# Irrational Numbers

## 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, 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 80168872420969807856 967187537694807317667… | 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 360287471352662497757 24709369995957496696 762772407663… | 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 20458683436563811772 03091798057628621… | A001622 | 1.618033988749895e+0 | 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 46264338327950288419 71693993751058209749 44592307816406286208 99862803482534211706 79821480865132823066 47093844609550582231 725359408128481117450 28410270193852110555 96446229489549303819… | 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 92528676655900576839 43387987502116419498 89184615632… | 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 997… | 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 96090885447809734… | 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”.

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- List of Commonly Used Abbreviations (acronyms)…
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