# Seeking AI-help Getting Up To Speed on Math

As someone who has always struggled with math, I understand that it is a skill that can be improved with practice and dedication. I recently found many online resources and self-teaching tools to help me learn at my own pace, such as the Mathematics Refresher Course on Udemy, a 16-hour Math Refresher course from Alamo Colleges, and 9 free online courses from edX. I'm also exploring the Maxwell Heaviside Schrödinger approach to physics.

## Exploring Online Resources and Self-Teaching Tools to Improve Math Skills

### By Michael Levin

#### Tuesday, March 7, 2023

I’ve never considered myself good at math. I was good at trig and analysis but not calculus or physics. I think I’m more of a visual person than a numbers person. In particular the chances in syntax between Algebra and Calculus have been difficult. How much do approach the subject of math to have the critical insights that are so difficult to me which seem to come natural to some people.

I understand that math can be challenging for some people, especially when it comes to different branches and levels of difficulty. However, math is also a skill that can be improved with practice, patience and perseverance.

According to various sources¹²³⁴⁵⁶, some of the ways to improve your math skills are:

- Understand the concepts rather than memorize the formulas or rules. Try to grasp the logic and reasoning behind each problem and solution.
- Practice math problems regularly and consistently. Use different sources and methods, such as textbooks, online games, worksheets, quizzes, etc. Review your mistakes and learn from them.
- Apply math to everyday scenarios and situations. Look for opportunities to use math in real life, such as calculating tips, budgets, distances, time zones, etc. This can help you develop your mental math skills and make math more relevant and fun.
- Use online resources or self-teaching tools to learn at your own pace and level. There are many websites and apps that offer free or affordable courses, tutorials, videos, podcasts, etc. on various topics and skills in math. You can also join online communities or forums where you can ask questions or share ideas with other learners or experts.
- Get a tutor or a mentor who can guide you through your learning process. A tutor can help you with specific problems or topics that you find difficult or confusing. A mentor can motivate you and provide feedback and encouragement along the way.

I hope these tips are helpful for you to improve your math skills. Remember that everyone has their own strengths and weaknesses in learning different subjects. The most important thing is to have a positive attitude and a growth mindset towards math.

Do you have any questions about these tips? 😊

Source: Conversation with Bing, 3/6/2023

```
(1) Basic Math Skills: Definitions, Examples and Improving Them. https://www.indeed.com/career-advice/career-development/basic-math-skills Accessed 3/6/2023.
(2) 12 Ways to Improve Math Skills Fast & Effectively - Study Equation. https://www.studyequation.com/12-ways-to-improve-math-skills-fast-effectively/ Accessed 3/6/2023.
(3) Helpful Concepts for Improving Students' Math Skills - ThoughtCo. https://www.thoughtco.com/steps-to-doing-well-in-math-2312095 Accessed 3/6/2023.
(4) 10 Ways to Improve Math Skills - wikiHow. https://www.wikihow.com/Improve-Math-Skills Accessed 3/6/2023.
(5) 9 Ways To Improve Math Skills Quickly | Prodigy. https://www.prodigygame.com/main-en/blog/improve-math-skills/ Accessed 3/6/2023.
(6) How to improve your math skills | Popular Science. https://www.popsci.com/story/diy/learn-math/ Accessed 3/6/2023.
```

There are many online courses that can help you refresh your math skills from basic algebra to linear algebra and calculus. Some of them are free and some of them may offer credits or certificates for a fee.

According to one source¹, some of the free online courses that cover these topics are:

- Introduction to Algebra (SchoolYourself)
- Introduction to Geometry (SchoolYourself)
- Precalculus (ASUx)
- College Algebra and Problem Solving (ASUX)
- Introduction to Probability (HarvardX)
- Pre-University Calculus (DelftX)
- Calculus 1A: Differentiation (MITX)
- Calculus Applied! (HarvardX)

You can find these courses on edX¹⁶, a platform that offers courses from various universities and institutions.

Another source² offers a paid course called Mathematics: Refresher Course, which covers basic algebra, logarithms, trigonometry, derivatives, and integration. You can find this course on Udemy², a platform that offers courses from various instructors.

Some colleges also offer refresher courses for math before the start of each semester. For example, Alamo Colleges³⁴ offer Math Refresher courses that are 16 hours long and cover topics such as fractions, decimals, percentages, ratios, proportions, exponents, radicals, linear equations, quadratic equations etc.

