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Hi Everyone
I have been doing further investigation into infinitesimals since I wrote my insight article.
I had an issue with the original article; the link to the foundations of natural numbers, integers, and rational numbers was somewhat advanced. I did need to write an insights article at a level more appropriate to a beginning student who has done some Calculus - but no abstract algebra.
At the same time, my article on infinitesimals had issues since learning more about the subject.
It took a while, but I completed an insights article rectifying both issues, which is now published - What Are Numbers.
https://www.physicsforums.com/insights/what-are-numbers/
I will leave my original article alone for now but will eventually delete it as this replaces it.
I will shortly publish an insights article based on an interesting book called Precalculus Made Difficult.
It is different from the usual precalculus text in that it covers in the US what is called Algebra 1, Geometry, Algebra 2, and Precalculus in one text. Instead of taking 4 years to cover the material, IMHO, an average student can do it in 2 years; a good student could do it in a year. The text has some 'issues' in that it uses things like 2^x without first defining 2^x, where x is a real number. There is only one way to fix this - with calculus. So when it reaches that point in the text, I introduce calculus and define it properly. This also includes an introduction to set theory and infinitesimals, but at a level, I believe appropriate for the audience.
After that is done, the rest of the text is studied. It, however, does not introduce complex numbers. To fix that, I included a section on complex numbers with a bit of a calculus flavour eg prove Euler's famous relation.
This provides, again, IMHO, good preparation to study the author's associated calculus text - Full Frontal Calculus.
Precalculus More Difficult is available for free. The associated calculus text Full Frontal Calculus was free but now costs $9.00 - still rather cheap. Both are available as books from Amazon and would be my preference unless money is an issue.
If the reader has not done calculus, that article, and Full Frontal Calculus, should be studied before tackling this article.
Thanks
Bill
I have been doing further investigation into infinitesimals since I wrote my insight article.
I had an issue with the original article; the link to the foundations of natural numbers, integers, and rational numbers was somewhat advanced. I did need to write an insights article at a level more appropriate to a beginning student who has done some Calculus - but no abstract algebra.
At the same time, my article on infinitesimals had issues since learning more about the subject.
It took a while, but I completed an insights article rectifying both issues, which is now published - What Are Numbers.
https://www.physicsforums.com/insights/what-are-numbers/
I will leave my original article alone for now but will eventually delete it as this replaces it.
I will shortly publish an insights article based on an interesting book called Precalculus Made Difficult.
It is different from the usual precalculus text in that it covers in the US what is called Algebra 1, Geometry, Algebra 2, and Precalculus in one text. Instead of taking 4 years to cover the material, IMHO, an average student can do it in 2 years; a good student could do it in a year. The text has some 'issues' in that it uses things like 2^x without first defining 2^x, where x is a real number. There is only one way to fix this - with calculus. So when it reaches that point in the text, I introduce calculus and define it properly. This also includes an introduction to set theory and infinitesimals, but at a level, I believe appropriate for the audience.
After that is done, the rest of the text is studied. It, however, does not introduce complex numbers. To fix that, I included a section on complex numbers with a bit of a calculus flavour eg prove Euler's famous relation.
This provides, again, IMHO, good preparation to study the author's associated calculus text - Full Frontal Calculus.
Precalculus More Difficult is available for free. The associated calculus text Full Frontal Calculus was free but now costs $9.00 - still rather cheap. Both are available as books from Amazon and would be my preference unless money is an issue.
If the reader has not done calculus, that article, and Full Frontal Calculus, should be studied before tackling this article.
Thanks
Bill