**Linear world**

Systems of Linear equations, part I: two unknowns

Systems of Linear equations, part II: three or more
unknowns

The Cartesian coordinate system and graph

__Comments__:

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**

Polynomials, expansion and
factoring

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 **

Composition of functions and
inverse functions

__Comments__:

The concept
of a function is very useful and important in mathematics.

**Analytic
Geometry **

Distances between points and
equations for circles

Tangent lines to parabolas and circles

__Comments__:

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.

**Arithmetic
series and geometric series**

Arithmetic series:
1+2+3+¡¦+99+100=?

Convergence and divergence of 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.