Linear world

Variables and Expressions

Linear equations

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.


Nonlinear world


Root, cube root and nth root

Polynomials, expansion and factoring

Polynomial division

Quadratic equation

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 Heisenbergs uncertainty principle in quantum mechanics.



What is a function?

Composition of functions and inverse functions



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


Analytic Geometry

The Pythagorean theorem

Distances between points and equations for circles

The triangle inequality

Tangent lines to parabolas and circles



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

Geometric 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.