JMathG

JMathG

JMathG is a tutoring service by Jonathan M Gerhard offering any level of math tutoring

Operating as usual

Graph Theory à la Frank Harary 14/09/2023

https://www.jmathg.com/post/graph-theory-a-la-frank-harary
My first attempt at a more biographical post, focused on the life and mathematics of Frank Harary, the Founding Father of Graph Theory. He was someone who loved mathematics and found wonderful ways to share that love with the world in the form of combinatorial games and outreach.

Graph Theory à la Frank Harary A biographical look at the life and mathematics of Frank Harary, the Founding Father of Graph Theory.

05/07/2023

Interactive Voronoi Diagram Maker in Pygame. Updates live as you add points!

05/07/2023

Interactive Voronoi Diagram Maker in Pygame.

01/07/2023

Building Voronoi Diagrams in Pygame

Photos from JMathG's post 01/07/2023
Image Processing Algorithms with OpenCV 29/06/2023

https://www.jmathg.com/post/image-processing-algorithms-with-opencv
We make a few Image Processing algorithms with OpenCV: Pixelate, Cutoffs, Brighten/Darken, Blending (with multiple methods), RGB channels, and Recursive Algorithms. Also addresses bounding to stop integer overflow and how cool it looks to let the algorithms go unbounded. All done on a picture of my cat, Lady, and a beautiful colorful image by Alexander Grey.

Image Processing Algorithms with OpenCV We make a few Image Processing algorithms with OpenCV: Pixelate, Cutoffs, Brighten/Darken, Blending, RGB channels, and Recursive Algorithms.

Photos from JMathG's post 29/06/2023

Can you see my cat in these images? New post coming soon!

Photos from JMathG's post 28/06/2023

Blog post on coloring Knot Diagrams and other general Image Processing stuff in OpenCV-Python coming soon!

27/06/2023

Building the Petersen Graph in our self-made GraphClass GUI in Python!

Making a Graph Class in Python 3 - More GUI 27/06/2023

https://www.jmathg.com/post/making-a-graph-class-in-python-3-more-gui
We add a lot of functionality to the GUI - Adding/Removing vertices and edges, returning graph info, and returning some graph matrices. Includes .py and .txt files if you want to run the code or an .exe if you just want to run the program.

Making a Graph Class in Python 3 - More GUI We add a lot of functionality to the GUI - Adding/Removing vertices and edges, returning graph info, and returning some graph matrices.

Posets and Python 05/06/2023

https://www.jmathg.com/post/posets-and-python
This is an exploration of Dunder Methods and how they can be used to create abstract posets like the divisibility or p-adic posets in Python

Posets and Python This is an exploration of Dunder Methods in Python and how they can be used to study abstract posets like the divisibility or p-adic posets.

Rectangles in Polar Coordinates 16/11/2021

Published two more fun blog posts! The first does something I've always found to be a fun exercise - solve a simple problem with BIG machinery. And that's why we're finding the area of a square using integrals in polar coordinates!
https://www.jmathg.com/post/rectangles-in-polar-coordinates

And the next one is a related one but deserved its own post. The integral of cosecant! Anyone who has had to do this probably gets told to multiply by some strange expression which allows some magic to happen. Well, I have always had the idea that learning math should be about exploration and discovery. So if we're stuck on finding the integral of cosecant, how can we figure out it out without being gifted this magic expression? Here's how!
https://www.jmathg.com/post/the-integral-of-csc-θ

Rectangles in Polar Coordinates Circles in rectangular coordinates...Rectangles in polar coordinates? This is a bit of a goofy one. Why? Well, I want to figure out something very simple using something much more complicated. There are many examples of this sort of thing - and I just find it to be a great aid in understanding the m...

Learning Pygame with Catan (Part 1) 06/11/2021

https://www.jmathg.com/post/learning-pygame-with-settlers-of-catan

Put out the first in a series of posts on programming in Python/Pygame with inspiration taken from the wonderful game Catan!

Learning Pygame with Catan (Part 1) An Intro to Catan Who else here used to play Settlers of Catan as a kid? The first memory I have with the game is happily placing roads down, not realizing I was supposed to pay for them with resources. It went on for way too long before my friends yelled at me for cheating! But I was just a noob. N...

My Tutoring Philosophy (Part 2) 26/10/2021

https://www.jmathg.com/post/my-tutoring-philosophy-part-2
Just posted Part 2 of my blog post series on my tutoring philosophy! It touches on my views on teaching vs tutoring and the importance of asking questions in the pursuit of learning.

My Tutoring Philosophy (Part 2) Teaching vs Tutoring What is the best way to tutor? That's naturally the golden question here. A teacher should give their students a glimpse into a subject and the inspiration to explore it more later. A tutor should help their tutee learn a subject in the most understandable, enjoyable, and produc...

