# Encyclopedia

.In mathematics, an irrational number is any real number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero and is therefore not a rational number.^ Quick definitions ( irrational number ) ▸ noun : a real number that cannot be expressed as a rational number .
• Definitions of irrational number - OneLook Dictionary Search 26 January 2010 1:52 UTC www.onelook.com [Source type: Reference]

^ An irrational number is a real number that cannot be expressed in the form , when a and b are integers ( b ≠ 0).
• Irrational Numbers 26 January 2010 1:52 UTC hotmath.com [Source type: Reference]

^ Together, the set of integers and the set of non-integer fractions make up the set of real numbers.
• Numbers 26 January 2010 1:52 UTC www.bsu.edu [Source type: Reference]

.Informally, this means that an irrational number cannot be represented as a simple fraction.^ Again it’s irrational as 3√2 cannot be written as a fraction.
• The difference between rational and irrational numbers in math. | Bukisa.com 26 January 2010 1:52 UTC www.bukisa.com [Source type: Reference]

^ Irrational numbers are numbers that cannot be expressed in the form  .
• aw_bennett_usingandun_2|Thinking Critically|Brief Review|Sets of Numbers 26 January 2010 1:52 UTC wps.aw.com [Source type: Reference]

^ Irrational numbers are numbers that are not rational ; that is, they cannot be represented in the form a / b where a and b are integers.
• Math Lair - Irrational Numbers 26 January 2010 1:52 UTC www.stormloader.com [Source type: FILTERED WITH BAYES]

.It can be proven that irrational numbers are precisely those real numbers that cannot be represented as terminating or repeating decimals, although mathematicians do not take that to be the definition.^ Quick definitions ( irrational number ) ▸ noun : a real number that cannot be expressed as a rational number .
• Definitions of irrational number - OneLook Dictionary Search 26 January 2010 1:52 UTC www.onelook.com [Source type: Reference]

^ Definition of irrational number .
• irrational number - Dictionary definition and pronunciation - Yahoo! Education 26 January 2010 1:52 UTC education.yahoo.com [Source type: Reference]

^ An irrational number is a real number that cannot be expressed in the form , when a and b are integers ( b ≠ 0).
• Irrational Numbers 26 January 2010 1:52 UTC hotmath.com [Source type: Reference]

.As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational.^ The real line consists of the union of the rational and irrational numbers.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

^ Is √9 a rational or irrational a number?
• The difference between rational and irrational numbers in math. | Bukisa.com 26 January 2010 1:52 UTC www.bukisa.com [Source type: Reference]

^ An irrational number is a real number which is not a rational number , meaning it cannot be expressed as a/b where a,b are integer .
• irrational number@Everything2.com 26 January 2010 1:52 UTC www.everything2.com [Source type: Reference]

[1] .Perhaps the best-known irrational numbers are π, e and √2.^ Many famous numbers are known to be irrational.

^ The best approximations to rational and irrational numbers.
• Math 126 Number Theory 26 January 2010 1:52 UTC babbage.clarku.edu [Source type: Reference]
• Math 126 Number Theory 26 January 2010 1:52 UTC aleph0.clarku.edu [Source type: Reference]

^ For close calls, when determining the proximity of xlna to numbers 0, 1 , Hurwitz's irrational number theorem gives the best rational approximation for an arbitrary irrational number xlna.
• Exponential Series, Irrational Numbers 26 January 2010 1:52 UTC www.coolissues.com [Source type: Reference]

[2][3][4]
.When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common.^ The Greeks perceived that there are segments which are not in the ratio of two whole numbers, and therefore have no common measure or are incommensurable.
• Heinrich Tietze on Numbers, Part 2 26 January 2010 1:52 UTC www.gap-system.org [Source type: FILTERED WITH BAYES]

^ We can say, then, that an irrational number is a number that has no common measure with 1.
• Irrational numbers. Evolution of the real numbers. 26 January 2010 1:52 UTC www.themathpage.com [Source type: FILTERED WITH BAYES]

^ Irrational numbers on a number line – Where are they?
• Faculty of Education Research Profiles » Dr. Rina Zazkis 26 January 2010 1:52 UTC www.educ.sfu.ca [Source type: Academic]

.A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as I.^ Inasmuch as numbers name the lengths of lines, then is a number.
• Rational and irrational numbers - Topics in precalculus 26 January 2010 1:52 UTC www.themathpage.com [Source type: Reference]

^ In the same way can be represented on the number line.
• Real Numbers | TutorVista 26 January 2010 1:52 UTC www.tutorvista.com [Source type: Reference]

^ The students have studied what it means for a number to be irrational, and they cannot understand how that can "measure" a line that obviously stops.

The number $\scriptstyle\sqrt{2}$ is irrational.

## History

.The concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c.^ Since somewhere around 500 BC, beginning with the counting numbers, 1,2,3...up to very many, mathematicians have expanded the concept of number to include negative numbers, zero, rational numbers, irrational numbers, real numbers, imaginary numbers, complex numbers and the list goes on.
• Complex Numbers 26 January 2010 1:52 UTC mcanv.com [Source type: Reference]

^ According to legend, the ancient Greek Hippasus, from the 5th century BC, discovered the existence of numbers which can't be represented by a fraction, called irrational numbers.
• http://www.worldalmanacforkids.com/WAKI-ViewArticle.aspx?pin=x-nu068900a&article_id=486&chapter_id=9&chapter_title=Numbers&article_title=Number_Systems 26 January 2010 1:52 UTC www.worldalmanacforkids.com [Source type: Reference]

^ For many centuries prior to the actual proof, mathematicians had thought that pi was an irrational number.

