Among the *real* numbers, some are *integer*.

Others are *rational*, i.e. they are solutions of a linear equation such as

$$px=q$$ where $p,q$ are integer. The numbers not falling in this scheme are called *irrational*.

A rather obvious generalization of this principle are numbers that are solutions of a polynomial equation such as

$$px^3+qx^2+rx+s=0$$ where $p,q,r,s$ are integer (any other degree can do). These numbers are called *algebraic*, which is the converse of *transcendental*.

The algebraic numbers enjoy a special property: even though there is an infinity of them, they can be numbered (they are said to be countable). By contrast, the transcendental numbers cannot, there is a "larger" infinity of them.

You easily understand that all integers are rational and all rationals are algebraic.

Among the functions of the real variable, some are *polynomials*.

A *rational fraction* is the quotient of two polynomials, i.e. a function $y=\dfrac{Q(x)}{P(x)}$, that verifies an equation like

$$P(x)y=Q(x).$$

More generally, an *algebraic function* $y=f(x)$ is such that it can be expressed as the root of a polynomial with coefficients that are themselves polynomials in $x$:

$$P(x)y^3+Q(x)y^2+R(x)y+S(x)=0.$$

A function that is not algebraic is called *transcendental*.

Looking closer, one can observe that algebraic items are defined from equations that use a finite number of additions and multiplications. Transcendental items require "stronger" tools (such as an infinite number of terms).