Algebra
All About Root Systems
A visual introduction
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Analysis
Proving the Mean Value Theorem
Let’s prove one of the most important theorems in elementary analysis! The only requirement is an elementary understanding of derivatives.
Classical Mechanics
The Curious Case of the Inverted Pendulum
Everyone who has studied basic physics, mechanics, dynamics or control system have encountered the inverted pendulum.
Mathematics
The Duties of John von Neumann’s Assistant in the 1930s
The year is 1933. A 26 year-old former graduate student at Columbia University, Edgar R. Lorch (1907–1990) has just completed his Ph.D. in mathematics and is, in his own words “by some miracle” awarded a National Research Council Fellowship for a year of postdoctoral studies at Harvard University.
Probability Theory
What is Probability For?
To start with an analogy, what is money? Textbooks and Wikipedia say that money is a medium of exchange, and a store of value, and a unit…
Graph Theory
Turán’s Theorem for Graphs
Using the Probabilistic Method and the Cauchy-Schwarz inequality
Linear Algebra
A Method of Counting The Number of Solutions
Linear algebraic equations are one of the simplest equations that we can solve. If there is only one variable, then the solutions are…
Analysis
Four Curious, Counter-Intuitive Mathematical Truths
In mathematics, it is often the case that we derive things through proof or previous results that we don’t fully understand.
Prime Numbers
How to Prove the Infinity of Primes
Two simple proofs about the infinity of primes
Geometry
Pythagoras’ Theorem With Proof
Where the sum of two squares meets the third.
Set Theory
What Is Zorn’s Lemma?
The quick and dirty guide to one of the stranger mathematical curiosities.
Number Theory
On Bernoulli Numbers
My Journey of Rediscovering Some of Mathematics’ Most Beautiful Numbers