.750–690 BC) believed that the square roots of certain numbers such as 2 and 61 could not be exactly determined.^ The square root of 2 is an irrational number.
• Introduction to Proofs 26 January 2010 1:52 UTC zimmer.csufresno.edu [Source type: Original source]

^ Only the square roots of square numbers.
• Irrational numbers. Evolution of the real numbers. 26 January 2010 1:52 UTC www.themathpage.com [Source type: FILTERED WITH BAYES]

^ Find the square root of each of the following numbers: .
• Rational and Irrational Numbers 26 January 2010 1:52 UTC www.tpub.com [Source type: Reference]

[5] .Another source suggests that irrational numbers were noticed as early as 1500 BC in India by Nilakantha.^ Euler's Number (e) is another famous irrational number.
• Maths: Rational and Irrational Numbers - e Homework (UK) 26 January 2010 1:52 UTC www.ehomework.co.uk [Source type: FILTERED WITH BAYES]

^ The number e ( Euler's Number ) is another famous irrational number.
• Irrational Numbers 26 January 2010 1:52 UTC www.mathsisfun.com [Source type: FILTERED WITH BAYES]

^ According to legend, the ancient Greek Hippasus, from the 5th century BC, discovered the existence of numbers which can't be represented by a fraction, called irrational numbers.
• http://www.worldalmanacforkids.com/WAKI-ViewArticle.aspx?pin=x-nu068900a&article_id=486&chapter_id=9&chapter_title=Numbers&article_title=Number_Systems 26 January 2010 1:52 UTC www.worldalmanacforkids.com [Source type: Reference]

[6]
.The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),[7] who probably discovered them while identifying sides of the pentagram.^ Irrational numbers, the Pythagorean Theorem, and a proof that the square root of 2 is irrational.
• VanWyk's 107 outline for Chapter 2 26 January 2010 1:52 UTC www.math.jmu.edu [Source type: Academic]

^ The discovery of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2.
• Dror Bar-Natan: Classes: 2004-05: Math 157 - Analysis I: History of the Theory of Irrational Numbers 26 January 2010 1:52 UTC www.math.utoronto.ca [Source type: General]

^ Proof that the square root of 2 is irrational number .
• Proof that the square root of 2 is irrational number 26 January 2010 1:52 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

[8] .The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other.^ What other type of length could there be?
• Our Playground: The Real Numbers and Their Development - Wikiversity 26 January 2010 1:52 UTC en.wikiversity.org [Source type: FILTERED WITH BAYES]

^ Although in some cases the mandatory penalty may well be appropriate, according to Carnes, in many other cases the prescribed sentence will be disproportionate to the offense that was committed.
• http://www.uscourts.gov/ttb/2009-08/article04.cfm?WT.cg_n=TTB&WT.cg_s=Aug09_article04_rss 26 January 2010 1:52 UTC www.uscourts.gov [Source type: FILTERED WITH BAYES]

^ He would cite Cantor for the ‘extraordinary claim that the set of natural numbers is the same size as the set of rational numbers, but that the set of real numbers is larger than the set of rationals.’ Another of Cantor’s daring suggestions was that there were different kinds of infinity, and that one infinity could be larger than another.

.However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction.^ In fact there is no such thing as too much practice.
• Beginning Algebra Tutorial on Symbols and Sets fo Numbers 26 January 2010 1:52 UTC www.wtamu.edu [Source type: Reference]

^ We do know that, whoever he was, he probably lived in the fourth century BC. There is an ancient tradition that says that he was murdered for his pains.

^ It is said that the first person known to have proved the existence of irrational numbers was a member of the Pythagorian school/cult/social organization named Hippasus in the fifth century, BC. .
• What are Irrational Numbers?: Real Numbers Which Cannot Be Fully Expressed as Fractions 26 January 2010 1:52 UTC mathchaostheory.suite101.com [Source type: FILTERED WITH BAYES]

.He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with an arm, then that unit of measure must be both odd and even, which is impossible.^ For if the sides of an isosceles right triangle are called 1, then we will have 1² + 1² = 2, so that the hypotenuse is .
• Rational and irrational numbers - Topics in precalculus 26 January 2010 1:52 UTC www.themathpage.com [Source type: Reference]

^ Then a and b must both be measurable by 1.
• Exactly what is an irrational number? - a knol by John Gabriel 26 January 2010 1:52 UTC knol.google.com [Source type: Reference]

^ If m were odd, then m 2 would be odd, so m must be even.
• Irrational number - encyclopedia article - Citizendium 26 January 2010 1:52 UTC en.citizendium.org [Source type: Academic]

His reasoning is as follows:
.
• The ratio of the hypotenuse to an arm of an isosceles right triangle is a:b expressed in the smallest units possible.
• By the Pythagorean theorem: a2 = 2b2.
• Since a2 is even, a must be even as the square of an odd number is odd.
• Since a:b is in its lowest terms, b must be odd.
• Since a is even, let a = 2y.
• Then a2 = 4y2 = 2b2
• b2 = 2y2 so b2 must be even, therefore b is even.
• However we asserted b must be odd.^ Indeed, apply the Pythagorean Theorem to the right-angled triangle with sides 1 and 2.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ Now, the square of an even number is even, and the square of an odd number is odd.
• Proof that the Square Root of 2 is Irrational 26 January 2010 1:52 UTC www.marts100.com [Source type: FILTERED WITH BAYES]

^ In that case p must be an even number as well because the square of an odd number is still odd.

.Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible.^ William Jones, a self-taught English mathematician born in Wales, is the one who selected the Greek letter for the ratio of a circle's circumference to its diameter in 1706.
• Interesting Facts about Pi 26 January 2010 1:52 UTC www.arcytech.org [Source type: FILTERED WITH BAYES]

^ Hence mathematicians studied magnitudes which had lengths which, in modern terms, could be formed from positive integers by addition, subtraction, multiplication, division and taking square roots.

.Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”[10] Another legend states that Hippasus was merely exiled for this revelation.^ So we can choose to write p = 2 r , where r is another whole number.

