**Middle
School Mathematics**

Addition and subtraction of negative numbers

Multiplication and division of negative numbers

Exponents

Root,
cube root and nth root

Scientific notation

__Comments__:

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.

**Linear world**

Symbols and Expressions

Linear equations

Linear
inequalities

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 7^{th}~9^{th}
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.

**Nonlinear
world**

Root, cube root and nth root, revisited

Polynomials,
expansion, and factoring

Polynomial
division

Quadratic
equation

Graphs
of quadratic polynomials

Quadratic
inequalities and the Cauchy-Schwarz inequality

__Comments__:

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

Set theory

De Morgan¡¯s laws and the contrapositive

The concepts
of a function and a set are very useful and important in mathematics.

**Triangles**

Congruence
of triangles

Similarity
of triangles

__Comments__:

The first article
is needed to understand ¡°Lightray reflecting on a mirror¡± in ¡°Some geometries.¡±

**Analytic
geometry **

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.

**Some
geometries**

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

__Comments__:

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

Geometric
series

__Comments__:

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.