Irrational Numbers List & FAQ About Number Types
Irrational numbers are real numbers which cannot be written as a fraction. The decimal expansion of irrational numbers, like Pi (π=3.141592653589793), never end and never repeat. ADDucation’s list of irrational numbers also includes constants, algebraic numbers, transcendental numbers, two mysterious morphic numbers and FAQs about number types.
ADDucation Tips: Click arrows in column headings to sort irrational numbers. Reload page for original sort order. Click the + icon to expand columns on tablets and mobiles. Set your browser to full screen and zoom out to display all irrational numbers. Start typing in the Filter table box to find anything inside the table.
|Irrational Numbers||Symbol/s||Number type/s||Decimal expansion||OEIS*||E Notation / Scientific Notation Value||Irrational Numbers Key Facts & Info|
|√2 (aka Pythagorean constant, the square root of 2 and (1/2)th power of 2)||√2||irrational number, algebraic number||1.414213562373095048|
|A002193||1.4142135623730951e+0||The square root of 2 is also known as the Pythagorean constant, after Hippasus of Metapontum, a Pythagorean philosopher who, around 500 BCE, demonstrated the square root of 2 could not be expressed as a ratio of integers so this was probably the first number to be proven to be irrational. In trigonometry √2 is the length of the hypotenuse of a right triangle, when the length of the other two sides are 1.|
There exist many different mathematical proofs of √2’s irrationality, including one so concise that it fits in a tweet:
|e (aka Euler’s number and Napier’s constant)||e||irrational number, transcendental number (not yet proven)||2.718281828459045235|
|A001113||2.718281828459045e+0||The history of the existence of e as a mathematical constant, irrational number and transcendental number (not yet proven) is interesting and the story continues to unfold today. As a mathematical constant e was first implied by John Napier in a list of logarithms calculated from the constant, published in 1618. Jacob Bernoulli, a Swiss mathematician, is credited with the discovery of the constant in 1683 when studying compound interest. Using the binomial theorem Bernoulli calculated the value of e was between 2 and 3. In 1748 Leonhard Euler (pronounced “Oiler”), another Swiss mathematician, published “Introduction to Analysis of the Infinite” which included the value of e to 18 decimal places (e = 2.718281828459045235) and popularized the use of the letter e to represent it. As a mathematical constant e (pronounced “ee”) also appears in calculus, number and probability theory.|
|φ (“Phi” aka golden number, golden ratio, divine proportion)||φ||irrational number, algebraic number, morphic number||1.618033988749894848|
The irrational number φ is well known as the golden ratio and divine proportion. In geometry φ can be expressed as the ratio of a regular pentagon’s diagonal to the length of a side. Any two numbers (e.g. x and y) are considered in the golden ratio if (x+y)/y = φ.
|pi||π||irrational number, transcendental number||3.141592653589793238|
|A000796||3.141592653589793e+0||Pi is a constant number which can be calculated by dividing the circumference of a circle by its diameter (or radius squared).|
|tau||τ||irrational number, transcendental number||6.283185307179586476|
|A000796||6.283185307179586e+0||Tau is a circle constant representing the ratio between circumference and radius and is preferred to 2π by Michael Hartl and other Tauists who have sparked plenty of π versus τ media coverage rebuffed by Pi-elitists and The Pi Manifesto.|
|Apéry’s constant||ζ(3)||irrational number||1.2020569031595942853|
|A002117||1.2020569031595942e+0||Apéry’s Constant was proven to be irrational in 1978 by French mathematician Roger Apéry. It’s still not known if Apéry’s Constant is transcendental. ζ is the Riemann Zeta Function.|
|Plastic number (aka plastic constant, plastic ratio, Siegel’s number, minimal Pisot number, platin number, le nombre radiant “the radiant number”)||ρ||irrational number, morphic number||1.324717957244746025|
|A060006||1.324717957244746e+0||The Plastic number is a mathematical constant which was discovered in 1924 by French engineer Gérard Cordonnier. He named the constant “le nombre radiant” (the radiant number). In 1928 it was named “het plastische getal” (the plastic number) by Dutch architect Dom Hans van der Laan using the adjective “plastic” to describe its three dimensional properties.|
The Exact Values of Irrational Numbers
*For the latest news and values of irrational numbers visit the On-Line Encyclopedia of Integer Sequences (OEIS) originally called “Sloane’s”.
Related ADDucation Lists:
- List of Commonly Used Abbreviations (acronyms)…
- Please help us improve ADDucation’s list of irrational numbers or correct errors and omissions by adding your comments below. Thank you.