I Math Myth: Reducing an equation

  • I
  • Thread starter Thread starter Greg Bernhardt
  • Start date Start date
AI Thread Summary
The discussion highlights the common misconception that if \( ab = ac \), then \( b = c \), emphasizing that this is often incorrect due to the potential for \( a \) to equal zero. It advocates for a more careful approach to algebraic manipulation, suggesting that avoiding division until necessary can prevent mistakes. The conversation also touches on the mathematical structures involved, noting that division requires a group while multiplication only requires a ring, which complicates the implications of the equation. Additionally, it suggests using visual aids to help students understand why multiplication by zero cannot be reversed, as it maps multiple inputs to zero. Overall, the dialogue stresses the importance of precision in mathematical reasoning to avoid fundamental errors.
Messages
19,781
Reaction score
10,734
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.
 
Last edited:
Mathematics news on Phys.org
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.
 
It seems to be some circular argumentation with your other point:
https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/ said:
#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?
 
jack action said:
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.
 
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.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
5
Views
2K
Replies
2
Views
1K
Replies
14
Views
2K
Replies
142
Views
9K
Replies
3
Views
2K
Replies
17
Views
2K
Back
Top