Math Myth: Reducing an equation

[Greg Bernhardt]

From @fresh_42's Insight
https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

This time I'm hopefully in accordance with teachers. The standard procedure is

$$ab=ac \;\Longrightarrow \;b=c$$

It is not only sloppy, it is even wrong sometimes, and belongs to the standard mistakes in class. If we all would take the time and write it down more carefully, this could really avoid the mistake: \begin{align*}ab=ac \;\Longrightarrow \;ab-ac=0\;\Longrightarrow \;a\cdot (b-c)=0\;\Longrightarrow \;a=0\quad\text{ or }\quad b=c\end{align*}

The second possibility is all of a sudden evident. The advice behind it is simple: avoid divisions as long as you can. And if, make sure you are allowed to.

 

[.Scott]

The "avoid division" advice is one I began following very early in my programming career. Unless the denominator is known to be well clear of zero (for example, the constant pi), it needs to be checked. And if the check fails, you need to figure out what the computer should do next.

 

[jack action]

It seems to be some circular argumentation with your other point:

#3. You cannot divide by $0$.

[...] $0$ hasn't the least to do with multiplication and even less with division. The question to divide by $0$ doesn't even arise! The neutral element for multiplication is $1,$ not $0$. That belongs to addition. [...]

So, do we consider $0$ as part of multiplication or not?

 

[fresh_42]

It seems to be some circular argumentation with your other point:

So, do we consider $0$ as part of multiplication or not?

Division needs a group, multiplication only a ring.

 

[StoneTemplePython]

The implication here of $ab = ac \implies ab-ac =0$ seems to assume we are in a ring not a (semi?)group but never states anything about groups vs rings, so I found this a touch problematic.

In both cases the underlying idea is to consider whether the mapping given by left multiplication with $a$ is injective.

I think the core of this can be conveyed to early school kids with a diagram-- in particular why you can't "undo" multiplication by zero because it maps multiple elements (presumably in $\mathbb Q$ or some other ring that isn't the zero ring) to zero. In particular multiplication by 0 maps 0 to 0 and 1 to 0.