^ Remember that the Pythagoreans believed that everything was either a whole number or a ratio.
• Number: What Is "This Many?" -- Platonic Realms MiniText 26 January 2010 1:52 UTC www.mathacademy.com [Source type: FILTERED WITH BAYES]
• Number: What Is "This Many?" -- Platonic Realms MiniText 26 January 2010 1:52 UTC www.mathacademy.com [Source type: FILTERED WITH BAYES]

^ A rational number is one which can be expressed as p/q, where p and q are whole numbers and q≠0.
• Ancient Greece and Irrational Numbers 26 January 2010 1:52 UTC www.mlahanas.de [Source type: Reference]

.Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable–a foundation of their theory.^ Complex numbers and geometry - the third in the series.
• Hanna Uscka-Wehlou, Research 26 January 2010 1:52 UTC www.wehlou.com [Source type: FILTERED WITH BAYES]

^ The discovery of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2.
• Dror Bar-Natan: Classes: 2004-05: Math 157 - Analysis I: History of the Theory of Irrational Numbers 26 January 2010 1:52 UTC www.math.utoronto.ca [Source type: General]

^ Dedekind’s work ( in Richard Dedekind (German mathematician) ) Greek theories ( in foundations of mathematics: Arithmetic or geometry ) properties .
• real number (mathematics) -- Britannica Online Encyclopedia 26 January 2010 1:52 UTC www.britannica.com [Source type: Reference]

.Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because the algebra he used couldn't be applied to the square root of 17.[11] It wasn't until Eudoxus developed a theory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundation of irrational numbers was created.^ Prove that the square root of 17 is irrational.

^ How then can we be expected to believe that there are as many whole numbers as there are rationals?
• http://deron.meranda.us/topics/infinity/ 26 January 2010 1:52 UTC deron.meranda.us [Source type: Original source]

^ Algebra: Real numbers, Irrational numbers, etc Algebra: Real numbers, Irrational numbers, etc .
• Algebra: Real numbers, Irrational numbers, etc 26 January 2010 1:52 UTC www.algebra.com [Source type: FILTERED WITH BAYES]

[12] .A magnitude "was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously.^ In Euclid, irrational numbers are handled by basing algebra on line segments rather than numbers, so that a product of two numbers was an area and a product of three numbers was a volume and a product of four numbers was impossible.
• Real and Complex Numbers 26 January 2010 1:52 UTC www.math.psu.edu [Source type: FILTERED WITH BAYES]

^ If on the other hand we discover a set of sequences such as the 'number' of twists a rubix cube must make to achieve some given orientation, then this would be a rational sequence.
• S.O.S. Mathematics CyberBoard :: View topic - Rational/Irrational Sequences 26 January 2010 1:52 UTC www.sosmath.com [Source type: General]

^ Real numbers are used in measurements of continuously varying quantities such as size and time, in contrast to the natural numbers 1, 2, 3, …, arising from counting.
• real number (mathematics) -- Britannica Online Encyclopedia 26 January 2010 1:52 UTC www.britannica.com [Source type: Reference]

.Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5."[13] Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible.^ Proposition X.5 Commensurable magnitudes have to one another the ratio which a number has to a number.

^ The development of geometry indicated the need for more numbers; the length of the diagonal of a square with sides one unit long cannot be expressed as a rational number.
• http://www.worldalmanacforkids.com/WAKI-ViewArticle.aspx?pin=x-nu068700a&article_id=483&chapter_id=9&chapter_title=Numbers&article_title=Number 26 January 2010 1:52 UTC www.worldalmanacforkids.com [Source type: Reference]

^ Similarly, at least from where this discussion has arrived at, the set of finite subsets of the natural numbers is countable, whereas the set of infinite subsets of the natural numbers is uncountable.
• xkcd • View topic - Continued Fractions and Irrational Numbers. 26 January 2010 1:52 UTC forums.xkcd.com [Source type: General]

.Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios.^ Book X considers commensurable and incommensurable magnitudes.

^ Proportion is an equation that states that two ratios are equal, for example 2:1 = 6:3.
• Number Definitions 26 January 2010 1:52 UTC www.learner.org [Source type: Reference]
• Number Definitions 26 January 2010 1:52 UTC learner2.learner.org [Source type: Reference]

^ Pi is an irrational number because it cannot be expressed as a ratio (fraction) of two integers: it has no exact decimal equivalent, although 3.1415926 is good enough for many applications.

.By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a number.^ Irrational Number Number that cannot be expressed as a fraction.
• Numbers - I Do Maths 26 January 2010 1:52 UTC www.idomaths.com [Source type: Reference]

^ An irrational number cannot be so expressed, e.g.
• Ancient Greece and Irrational Numbers 26 January 2010 1:52 UTC www.mlahanas.de [Source type: Reference]

^ The number can't be expressed as a fraction, which makes it an irrational number.
• Montshire Museum: 3.141592... 26 January 2010 1:52 UTC www.montshire.org [Source type: FILTERED WITH BAYES]

“Eudoxus’ theory enabled the .Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios.”[14] Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes.^ Euclid (Greek) publishes Elements ; basis for Euclidean Geometry.
• Milestones in Mathematics History 26 January 2010 1:52 UTC www.ccsdk12.org [Source type: Reference]

^ To express the ratios of incommensurable magnitudes.
• Irrational numbers. Evolution of the real numbers. 26 January 2010 1:52 UTC www.themathpage.com [Source type: FILTERED WITH BAYES]

^ Essays on the Theory of Numbers by Richard Dedekind Two classic essays by great German mathematician: one provides an arithmetic, rigorous foundation for the irrational numbers, the other is an attempt to give the logical basis for transfinite numbers and properties of the natural numbers.

### Middle Ages

.In the Middle ages, the development of algebra by Arab mathematicians allowed irrational numbers to be treated as "algebraic objects".[15] Arab mathematicians also merged the concepts of "number" and "magnitude" into a more general idea of real numbers, criticized Euclid's idea of ratios, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude.^ Real numbers measure continuous quantities.
• WikiSlice 26 January 2010 1:52 UTC dev.laptop.org [Source type: Reference]

^ The real numbers are the rational numbers and the irrational numbers combined.
• Real Numbers and Notation 26 January 2010 1:52 UTC www.pocketmath.net [Source type: Reference]

^ The theory of irrational numbers belongs to Calculus.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]
• irrational numbers - Bing 26 January 2010 1:52 UTC search.ninemsn.com.au [Source type: Reference]

[16] In his commentary on Book 10 of the Elements, the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. .He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers.^ Is 3√2 a rational or irrational number?
• The difference between rational and irrational numbers in math. | Bukisa.com 26 January 2010 1:52 UTC www.bukisa.com [Source type: Reference]