I hope this helps you find a suitable math refresher course for your needs. Do you have any questions about these courses? 😊

Source: Conversation with Bing, 3/6/2023

```
(1) Improve Your Math Skills with 9 Free Online Courses - edX. https://blog.edx.org/improve-your-math-skills-with-5-free-online-courses Accessed 3/6/2023.
(2) Learn Math With Online Courses, Classes, and Lessons | edX. https://www.edx.org/learn/math Accessed 3/6/2023.
(3) Mathematics: Refresher Course | Udemy. https://www.udemy.com/course/mathematics-refresher-course/ Accessed 3/6/2023.
(4) Refresher Courses | Alamo Colleges. https://www.alamo.edu/academics/academic-resources/refresher-courses/ Accessed 3/6/2023.
(5) NVC : Refresher Course | Alamo Colleges. https://www.alamo.edu/nvc/admissions-and-aid/how-to-apply/ftic/refresher-course/ Accessed 3/6/2023.
(6) Improve Your Math Skills with 9 Free Online Courses - edX. https://blog.edx.org/improve-your-math-skills-with-5-free-online-courses Accessed 3/6/2023.
```

What is the Maxwell Heaviside Schrödinger approach to physics as in an alternative to quantum.

The Maxwell Heaviside Schrödinger approach to physics is a hybrid method that combines classical electromagnetism and quantum mechanics to describe the interaction between electromagnetic fields and charged particles¹³. It uses the Maxwell equations for the electromagnetic field and the Schrödinger equation for the wave function of the particle². It is not an alternative to quantum physics, but rather a way of applying it to certain systems⁴.

The Heaviside notation is a way of simplifying Maxwell’s equations using vector calculus⁵⁶. It reduces the number of equations from 20 to 4 and makes them easier to manipulate. It was introduced by Oliver Heaviside in the late 19th century⁶.

Source: Conversation with Bing, 3/5/2023

```
(1) Comparison Between Maxwell–Schrödinger and Maxwell ... - ResearchGate. https://www.researchgate.net/publication/260947187_Comparison_Between_Maxwell-Schrodinger_and_Maxwell-Newton_Hybrid_Simulations_for_Multi-Well_Electrostatic_Potential Accessed 3/5/2023.
(2) Derivation of the Maxwell–Schrödinger equations: A note on the infrared .... https://aip.scitation.org/doi/10.1063/5.0093786 Accessed 3/5/2023.
(3) Comparison Between Maxwell–Schrödinger and Maxwell ... - ResearchGate. https://www.researchgate.net/publication/260947187_Comparison_Between_Maxwell-Schrodinger_and_Maxwell-Newton_Hybrid_Simulations_for_Multi-Well_Electrostatic_Potential Accessed 3/5/2023.
(4) A unified Hamiltonian solution to Maxwell–Schrödinger equations for .... https://www.sciencedirect.com/science/article/abs/pii/S0010465517300486 Accessed 3/5/2023.
(5) A derivation of Maxwell's equations using the Heaviside notation. https://royalsocietypublishing.org/doi/10.1098/rsta.2017.0447 Accessed 3/5/2023.
(6) History of Maxwell's equations - Wikipedia. https://en.wikipedia.org/wiki/History_of_Maxwell%27s_equations Accessed 3/5/2023.
```

How did Planck derive h and k. How could this have been derived wrong?

Planck derived h and k by trying to explain the phenomenon of blackbody radiation, which is the emission of electromagnetic radiation by a perfect absorber of heat. He assumed that the radiation was emitted by discrete units of energy called quanta, whose energy was proportional to their frequency. He then used statistical mechanics to derive a formula for the intensity of radiation as a function of frequency and temperature³⁵.

The formula he obtained was:

`B(ν,T) = (2hν³/c²) / (e^(hν/kT) - 1)`

where B(ν,T) is the spectral radiance per unit frequency, ν is the frequency, T is the absolute temperature, c is the speed of light, h is Planck’s constant and k is Boltzmann’s constant⁵.

He then compared his formula with experimental data and found that it fitted very well. He also used his formula to derive two other laws: Wien’s displacement law and Stefan-Boltzmann law⁵. Wien’s displacement law states that the peak wavelength of blackbody radiation is inversely proportional to its temperature⁴. Stefan-Boltzmann law states that the total power emitted by a blackbody is proportional to its surface area and the fourth power of its temperature⁵.

