Extra (optional) exercises – Short Day

These exercises are optional extra material for students who have finished the exercises of the day and have the time and interest for more.

Extra exercise 1: Bad “proofs”

Question a

Somebody proposes the following “proof” of the claim \(1=0\):

\[\begin{split} \begin{array}{ll} a=b & \text{Assumption: $a$ and $b$ are real with $a=b$}\\ & \text{Multiply by $a$}\\ a^2=ab & \\ & \text{Subtract $b^2$}\\ a^2-b^2=ab-b^2 & \\ & \text{Factorise}\\ (a+b)(a-b)=(a-b)b & \\ & \text{Divide by $a-b$}\\ a+b=b & \\ & \text{Use the assumption that $a=b$}\\ 2a=a & \\ & \text{Divide by $a$}\\ 2=1 & \\ & \text{Subtract $1$}\\ 1=0 & \\ \end{array} \end{split}\]

Obviously something has gone wrong. Where is the mistake?

Question b

Here is another “proof” of the claim \(1=0\):

\[\begin{split} \begin{array}{ll} -20=-20 & \\ & \text{Rewrite}\\ 25-45=16-36 & \\ & \text{Factorise terms}\\ 5^2-9\cdot 5=4^2-9\cdot 4 & \\ & \text{Add $\left(\frac92\right)^2$}\\ 5^2-9\cdot 5+\left(\frac92\right)^2=4^2-9\cdot 4+\left(\frac92\right)^2 & \\ & \text{Complete the square}\\ \left( 5-\frac{9}{2}\right)^2=\left(4-\frac{9}{2}\right)^2 & \\ & \text{Take the square root}\\ 5-\frac{9}{2}=4-\frac{9}{2} & \\ & \text{Subtract $4-\frac{9}{2}$}\\ 1=0 & \\ \end{array} \end{split}\]

Where is the mistake this time?