JMathG | Free Written Math Lessons | Virginia 25/10/2021

https://www.jmathg.com/free-math-lessons
Added a few more free lessons on Linear Algebra, Vector Spaces and Subspaces, and Gauss-Jordan Elimination/Row Reduction.

One thing I'll explain in my next blog post on tutoring/teaching is that I think the best way to learn a topic is to motivate it with a problem. For example, most of my written lessons are going to be me working through a problem with a detailed explanation.

Another example is an upcoming series of blog posts where we'll learn some Python (specifically Pygame) with the motivation of creating the Settlers of Catan board and running some stats on it.

Thanks for reading!

JMathG | Free Written Math Lessons | Virginia Free written math lessons for a variety of topics and levels, including educational activities for getting kids excited about math and science!

My Tutoring Philosophy (Part 1) 19/10/2021

After writing a few blog posts on the fun mathematics of the Arithmetic Derivative, I've finally published my first post of another category - Tutoring/Educational. This is where I will blog about my tutoring/teaching philosophy and other educational concepts and ideas that I find interesting!

https://www.jmathg.com/post/my-tutoring-philosophy-part-1

My first post touches on thinking/learning styles and the next will talk about asking questions and the Socratic Method!

My Tutoring Philosophy (Part 1) Tutoring is such a dynamic activity between a tutor and a tutee - to put this relationship into words would depend on exactly who the two people working together are - their existing knowledge, their desired knowledge, their learning/thinking patterns, and their teaching patterns. In a sense, this d...

JMathG | Free Written Math Lessons | Virginia 19/10/2021

Just added a few more free lessons to the site! https://www.jmathg.com/free-math-lessons
Including some short lessons on Linear Algebra, Eigenvalues, and SVD; the Euclidean Algorithm in Number Theory; and a theorem on Triangle-Free Graphs in Graph Theory and Discrete Math!

JMathG | Free Written Math Lessons | Virginia Free written math lessons for a variety of topics and levels, including educational activities for getting kids excited about math and science!

The Arithmetic Derivative (Part 4) 17/10/2021

Just made my fourth post on the Arithmetic Derivative - it includes a proof of the irrationality of sqrt(2), which is fun!

https://www.jmathg.com//post/the-arithmetic-derivative-part-4

The Arithmetic Derivative (Part 4) What do the differential equations mean? Usually, we are motivated to study differential equations by situations in everyday life, nature, physics, and math. What motivates studying this arithmetic derivative? The first answer is that it's a small adjustment to something much more commonly studied c...

The Arithmetic Derivative (Part 2) 12/10/2021

Just wrote my second blog post on Arithmetic Derivatives! https://www.jmathg.com/post/the-arithmetic-derivative-part-2
This is what you get if you take Leibniz's Product Rule and apply it to the natural numbers:
D(0) = 0
D(1) = 0
D(p) = 1 for prime p
D(ab) = aD(b) + D(a)b
The cool thing in this post is proving the infinitude of primes using the arithmetic derivative - a small adaption from Euclid's usual proof but fun nonetheless! =-) Next post will be more stats/counting focused.

It's also nice because it has examples of a proof by induction and a proof by contradiction, so it could be helpful for those trying to get comfortable with these concepts.

The Arithmetic Derivative (Part 2) Our first pattern So we've taken the usual rules for differentiation and applied them to natural numbers: D(ab) = aD(b) + D(a)b with initial conditions D(0) = 0 D(1) = 0 D(p) = 1 for all primes p We can use the following code to generate a graph of natural numbers and their derivatives: k = 10000 N....

The Arithmetic Derivative 09/10/2021

Just finished my first blog post for JMathG! It's an old favorite topic of mine - what happens if you try to define a derivative for natural numbers? Lots of FUN, that's what!

https://www.jmathg.com/post/the-arithmetic-derivative

The Arithmetic Derivative An introduction If you've done calculus, then you've talked about differentiation. We usually talk about this in terms of "linear approximations" and "limit definitions" to try to understand functions of a real variable - those smooth curves we can draw nicely, like polynomials or exponential functi...

06/10/2021

Just added a cool feature to the site that I've wanted since the beginning! Members now have a "My Lessons" page where their past lessons (with a lesson summary) will appear as PDFs.

28/09/2021

Just set up my page for free written math lessons! First one is on Sequences and Series (at a precalc level). Planning on doing a Calc 2 level one and a Combinatorics level one soon!

https://www.jmathg.com/free-math-lessons

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Videos (show all)

Interactive Voronoi Diagram Maker in Pygame. Updates live as you add points!
Interactive Voronoi Diagram Maker in Pygame.
Building Voronoi Diagrams in Pygame
Building the Petersen Graph with our self-made GraphClass GUI in Python