^ What makes an irrational number different from a rational number?
• Math 23 Activity 2: The Real Number System 26 January 2010 1:52 UTC academic.cuesta.edu [Source type: Reference]

^ Numbers that are not rational are called irrational numbers.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

He dealt with them freely but explains them in geometric terms as follows:[17]
.
"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively.^ The number 12 is convenient as a base because it is exactly divisible by 2, 3, 4, and 6; for this reason, some mathematicians have advocated adoption of base 12 in place of the base 10.
• http://www.worldalmanacforkids.com/WAKI-ViewArticle.aspx?pin=x-nu068900a&article_id=486&chapter_id=9&chapter_title=Numbers&article_title=Number_Systems 26 January 2010 1:52 UTC www.worldalmanacforkids.com [Source type: Reference]

^ Rational numbers CAN be expressed as a fraction and therefore include all values greater than 0 and less than 1.
• Encyclopedia Uselessia -- NUMBERS (math) 26 January 2010 1:52 UTC www.greatplay.net [Source type: Reference]

^ So all integers are rational, since 5, 12, 18, 25 etc.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

.What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively.^ Erin McKeon pose la question : Why does the letter J represent the set of integers, the letter Q represent a set of rational numbers and the letter P represent a set of irrational numbers?

^ Numbers and quantities are represented using rational and irrational numbers as appropriate to the context.  .

^ Can we ask the question: Are their both irrational and rational sequences of values?
• S.O.S. Mathematics CyberBoard :: View topic - Rational/Irrational Sequences 26 January 2010 1:52 UTC www.sosmath.com [Source type: General]

For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc."
.In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes.^ Prove that the square root of 17 is irrational.

^ The square root of 2 is an irrational number.
• Introduction to Proofs 26 January 2010 1:52 UTC zimmer.csufresno.edu [Source type: Original source]

^ The square root of 4 is rational .
• Is It Irrational? 26 January 2010 1:52 UTC www.mathsisfun.com [Source type: Original source]

He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes:[17]
"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."
The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. .850–930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.^ Know the squares of numbers from 1 to 12 and the cubes of numbers from 1 to 5.
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]

^ Imaginary Number     The square root of a negative number.
• CBofN - Glossary - I 26 January 2010 1:52 UTC mitpress.mit.edu [Source type: Academic]

^ Prove that the square root of 17 is irrational.

[18] .In the 10th century, the Iraqi mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions.^ Other functions tending to return irrational numbers (non-exhaustive) .
• Exploration of Cryptography Based on Irrational Numbers 26 January 2010 1:52 UTC www.rossbach.to [Source type: Reference]

^ Other irrational numbers .

^ In other words, there are more irrational numbers than there are rational.
• Fascinating irrational numbers: Pi and square roots 26 January 2010 1:52 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

[19] Abū Ja'far al-Khāzin (900–971) provides a definition of rational and irrational magnitudes, stating that if a definite quantity is:[20]
.
"contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to a rational number.^ If on the other hand we discover a set of sequences such as the 'number' of twists a rubix cube must make to achieve some given orientation, then this would be a rational sequence.
• S.O.S. Mathematics CyberBoard :: View topic - Rational/Irrational Sequences 26 January 2010 1:52 UTC www.sosmath.com [Source type: General]

^ Show that a given interval on the real number line, no matter how small, contains both rational and irrational numbers.
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]

^ The experiment was along the lines of having a certain number of people enter a building containing grain (that attracted the bird) and then having them leave one at a time over a long period of time.
• Our Playground: The Real Numbers and Their Development - Wikiversity 26 January 2010 1:52 UTC en.wikiversity.org [Source type: FILTERED WITH BAYES]

. . . .Each time when this (latter) magnitude comprises a half, or a third, or a quarter of the given magnitude (of the unit), or, compared with (the unit), comprises three, five, or three fifths, it is a rational magnitude.^ The latter is rational since it's equal to 2 divided by three, but it's not finite.
• CR4 - Thread: Irrational Constants 26 January 2010 1:52 UTC cr4.globalspec.com [Source type: General]

^ Second, this index and the others here are not foolproof: In Beneishs research they correctly identified predictors in about one-half to three-quarters of the cases.
• Irrational Ratios 26 January 2010 1:52 UTC www.buec.udel.edu [Source type: FILTERED WITH BAYES]

^ The fifth generation of the snowflake requires snowflake 3517 to be surrounded by an outer ring, comprising six snowflakes of 1279 units.

.And, in general, each magnitude that corresponds to this magnitude (i.e. to the unit), as one number to another, is rational.^ To understand the Real numbers, we  need to understand the concept of repeating and non-repeating decimals.   As it turns out, every rational number can be represented in one of these ways.   .
• Exploring Numbers 26 January 2010 1:52 UTC k12math.com [Source type: Reference]

^ Traditionally, the real numbers are thought of as living on the horizontal axis and the imaginary numbers on the vertical axis, with i one unit up and -i one unit down from 0, which lives at the origin.
• Metaphor for imaginary numbers? - Straight Dope Message Board 26 January 2010 1:52 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ The Approximations Of Pi, Phi and e Now we come to the calculations themselves, which are simple divisions of one of the derived numbers by another.

If, however, a magnitude cannot be represented as a multiple, a part (l/n), or parts (m/n) of a given magnitude, it is irrational, i.e. it cannot be expressed other than by means of roots."
.Many of these concepts were eventually accepted by European mathematicians sometime after the Latin translations of the 12th century.^ But you can't keep a good mathematician down, and eventually these transcendental numbers were accepted into the numbering system, creating the real numbers.
• http://www.bbc.co.uk/dna/h2g2/A59591 26 January 2010 1:52 UTC www.bbc.co.uk [Source type: FILTERED WITH BAYES]

^ By the sixteenth century rational numbers and roots of numbers were becoming accepted as numbers although there was still a sharp distinction between these different types of numbers.

