Middle School Mathematics
Addition and subtraction of negative numbers
Multiplication and division of negative numbers
Root, cube root and nth root
Even though the concepts dealt in these articles are treated in middle school, I included these articles to remind the adults who have forgotten them.
Symbols and Expressions
Systems of Linear equations, part I: two unknowns
Systems of Linear equations, part II: three or more unknowns
The Cartesian coordinate system and graph
In South Korea, after six years of drilling addition, subtraction, multiplication, and division in elementary school, you enter middle school (equivalent to 7th~9th grades). There, you learn somewhat abstract mathematics for the first time and how to express unknown quantities in terms of alphabets such as x, y, and z. The first article talks about this abstraction. The second to fourth articles are about equations. Equations involve unknowns, and by solving them, you find the unknowns. Even though you may know how to solve systems of linear equations, I encourage you to read ¡°Systems of Linear equations, part II¡± as I talk about an important concept called ¡°linear independence¡± in the final comment there. The last article is about the Cartesian coordinate system, first introduced by and named after the great French philosopher René Descartes. (Yes, the one who famously said ¡°I think, therefore I am.¡±) All the equations considered in these articles are linear equations. You will find the meaning of ¡°linear¡± in our article ¡°Polynomials, expansion and factoring¡¯ in the ¡°Nonlinear world¡¯¡¯ section.
Root, cube root and nth root, revisited
Polynomials, expansion, and factoring
Graphs of quadratic polynomials
Quadratic inequalities and the Cauchy-Schwarz inequality
Having introduced how to express unknowns abstractly and how to solve linear equations, I introduce how to solve more difficult equations (such as quadratic equations) and how to manipulate the variables arithmetically. In the last article, we introduce the Cauchy-Schwarz inequality, which is essential to prove Heisenberg¡¯s uncertainty principle in quantum mechanics.
Function and set theory
What is a function?
Composition of functions and inverse functions
De Morgan¡¯s laws and the contrapositive
The concepts of a function and a set are very useful and important in mathematics.
Congruence of triangles
Similarity of triangles
The first article is needed to understand ¡°Lightray reflecting on a mirror¡± in ¡°Some geometries.¡±
The Pythagorean theorem
Distances between points and equations for circles
The triangle inequality
Tangent lines to parabolas and circles
Slopes of two perpendicular lines
In the first article, we explain the famous theorem named after the Greek philosopher Pythagoras who died about five centuries before Jesus Christ. In the next two articles, using the Pythagorean theorem, we will see the great utility of the Cartesian coordinate system; it can translate geometry to numbers. In the last article, we will apply the knowledge we learned in ¡°Quadratic equation¡± to geometry.
Lightray reflecting on a mirror
Inscribed triangle in a circle
Inscribed quadrilateral in a circle
Inscribed circle in a triangle
The center of mass of a triangle
The three altitudes of a triangle always meet at a point
A geometric proof for an algebraic problem
Except for the first one, the articles here are by no means directly helpful to understand high-level physics or high-level math. But, I included them here as they can be good exercises for your brain. The second to the last article deals with the problem that impressed Albert Einstein when he was 12 years old.
Arithmetic series and geometric series
Arithmetic series: 1+2+3+¡¦+99+100=?
Convergence and divergence of series
What would be the sum of all the natural numbers from 1 to 100? The first article introduces a clever trick to answer this problem, discovered by Gauss when he was nine years old. The third article is necessary to understand our later article ¡°composition,¡± which in turn is important to understand ¡°Approximation of the naïve black hole degeneracy,¡± my first published research article.