Rational vs. irrational vs. algebraic vs. transcendental numbers
Real numbers are classified in two independent ways. Every real number is either rational or irrational. Separately, every number is either algebraic or transcendental. Irrational does not mean transcendental: $\sqrt{2}$ is irrational but algebraic.
rational numbers ⊂ algebraic numbers and transcendental numbers ⊂ irrational numbers (for real numbers).
Definitions
A rational number is a number that can be written as a fraction of whole numbers, or the negative of one.
A rational number is a number that can be written as $p/q$, where $p$ and $q$ are integers and $q \neq 0$.
A rational number is an element of $\mathbb{Q}$, constructed as equivalence classes of pairs $(p, q)$ of integers such that $q \neq 0$, where $(p, q) \sim (r, s)$ if $p s = q r$.
Examples
-
$\frac{1}{2}$ and $\frac{2}{4}$
$\frac{1}{2}$ and $\frac{2}{4}$ represent the same rational number.
-
$1/2$
$1/2$ is a rational number because it can be written as a ratio of $1$ and $2$.
-
$2$
$2$ is rational number because it can be written as a ratio of two integers: $2/1$.
Non-examples
-
$\pi$
$\pi$ is not a rational number, because it is not a ratio of two integers.
-
$i = \sqrt{-1}$
$i$ not rational because is not a real number.
- $\sqrt{2}$
Properties
- A real number is rational if and only if its decimal expansion either ends or repeats.
- Between every two real numbers there is a rational number: $\mathbb{Q}$ is dense in $\reals$.
An irrational number is a number that cannot be written as a fraction of whole numbers (or negative whole numbers).
An irrational number is a real number that cannot be written as a fraction of integers.
Examples
- $\pi$
- $e$
-
$\sqrt{2}$
$\sqrt{2}$ is irrational, because it is not a ratio of integers.
Non-examples
-
$i = \sqrt{-1}$
$i$ is not irrational because is not a real number.
- $1/2$
-
$2$
$2$ is not irrational because it is rational.
Properties
- The decimal expansion of an irrational number neither ends nor repeats.
- The irrational numbers are uncountable, while the rational numbers are countable.
A real number is any number on the number line. Every real number can be written as a (possibly infinite) decimal.
A real number is an element of $\reals$, the unique complete ordered field.
Examples
-
$\pi$
$\pi$ is a real number, because it is the ratio of a circle’s circumference to its diameter, a point on the number line.
- $e$
- $1/2$
- $\sqrt{2}$
- $2$
Non-examples
Properties
- Every nonnegative real number $x$ can be written as a decimal $x = n.d_1 d_2 d_3 \ldots$, where $n$ is a natural number and each digit $d_k$ is one of $0, 1, \dots, 9$, which means that \[x = n + \sum_{k = 1}^{\infty} \frac{d_k}{10^k}.\] The digits are found one at a time: $n$ is the largest whole number at most $x$, and each digit $d_k$ is the largest digit such that the decimal $n.d_1 \ldots d_k$ is at most $x$. The decimals $n.d_1 \ldots d_k$ then lie within $10^{-k}$ of $x$, so the decimals converge to $x$. A negative real number is written as the negative of a decimal.
- A real number is rational exactly when its decimal expansion either ends or repeats, such as $\frac{1}{4} = 0.25$ and $\frac{1}{3} = 0.333\ldots$, so the decimal expansion of an irrational number, such as $\sqrt{2} = 1.41421\ldots$, neither ends nor repeats.
- Some real numbers have two decimal expansions: a decimal that ends, such as $1.0$, equals the decimal that instead ends with repeating $9$s, such as $0.999\ldots$, because $\sum_{k = 1}^{\infty} \frac{9}{10^k} = 1$.
- $\reals$ can be constructed from $\mathbb{Q}$ using Dedekind cuts or Cauchy sequences.
- Every nonempty set of real numbers that is bounded above has a least upper bound (the completeness of $\reals$).
A number $a$ is an algebraic number if $a$ is a solution to a polynomial equation whose coefficients are all integers.
Consider a real or complex number $a$. The number $a$ is an algebraic number if $a$ is a root of a nonzero polynomial with integer coefficients.
Examples
-
$i = \sqrt{-1}$
$i$ is algebraic, because it is a root of $x^2 + 1$, so it is not transcendental.
-
$1/2$
$1/2$ is algebraic because it is a root of $2x - 1$.
-
$\sqrt{2}$
$\sqrt{2}$ is an algebraic number, because it solves $x^2 = 2$, or equivalently, it is a root of $x^2 - 2$.
-
$2$
$2$ is algebraic because it is a root of $x - 2$.
Non-examples
A transcendental number is a number that does not solve any polynomial equation with integer coefficients.
A transcendental number is a (real or complex) number that is not algebraic.
Examples
Non-examples
- $i = \sqrt{-1}$
- $1/2$
- $\sqrt{2}$
-
$2$
$2$ is not transcendental because it is algebraic.
Examples
| Example | rational number | irrational number | real number | algebraic number | transcendental number | Why |
|---|---|---|---|---|---|---|
| $2$ | ✓ | ✗ | ✓ | ✓ | ✗ | |
| $1/2$ | ✓ | ✗ | ✓ | ✓ | ✗ | |
| $\sqrt{2}$ | ✗ | ✓ | ✓ | ✓ | ✗ | |
| $\pi$ | ✗ | ✓ | ✓ | ✗ | ✓ | |
| $i = \sqrt{-1}$ | ✗ | ✗ | ✗ | ✓ | ✗ |
Notes
- “Irrational” only applies to real numbers. A complex number such as $1 + i$ is neither rational nor irrational.
- The algebraic numbers are countable, so almost every real number is transcendental, even though it is hard to prove that any particular number (such as $\pi$) is transcendental.