^ Many mathematicians then could not accept the idea of an irrational number.
• Milestones in Mathematics History 26 January 2010 1:52 UTC www.ccsdk12.org [Source type: Reference]

.Al-Hassār, an Arabic mathematician from the Maghreb (North Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated by a horizontal bar.^ Rationalising Surds - This is a way of modifying surd expressions so that the square root is in the numerator of a fraction and not in the denominator.

^ The numerator and denominator of the fraction must be whole numbers.
• The difference between rational and irrational numbers in math. | Bukisa.com 26 January 2010 1:52 UTC www.bukisa.com [Source type: Reference]

^ SPI 0306.2.13 Recognize, compare, and order fractions (benchmark fractions, common numerators, or common denominators).
• Standard 2 : Number and Operations (Mathematics) 26 January 2010 1:52 UTC www.stemresources.com [Source type: Reference]

.This same fractional notation appears soon after in the work of Fibonacci in the 13th century.^ These numbers all belong to a sequence named for the 13th-century Italian mathematician Fibonacci.
• Ivars Peterson's MathTrek - Golden Blossoms, Pi Flowers 26 January 2010 1:52 UTC www.maa.org [Source type: Reference]

[21] .During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as pi and certain irrational values of trigonometric functions.^ Mathematicians have proved that certain special numbers are irrational, for example Pi and e .
• Fascinating irrational numbers: Pi and square roots 26 January 2010 1:52 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

^ Expressions such as and have irrational numbers in the denominator.
• Rational and Irrational Numbers 26 January 2010 1:52 UTC www.tpub.com [Source type: Reference]

^ The irrational numbers are uncountably infinite .
• PRIME Mathematics Encyclopedia 26 January 2010 1:52 UTC www.mathacademy.com [Source type: Reference]

.Jyesthadeva provided proofs for these infinite series in the Yuktibhasa.^ They identify subsets of these as discrete or continuous, finite or infinite and provide examples of their elements and apply these to functions and relations and the solution of related equations.
• Level 6 | Mathematics | Discipline-based Learning | Prep to Year 10 Curriculum and Standards | Victorian Essential Learning Standards 26 January 2010 1:52 UTC vels.vcaa.vic.edu.au [Source type: Reference]

[22]

### Modern period

.The 17th century saw imaginary numbers become a powerful tool in the hands of Abraham de Moivre, and especially of Leonhard Euler.^ This little complex number, corresponding to the point $(0,1)$, not only allows all polynomial equations to have solutions but gives a powerful tool for working with rotations.
• nrich.maths.org :: Mathematics Enrichment :: What Are Numbers? 26 January 2010 1:52 UTC nrich.maths.org [Source type: FILTERED WITH BAYES]

^ Leonhard Euler (1707-1783) in a paper published posthumously, showed that every even perfect number has Euclidean form.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ By the sixteenth century rational numbers and roots of numbers were becoming accepted as numbers although there was still a sharp distinction between these different types of numbers.

.The completion of the theory of complex numbers in the nineteenth century entailed the differentiation of irrationals into algebraic and transcendental numbers, the proof of the existence of transcendental numbers, and the resurgence of the scientific study of the theory of irrationals, largely ignored since Euclid.^ Algebraic number theory .
• The educational encyclopedia, mathematics: number theory, Sorting algorithms 26 January 2010 1:52 UTC www.educypedia.be [Source type: Reference]

^ The theory of irrational numbers belongs to Calculus.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ It is called a transcendental number; i.e., a real number which is not an algebraic number.
• The Nature of the Real Numbers 26 January 2010 1:52 UTC www.sjsu.edu [Source type: Reference]

.The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind.^ Very little study was given to irrationals from this time until the late 1700s and 1800s where Johann Heinrich Lambert , Paolo Ruffini , Karl Weierstrass , Heine , Georg Cantor , Richard Dedekind , and numerous other mathematicians all studied and wrote about them.
• irrational number@Everything2.com 26 January 2010 1:52 UTC www.everything2.com [Source type: Reference]

^ Author: Dedekind, Richard, 1831-1916 Subject: Irrational numbers ; Number theory Publisher: Chicago Open Court Pub.
• Internet Archive: Free Download: Essays in the theory of numbers, 1. Continuity of irrational numbers, 2. The nature and meaning of numbers. Authorized translation by Wooster Woodruff Beman 26 January 2010 1:52 UTC www.archive.org [Source type: General]

^ Cantor, Dedekind, Frege, Peano, Russell, and Whitehead) turned to a new (at the time) branch of mathematics called set theory .
• Number: What Is "This Many?" -- Platonic Realms MiniText 26 January 2010 1:52 UTC www.mathacademy.com [Source type: FILTERED WITH BAYES]
• Number: What Is "This Many?" -- Platonic Realms MiniText 26 January 2010 1:52 UTC www.mathacademy.com [Source type: FILTERED WITH BAYES]

.Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle in 1880,[23] and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by Paul Tannery (1894).^ This working-through of one which becomes a repetition of the other in a formal scheme is an important feature of Alain Badiou’s philosophy of existence, as “theory of the subject”.
• irrational numbers: August 2005 Archives 26 January 2010 1:52 UTC blog.urbanomic.com [Source type: Original source]

^ By late autumn of that same year he had evolved a systematic theory, but it was not until 1872 that he published it, under the stimulus of the appearance of other theories of irrational numbers.
• Heinrich Tietze on Numbers, Part 2 26 January 2010 1:52 UTC www.gap-system.org [Source type: FILTERED WITH BAYES]

^ Rationals + Irrationals All points on the number line Or all possible distances on the number line When we put the irrational numbers together with the rational numbers, we finally have the complete set of real numbers.
• The Real Number System 26 January 2010 1:52 UTC www.jamesbrennan.org [Source type: FILTERED WITH BAYES]

.Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties.^ The real numbers are the rational numbers and the irrational numbers combined.
• Real Numbers and Notation 26 January 2010 1:52 UTC www.pocketmath.net [Source type: Reference]

^ Set of all real numbers .