Planck’s derivation of h and k was not wrong, but it was based on some assumptions that were later challenged by quantum mechanics. For example, he assumed that quanta could only be emitted or absorbed in whole numbers, which implied that energy was discrete rather than continuous. He also assumed that quanta could have any frequency, which implied that there was no limit to how much energy they could have. These assumptions were later modified by Einstein and Bohr, who introduced concepts such as photons, energy levels and quantum numbers⁵.

I hope this answers your question about Planck’s derivation of h and k. Do you have any follow-up questions? 😊

Source: Conversation with Bing, 3/6/2023

```
(1) Learning from the Past: Planck's theory and thermodynamics. https://pubsapp.acs.org/subscribe/archive/ci/31/i02/html/02learning.html Accessed 3/6/2023.
(2) Planck's law - Wikipedia. https://en.wikipedia.org/wiki/Planck%27s_law Accessed 3/6/2023.
(3) Wien's displacement law - Wikipedia. https://en.wikipedia.org/wiki/Wien%27s_displacement_law Accessed 3/6/2023.
(4) planck function - New Mexico State University. https://bing.com/search?q=Planck+derivation+of+h+and+k Accessed 3/6/2023.
(5) planck function - New Mexico State University. https://pds-atmospheres.nmsu.edu/education_and_outreach/encyclopedia/planck_function.htm Accessed 3/6/2023.
(6) Planck’s radiation law | Definition, Formula, & Facts | Britannica. https://www.britannica.com/science/Plancks-radiation-law Accessed 3/6/2023.
```

if someone were challenging quantum theory based on these assumptions being wrong about whole numbers, how might they articulate their argument do someone who is not particularly good at math. Put it in simple terms and use analogies and metaphors if necessary.

One possible way to challenge quantum theory based on the assumption of whole numbers is to use the analogy of a staircase. Imagine that you are standing on a staircase and you want to go up or down. You can only move by taking one step at a time, and each step has the same height. This is similar to how Planck assumed that quanta could only be emitted or absorbed in whole numbers, and each quantum had the same amount of energy. Now imagine that someone tells you that the staircase is actually not made of steps, but of a smooth ramp. You can move up or down by any amount you want, and your energy changes continuously as you do so. This is similar to how quantum theory later showed that energy was not discrete but continuous, and quanta could have any frequency. You might find this hard to believe, because you have always seen the staircase as made of steps. You might ask for evidence or proof that the ramp exists. You might also wonder how the ramp explains some phenomena that seem to depend on steps, such as why there is a limit to how hot an object can get (the Planck temperature) or why some atoms emit light only at certain wavelengths (the spectral lines). To answer these questions, quantum theory would have to show you some experiments or observations that support the ramp model over the step model. For example, it would have to show you how light behaves like both a wave and a particle (the wave-particle duality), how electrons can jump between different energy levels in atoms (the quantum jumps), or how two particles can influence each other even when they are far apart (the quantum entanglement). I hope this analogy helps you understand how someone might challenge quantum theory based on the assumption of whole numbers. Do you have any questions about this analogy? 😊

is it common for people to challenge quantum theory this way? if so, what scientists names are associated with it and where can I learn more. and is it considered credible?

It is not very common for people to challenge quantum theory this way, because quantum theory has been very successful in explaining many phenomena and experiments at the atomic and subatomic level. However, there are some unsolved problems and open questions in quantum theory that some people try to address or challenge. For example¹:

- Questions of the interpretation of quantum theory, which are of more interest to philosophers of science than many practicing physicists. These questions deal with issues such as what does it mean for a quantum system to be in a superposition of states, how does a measurement collapse the wave function, and what is the role of the observer in quantum theory⁵.
- Questions of how to include gravity in quantum theory, which actually is a critically important and active field of research today. This question deals with how to reconcile quantum theory with general relativity, which is the classical theory of gravity. Quantum theory and general relativity seem to be incompatible at very high energies or small distances, such as near black holes or during the Big Bang².

Some scientists names that are associated with challenging quantum theory are Albert Einstein, Niels Bohr, Erwin Schrödinger, Werner Heisenberg, David Bohm, John Bell, Hugh Everett III and Roger Penrose⁶. You can learn more about their contributions and debates by reading books or articles about them.

Quantum theory is considered credible by most physicists because it has been tested and confirmed by many experiments and observations. However, it is also considered incomplete or provisional by some physicists because it does not explain everything and it has some paradoxes or puzzles that remain unresolved.

