irrational numbers

Irrational Numbers π

Irrational Numbers List πφτ 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.

  • ADDucation’s irrational numbers list compiled by Joe Connor and last updated Nov 7, 2022 @ 12:28 pm
Frequently Asked Questions About Irrational Numbers FAQ 🔽
What are Integers?
Integers are “whole numbers” which can be positive or negative but NOT fractions. For example: 1,2,3,-1,-2,-3 etc.
What are Fractions?
A fraction is part of a whole and can be written as:
  • A fraction: e.g. 1/4
  • A percentage: e.g. 25%
  • In decimal: e.g. 0.25

What are Real Numbers?
Real numbers exist and have a physical value which can be expressed in numerals.

  • Real numbers can be integers.
  • Real numbers can be fractions.
  • Infinity ∞ is NOT a real number but affinely extended real numbers systems makes it possible to do arithmetic by treating infinity as if it’s a real number.

What are Imaginary Numbers / Unreal / Non-real Numbers?
Imaginary numbers do not have a physical value and cannot be expressed in numbers so they are notated using “i” (for imaginary).
Any positive or minus number squared (multiplied by itself) results in a positive number (or zero) so squaring never results in a negative number and because square roots undo squaring it follows negative numbers cannot have square roots.
However consider i x i = -1 then take the square root of both sides and the result is i = √-1 which cannot exist so it’s called an imaginary number.

Any real number multiplied by the square root of -1 (√-1) is an imaginary number. For example: 4i is an imaginary number and its square is -16 (4i x 4i = −16).
Infinity ∞ is an imaginary number (except when expressed as a real number).
Imaginary numbers makes algebra more elegant and makes solving some real world calculations easier.

What are Complex Numbers?
Complex numbers combine a real number and an imaginary number.
Because either number could be 0 (zero) both real numbers and imaginary numbers can also be complex numbers. These examples cover each possibility:

Example Complex Numbers
Complex Number
Real Part
Imaginary Part
Complex number
5 + 3i
5
3
Complex number and real number
9
9
0
Complex number and imaginary number
−4i
0
−4

What are Rational Numbers?
Rational numbers are a small subset of real numbers which can be written as a fraction or ratio. For example:

  • 3 is a rational number because it can be written as a ratio 3/1
  • 1.5 is rational number because it can be written as a fraction 3/2
  • 0.333… is a rational number because it can be written as the fraction 1/3

What are Irrational Numbers?
Irrational numbers are real numbers which cannot be written as a fraction. The decimal expansion of an irrational number never ends and never repeats. Pi for example: π=3.141592653589793…

Are there more irrational numbers or more rational numbers?
There are infinitely more irrational numbers than rational numbers. Why? That’s beyond the scope of this numbers FAQ but this convinced us: Between any two rational numbers there will be an infinite number of irrational numbers.

What are Transcendental Numbers?
A transcendental number is a real number or an imaginary number that is not an algebraic number. There are more transcendental numbers than algebraic numbers but it’s not easy to prove a specific number is transcendental. Joseph Liouville (1809-1882) “Liouville’s theorem” first proved the existence of transcendental numbers in 1844. Charles Hermite proved that the number e was transcendental in 1873. Ferdinand von Lindemann proved that pi was transcendental in 1882.

What are Algebraic Numbers?
Here are a few pointers:

  • All rational numbers and integers are algebraic.
  • Irrational numbers may or may not be algebraic
  • If a number is not an algebraic number then it is a transcendental number.

What are Morphic Numbers?
There are only two morphic numbers; the golden number and the plastic number, which are both irrational numbers. It has been shown there are no other real morphic numbers greater than 1.

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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.
This ratio can also be derived from the Fibonacci sequence, in which each term is the sum of the previous two terms i.e. 1,1,2,3,5,8,13,21, etc. The further along the Fibonacci sequence, the closer the ratio of consecutive terms gets to 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
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 two times the radius).
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 Proper 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.
See also: ⏩ Fundamental physical constants list… | 🔀 Units of measurement list…

* Visit the On-Line Encyclopedia of Integer Sequences (OEIS) for the latest news and exact values of irrational numbers. The OEIS was originally called “Sloane’s”.

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