^ A rational number is is any number that can be written as a ration of two integers.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray.
.Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Lagrange.^ Regular patterns in the continued fractions of some irrational numbers.
• Numbers and Functions as Continued Fractions - Numericana 26 January 2010 1:52 UTC home.att.net [Source type: Reference]

^ Any irrational number has a continued fraction expansion, although it won't terminate.
• xkcd • View topic - Continued Fractions and Irrational Numbers. 26 January 2010 1:52 UTC forums.xkcd.com [Source type: General]

^ Quadratic surds are irrational numbers which have periodic continued fractions.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
.Lambert proved (1761) that π cannot be rational, and that en is irrational if n is rational (unless n = 0).^ For example, Dirichlet defined the function f(x) = lim [m-> inf] lim [n-> inf] { cos(m!*pi*x) }^(2n) and proved that f(x) equals 1 for all rational x and 0 for all irrational x.
• Convergence of Series (How NOT to Prove PI Irrational) 26 January 2010 1:52 UTC www.mathpages.com [Source type: FILTERED WITH BAYES]

^ Lambert (German) proves pi is irrational.
• Milestones in Mathematics History 26 January 2010 1:52 UTC www.ccsdk12.org [Source type: Reference]

^ An irrational number is a real number which is not a rational number , meaning it cannot be expressed as a/b where a,b are integer .
• irrational number@Everything2.com 26 January 2010 1:52 UTC www.everything2.com [Source type: Reference]

[24] .While Lambert's proof is often said to be incomplete, modern assessments support it as satisfactory, and in fact for its time it is unusually rigorous.^ A related phenomenon is that we have an easier time remembering facts or experiences that support our beliefs than ones that fail to.
• Why People Are Irrational about Politics 26 January 2010 1:52 UTC home.sprynet.com [Source type: Original source]

.Legendre (1794), after introducing the Bessel–Clifford function, provided a proof to show that π2 is irrational, Whence it follows immediately that π is irrational also.^ When most of them did it, following the same method, I think that showed they didn’t really understand the key step of the proof.
• Rational and irrational numbers « The Math Less Traveled 26 January 2010 1:52 UTC www.mathlesstraveled.com [Source type: FILTERED WITH BAYES]

^ The proof that exponential series e x is irrational, with x rational or irrational 3 , is obtained following the proof that exponential series e is irrational 4 , and using the comparison test with exponential series e .
• Exponential Series, Irrational Numbers 26 January 2010 1:52 UTC www.coolissues.com [Source type: Reference]

^ He provided the proof of the irrationality of all integer numbers between 3 and 17 except the square numbers 4, 9 and 16 (the case for n = 2 was well-known before him).
• Ancient Greece and Irrational Numbers 26 January 2010 1:52 UTC www.mlahanas.de [Source type: Reference]

The existence of transcendental numbers was first established by Liouville (1844, 1851). .Later, Georg Cantor (1873) proved their existence by a different method, that showed that every interval in the reals contains transcendental numbers.^ In reality every number can be written in many different ways.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ It is called a transcendental number; i.e., a real number which is not an algebraic number.
• The Nature of the Real Numbers 26 January 2010 1:52 UTC www.sjsu.edu [Source type: Reference]

^ Real numbers that are not algebraic are called transcendental .
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

.Charles Hermite (1873) first proved e transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed the same for π.^ Hermite in 1873, and π by Lindemann in 1882.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that π is transcendental.
• WikiSlice 26 January 2010 1:52 UTC dev.laptop.org [Source type: Reference]

^ The second major discovery came in 1882 when Ferdinand von Lindemann proved that pi had another unusual feature: it was transcendental.
• The life of pi - Science, News - The Independent 26 January 2010 1:52 UTC www.independent.co.uk [Source type: News]

Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and was finally made elementary by Adolf Hurwitz and Paul Albert Gordan.

## Example proofs

### Square roots

.The square root of 2 was the first number to be proved irrational and that article contains a number of proofs.^ Prove that the square root of 17 is irrational.

^ The square root of 2 is one of the fundamental irrational numbers.
• Proof that the Square Root of 2 is Irrational 26 January 2010 1:52 UTC www.marts100.com [Source type: FILTERED WITH BAYES]

^ The include numbers such as i , the square root of -1.
• Numbers 26 January 2010 1:52 UTC www.bsu.edu [Source type: Reference]

.The golden ratio is the next most famous quadratic irrational and there is a simple proof of its irrationality in its article.^ In this article we will see a simple proof by contradiction that the square root of 2 is irrational.
• Proof that the Square Root of 2 is Irrational 26 January 2010 1:52 UTC www.marts100.com [Source type: FILTERED WITH BAYES]

^ In the section on applications there are a number of interactive programs that convert rationals (or quadratic irrationals) into a simple continued fraction, as well as the converse.

^ Some famous examples of irrational numbers are p i (curse my inability inscribe Greek letters onto my blog), the square root of 2 (similar curse for mathematical notation), the golden ratio.
• The Hypothetical Third Dimension: Real Men Love Real Numbers 26 January 2010 1:52 UTC xyzw.r3.nu [Source type: General]

.The square root of all non-square natural numbers is irrational and a proof may be found in quadratic irrationals.^ Prove that the square root of 17 is irrational.

^ The square root of 2 is one of the fundamental irrational numbers.
• Proof that the Square Root of 2 is Irrational 26 January 2010 1:52 UTC www.marts100.com [Source type: FILTERED WITH BAYES]

^ The include numbers such as i , the square root of -1.
• Numbers 26 January 2010 1:52 UTC www.bsu.edu [Source type: Reference]

.The irrationality of the square root of 2 may be proved by assuming it is rational and inferring a contradiction, called an argument by reductio ad absurdum.^ Prove that the square root of 17 is irrational.