I hope this answers your question about challenging quantum theory. Do you have any follow-up questions? 😊

Source: Conversation with Bing, 3/6/2023

```
(1) The 2 Biggest Unsolved Problems in Quantum Mechanics. https://www.huffpost.com/entry/the-2-biggest-unsolved-problems-in-quantum-mechanics_b_59f7fb10e4b094db8e76f831 Accessed 3/6/2023.
(2) Philosophical Issues in Quantum Theory - Stanford Encyclopedia of .... https://plato.stanford.edu/entries/qt-issues/ Accessed 3/6/2023.
(3) Quantum theory's last challenge | Nature. https://www.nature.com/articles/35047210 Accessed 3/6/2023.
(4) Einstein, Bohr and the war over quantum theory - Nature. https://www.nature.com/articles/d41586-018-03793-2 Accessed 3/6/2023.
(5) Why Has Science Stopped Trying to Understand Quantum Theory?. https://projectqsydney.com/why-has-science-stopped-trying-to-understand-quantum-theory/ Accessed 3/6/2023.
(6) Challenges to a Quantum-Theoretic Social Theory. https://journals.sagepub.com/doi/full/10.1177/0305829818781691 Accessed 3/6/2023.
```

How are aluminum clocks so accurate in relativity experiments?

Aluminum clocks are atomic clocks that use a single aluminum ion as their timekeeping element. They are very accurate because they have a high “Q factor”, which means they can absorb and retain optical energy very well²³⁴. This allows them to measure tiny effects of relativity, such as time dilation and gravitational redshift¹²³. These effects occur when the clock’s speed or position changes relative to another clock or observer. For example, a clock that is higher up or moving faster will run slightly faster than a clock that is lower down or moving slower¹²⁴.

Source: Conversation with Bing, 3/6/2023

```
(1) NIST pair of aluminum atomic clocks reveal Einstein's relativity at a .... https://www.sciencecodex.com/nist_pair_of_aluminum_atomic_clocks_reveal_einsteins_relativity_at_a_personal_scale Accessed 3/6/2023.
(2) Pair of aluminum atomic clocks reveal Einstein's relativity at a .... https://www.sciencedaily.com/releases/2010/09/100923142436.htm Accessed 3/6/2023.
(3) NIST Pair of Aluminum Atomic Clocks Reveal Einstein's Relativity at a .... https://www.nist.gov/news-events/news/2010/09/nist-pair-aluminum-atomic-clocks-reveal-einsteins-relativity-personal-scale Accessed 3/6/2023.
(4) NIST pair of aluminum atomic clocks reveal Einstein's relativity at a .... https://www.sciencecodex.com/nist_pair_of_aluminum_atomic_clocks_reveal_einsteins_relativity_at_a_personal_scale Accessed 3/6/2023.
(5) Three Experiments That Show Relativity Is Real - Forbes. https://www.forbes.com/sites/chadorzel/2015/07/22/three-experiments-that-show-relativity-is-real/ Accessed 3/6/2023.
```

The distance to the event horizon of a black hole is called the **Schwarzschild radius**⁴⁶. It is the distance from the center of the black hole where nothing can escape its gravity, not even light¹⁵. The Schwarzschild radius depends on the mass of the black hole. For example, the supermassive black hole at the center of our galaxy has a Schwarzschild radius of about **7.9 million miles (12.7 million km)**⁵.

Source: Conversation with Bing, 3/6/2023

```
(1) Schwarzschild Radius Calculator. https://www.omnicalculator.com/physics/schwarzschild-radius Accessed 3/6/2023.
(2) black holes - What's the proper distance from the event horizon to the .... https://physics.stackexchange.com/questions/526846/whats-the-proper-distance-from-the-event-horizon-to-the-singularity Accessed 3/6/2023.
(3) What happens beyond an event horizon? - UC Santa Barbara. http://scienceline.ucsb.edu/getkey.php?key=1842 Accessed 3/6/2023.
(4) What is a black hole event horizon (and what happens there)?. https://www.space.com/black-holes-event-horizon-explained.html Accessed 3/6/2023.
(5) What happens beyond an event horizon? - UC Santa Barbara. http://scienceline.ucsb.edu/getkey.php?key=1842 Accessed 3/6/2023.
(6) Telescopes Get Extraordinary View of Milky Way's Black Hole. https://www.jpl.nasa.gov/edu/news/2022/5/12/telescopes-get-extraordinary-view-of-milky-ways-black-hole/ Accessed 3/6/2023.
```