 5=4^2-9\cdot 4 & \\
& \text{Rewrite}\\
5^2-9\cdot 5=4^2-9\cdot 4 & \\
& \text{Rewrite}\\
\left( 5-\frac{9}{2}\right)^2=\left(4-\frac{9}{2}\right)^2 & \\
& \text{Take the square root}\\
5-\frac{9}{2}=4-\frac{9}{2} & \\
& \text{Subtract $4-\frac{9}{2}$}\\
1=0 & \\
\end{array}
\end{split}\]
Where is the mistake this time?