^ Numbers that are not rational are called irrational numbers.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

^ Then the square root is irrational.
• 11.1 Square Root Irrational 26 January 2010 1:52 UTC www.slideshare.net [Source type: Reference]

.The following argument appeals twice to the fact that the square of an odd integer is always odd.^ (In fact for any integer n, which is not a square of another integer, √ n is irrational .
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ Thus is even, and since the squares of odd numbers are always odd we can deduce that is also even.
• The origins of proof III: Proof and puzzles through the ages 26 January 2010 1:52 UTC plus.maths.org [Source type: FILTERED WITH BAYES]

^ Given two squares with integer sides, one twice the other .
• Square root of 2 is irrational from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Academic]

.If √2 is rational it has the form m/n for integers m, n not both even.^ The rational numbers are numbers that can be written in the form , where a and b are integers and b is not 0.
• Real Numbers and Notation 26 January 2010 1:52 UTC www.pocketmath.net [Source type: Reference]

^ It is conjectured that if there exists a real number x for which both 2^x and 3^x are integers, then is rational.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

^ In fact you can't even have the circumference and diameter both rational since the quotient of two rationals is again a rational.

Then m2 = 2n2, hence m is even, say m = 2p. Thus 4p2 = 2n2 so 2p2 = n2, hence n is also even, a contradiction.

### General roots

.The proof above for the square root of two can be generalized using the fundamental theorem of arithmetic which was proved by Gauss in 1798.^ Prove that the square root of 17 is irrational.

^ The proofs above, directly or indirectly, appeal to the Fundamental Theorem of Arithmetic .
• Square root of 2 is irrational from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Academic]

^ Help me prove that the square root of 6 is irrational .

.This asserts that every integer has a unique factorization into primes.^ Every even integer greater than 2 is the sum of two primes.
• Milestones in Mathematics History 26 January 2010 1:52 UTC www.ccsdk12.org [Source type: Reference]

^ Know and apply the Fundamental Theorem of Arithmetic, that every positive integer is either prime itself or can be written as a unique product of primes (ignoring order).
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]

^ Then factor 15 into prime numbers: 30 = 2  15 and 15 = 3  5.
• Number Definitions 26 January 2010 1:52 UTC www.learner.org [Source type: Reference]
• Number Definitions 26 January 2010 1:52 UTC learner2.learner.org [Source type: Reference]

.Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms there must be a prime in the denominator which does not divide into the numerator whatever power each is raised to.^ For example, he proved that there is no largest prime number.
• Milestones in Mathematics History 26 January 2010 1:52 UTC www.ccsdk12.org [Source type: Reference]

^ Since there is no rational number equal to the , this must be a number that is not rational.
• Number and Geometry 26 January 2010 1:52 UTC www.highhopes.com [Source type: FILTERED WITH BAYES]

^ A rational number p/q is said to have numerator p and denominator q .
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

.Therefore if an integer is not an exact kth power of another integer then its kth root is irrational.^ (In fact for any integer n, which is not a square of another integer, √ n is irrational .
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ Non-integer Powers and Exponents How do you find x^n, where n can be an integer, a fraction, a decimal, or an irrational number?
• Fascinating irrational numbers: Pi and square roots 26 January 2010 1:52 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

^ SR of 2 is Irrational, and the SR or 3 is irrational (they are prime so they have no factors other than 1 and itself therefore the root is irrational) since those two roots are irational, an Irrational times an irrational is an irrational, therefore, root 6 is irrational.
• Prove sqrt(6) is irrational - Page 2 26 January 2010 1:52 UTC www.physicsforums.com [Source type: FILTERED WITH BAYES]

### Logarithms

.Perhaps the numbers most easily proved to be irrational are certain logarithms.^ Mathematicians have proved that certain special numbers are irrational, for example Pi and e .
• Fascinating irrational numbers: Pi and square roots 26 January 2010 1:52 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

^ Most irrational numbers contain surds (roots) and constants like Pi or e.
• The difference between rational and irrational numbers in math. | Bukisa.com 26 January 2010 1:52 UTC www.bukisa.com [Source type: Reference]

^ While most mathematicians think of complex numbers as being the points of the plane, perhaps it is better, as an answer to the OP to think of them as rigid motions of the plane that preserve orientation.
• Metaphor for imaginary numbers? - Straight Dope Message Board 26 January 2010 1:52 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

.Here is a proof by reductio ad absurdum that log2 3 is irrational.^ Proofs that e, log(2) and pi are irrational are given.

^ "Reductio ad absurdum" is a fancy way to say "this is crazy and can't be true".

^ It borrows from Hippasus's initial assumption and also ends with reductio ad absurdum .

Notice that log2 3 ≈ 1.58 > 0.
Assume log2 3 is rational. For some positive integers m and n, we have
$\log_2 3 = \frac{m}{n}$
It follows that
$2^{m/n}=3\,$
$(2^{m/n})^n = 3^n\,$
$2^m=3^n\,$
.However, the number 2 raised to any positive integer power must be even (because it will be divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2).^ Use divisibility rules to factor numbers.
• Standard 2 : Number and Operations (Mathematics) 26 January 2010 1:52 UTC www.stemresources.com [Source type: Reference]

^ A prime number is a positive integer which has exactly two factors, 1 and itself.
• 2. Number Properties 26 January 2010 1:52 UTC www.intmath.com [Source type: Reference]

^ The number 1 is neither prime nor composite, since it has exactly one factor.
• 2. Number Properties 26 January 2010 1:52 UTC www.intmath.com [Source type: Reference]

.Clearly, an integer can not be both odd and even at the same time: we have a contradiction.^ For example, the sets of odd numbers, even numbers, complete squares or cubes, the set of integers greater than 1996 all can be brought into a 1-1 correspondence with the set of all integers (of which they are subsets.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

.The only assumption we made was that log2 3 is rational (and so expressible as a quotient of integers m/n with n ≠ 0).^ Given the second assumption (and given that individuals are usually instrumentally rational), most people will accept this cost only if they receive greater benefits from thinking rationally.
• Why People Are Irrational about Politics 26 January 2010 1:52 UTC home.sprynet.com [Source type: Original source]

^ Irrational numbers are numbers that can't be expressed as a quotient of two integers, such as pi or square roots; they can only be expressed as infinite, non-repeating decimals.
• Number Definitions 26 January 2010 1:52 UTC www.learner.org [Source type: Reference]
• Number Definitions 26 January 2010 1:52 UTC learner2.learner.org [Source type: Reference]

^ Rational Rational numbers are the real numbers that can be expressed as the quotient of two integers.
• Numbers 26 January 2010 1:52 UTC www.bsu.edu [Source type: Reference]

The contradiction means that this assumption must be false, i.e. log2 3 is irrational, and can never be expressed as a quotient of integers m/n with n ≠ 0.
Cases such as log10 2 can be treated similarly.

