Irrational Numbers List & FAQ About Number Types
Irrational numbers are real numbers which cannot be written as a fraction. The decimal expansions of irrational numbers, e.g. 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.
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|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. Many different mathematical proofs of √2’s irrationality exist, including one so concise it fitted into a 140 character limit tweet posted by @TinyProof.|
|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.
The irrational number φ has always fascinated mathematicians, astronomers, biologists and artists because the ratio it represents is thought to have aesthetic appeal. Ratios close to φ can be found in Egyptian pyramids and in architecture and artwork from the ancient Greeks and Mayans to the Gothic and Renaissance eras to modern pieces.
|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 two times the radius).|
||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, discovered in 1924 by French engineer Gérard Cordonnier is a mathematical constant. Cordonnier named the constant “le nombre radiant” (the radiant number). Independently it was named “het plastische getal” (the plastic number) in 1928 by Dutch architect Dom Hans van der Laan. He used 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)…
- ADDucation’s irrational numbers list compiled by Joe Connor, last updated 23 Apr 2020.
- Please help us improve ADDucation’s list of irrational numbers or correct errors and omissions by adding your comments below. Thank you.