Genady

Could you elabotate?

I just thought of something else, for a case when one sees one solution but not the other:

It is not a difficult equation. I wonder, how people approach it. Here it is: \[x+\frac{1}{x}=4\frac{1}{4}\] (Please, use spoiler in your response.)

I live with the clear, rigorous construction of rational numbers ("fractions") using equivalence classes for very long; this construction is obvious and intuitive to me. OTOH, I realize that it is not fit to school children. I also know many intelligent adults who think that math is just "following the rules", which I think is the outcome of poor presentation of mathematical concepts in schools. So, I wonder what justifications/explanations of these "rules" for school children are there, if at all.

Here (in "The Road to Reality") Roder Penrose asks the same question, but perhaps clearer than me:

If the same thing which is done to top and bottom is multiply or divide, it is ok. But if this thing is add or subtract, it is not. IOW, 4/14 is certainly not the same as 2/12 (subtracting 2 from top and bottom). How do we know that it is the same as 2/7 (dividing top and bottom by 2)?

Did they explain/justify it? I mean, why this action leaves the numbers unchanged.

I mean, ONE third is different from TWO sixths.

Not in my case. I was an only child.

Me too. But I remember that it bothered me, because how ONE piece of pie can be EQUAL TWO pieces of pie? Sure, they weigh the same, but they are different in so many ways...

I also had my share of bad teachers. Fortunately, I had two very good math teachers. They would never say anything like that.

How is it explained to school children that or why, e.g., 2/7=4/14=...?

To me, it is 'cold' when we need to close windows, and it would be 'extremely cold' if we needed to turn on a heater. (The latter never happened.)

Yes, it is. The temperature is down to 27oC.

Do you think this derivation, has a better chance?

No problem, I thought so. I don't know this.

They did not demonstrate that (my emphasis). They demonstrated this:

Energy and momentum cannot be both conserved in such a process.

Typo? 46.3

I have assumed only that multiplication is distributive (on the step you have marked): (a-b)×c=a×c-b×c In the case above, a=1, b=2, c=-1.

If we accept that -1 and +1 are additive inverses, and if we want to keep the distributive property of multiplication, then it appears that we don't have a choice but make (-1)×(-1)=+1. Here it goes: (-1)×(-1)=(1-2)×(-1)=1×(-1)-2×(-1)=(-1)-(-1)-(-1)=(-1)+(+1)+(+1)=+1 QED

Regardless of being a root of +1, -1 is a label for the additive inverse of +1. How come we use the same label?

https://www.amazon.com/Quantum-Field-Theory-Standard-Model/dp/1107034736/?tag=pfamazon01-20