Question

DISCUSS: Proof That $0=1$? The following steps appear to give equivalent equations, which seem to prove that $1=0$. Find the error.

$x=1$ Given

$x^2=x$ Multiply by $x$

$x^2-x=0$ Subtract $x$

$x\left(x-1\right)=0$ Factor

$\frac{x\left(x-1\right)}{x-1}=\frac{0}{x-1}$ Divide by $x-1$

$x=0$ Simplify

$1=0$ Given $x=1$

Short answer

The error in this process is thedivision by $0$ in the fifth step.

Step-by-step solution

Step 1. State the concept.

Division by$0$ is strictly prohibited because the result of division by$0$ is undefined.

Step 2. List the given data.

The steps that appear to prove $1=0$as follows:

role="math" localid="1643987481568" $\begin{array}{l}x=1\\ x^2=x\\ x^2-x=0\\ x\left(x-1\right)=0\\ \frac{x\left(x-1\right)}{x-1}=\frac{0}{x-1}\\ x=0\\ 1=0\end{array}$

Step 3. Determine the error.

The first step gives $x=1$. This implies that $x-1=0$.

However, in the fifth step; $\frac{x\left(x-1\right)}{x-1}=\frac{0}{x-1}$, both sides are being divided by $x-1$.

So, in effect, both sides are being divided by $0$, which is not allowed as division by$0$ is undefined. This is the error.