## Transcendental and algebraic irrationals

.Almost all irrational numbers are transcendental and all transcendental numbers are irrational: the article on transcendental numbers lists several examples.^ The set I of all irrational numbers is not countable.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ So, pi and e and sqr rt of 2 are all examples of irrational numbers.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

^ This result, called the Gelfond-Schneider Theorem , says that If α is an algebraic number other than 0 or 1, and β is an irrational algebraic number then α β is transcendental; i.e., non-algebraic.
• The Nature of the Real Numbers 26 January 2010 1:52 UTC www.sjsu.edu [Source type: Reference]

e r and π r are irrational if r ≠ 0 is rational; eπ is irrational.
Another way to construct irrational numbers is as irrational algebraic numbers, i.e. as zeros of polynomials with integer coefficients: start with a polynomial equation
$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0 \,$
where the coefficients ai are integers. .Suppose you know that there exists some real number x with p(x) = 0 (for instance if n is odd and an is non-zero, then because of the intermediate value theorem).^ Are there some differences of fundamental values?
• Why People Are Irrational about Politics 26 January 2010 1:52 UTC home.sprynet.com [Source type: Original source]

^ We know we have got the money because it is there.

^ There are many, you know.
• xkcd • View topic - What's your favourite irrational number? 26 January 2010 1:52 UTC forums.xkcd.com [Source type: General]

.The only possible rational roots of this polynomial equation are of the form r/s where r is a divisor of a0 and s is a divisor of an; there are only finitely many such candidates which you can all check by hand.^ Algebraic numbers are all the numbers that are solutions to polynomial equations where the polynomials have rational coefficients (e.g.
• Number Definitions 26 January 2010 1:52 UTC www.learner.org [Source type: Reference]
• Number Definitions 26 January 2010 1:52 UTC learner2.learner.org [Source type: Reference]

^ There are many, you know.
• xkcd • View topic - What's your favourite irrational number? 26 January 2010 1:52 UTC forums.xkcd.com [Source type: General]

^ So the rationals are all of the possible fractions that you can think of.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

If neither of them is a root of p, then x must be irrational. .For example, this technique can be used to show that x = (21/2 + 1)1/3 is irrational: we have (x3 − 1)2 = 2 and hence x6 − 2x3 − 1 = 0, and this latter polynomial does not have any rational roots (the only candidates to check are ±1).^ Also, every polynomial of odd degree admits at least one root: these two properties make R the premier example of a real closed field.
• WikiSlice 26 January 2010 1:52 UTC dev.laptop.org [Source type: Reference]

^ Erin McKeon pose la question : Why does the letter J represent the set of integers, the letter Q represent a set of rational numbers and the letter P represent a set of irrational numbers?

^ GLE 0806.2.3 Solve real-world problems using rational and irrational numbers.
• Standard 2 : Number and Operations (Mathematics) 26 January 2010 1:52 UTC www.stemresources.com [Source type: Reference]

.Because the algebraic numbers form a field, many irrational numbers can be constructed by combining transcendental and algebraic numbers.^ The real numbers are the rational numbers and the irrational numbers combined.
• Real Numbers and Notation 26 January 2010 1:52 UTC www.pocketmath.net [Source type: Reference]

^ "How many irrational numbers are there?"
• How many irrational numbers are there? 26 January 2010 1:52 UTC askville.amazon.com [Source type: General]

^ How many irrational numbers are there?
• How many irrational numbers are there? 26 January 2010 1:52 UTC askville.amazon.com [Source type: General]

.For example 3π + 2, π + √2 and e3 are irrational (and even transcendental).^ Most examples of irrational numbers involve the square roots of numbers, although cube roots, fourth roots, and even higher roots 5 are also possible.
• http://www.bbc.co.uk/dna/h2g2/A59591 26 January 2010 1:52 UTC www.bbc.co.uk [Source type: FILTERED WITH BAYES]

^ Not always though; for example, e + (- e ) = 0, and 0 is rational even though both e and -e are irrational.
• Fascinating irrational numbers: Pi and square roots 26 January 2010 1:52 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

## Decimal expansions

.The decimal expansion of an irrational number never repeats or terminates, unlike a rational number.^ Conversely, the expansion of any rational number is of this type.
• Numbers and Functions as Continued Fractions - Numericana 26 January 2010 1:52 UTC home.att.net [Source type: Reference]

^ Numbers that are not rational are called irrational numbers.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

^ What is a rational or irrational number?
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

To show this, suppose we divide integers n by m (where m is nonzero). When long division is applied to the division of n by m, only m remainders are possible. .If 0 appears as a remainder, the decimal expansion terminates.^ Irrational numbers have decimal expansions that neither terminate nor become periodic.
• What is a rational or irrational number? - Yahoo! Answers 26 January 2010 1:52 UTC answers.yahoo.com [Source type: General]

^ A fraction has a terminating decimal expansion if and only if its denominator in reduced form has only 2 and 5 as prime factors.
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]

.If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once.^ I'm rather more comfortable dealing with someone whose politics I disagree with, but I can see how they got to where they are, than someone who politics are in line with mine but who appear to have arrived at those politics without an intermediary step of, you know, thinking about those politics.
• Whatever: Irrationalists 26 January 2010 1:52 UTC www.scalzi.com [Source type: Original source]

^ Or more generally, why might using irrational numbers as tax rates be less distortionary than rational tax rates?

^ Most real numbers are irrational so there will probably be more irrational numbers than irrational questions!
• How many irrational numbers are there? 26 January 2010 1:52 UTC askville.amazon.com [Source type: General]

.After that, a remainder must recur, and then the decimal expansion repeats.^ Know that the decimal expansion of an irrational number never ends and never repeats.
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]

.Conversely, suppose we are faced with a recurring decimal, we can prove that it is a fraction of two integers.^ The students have demonstrated proficiency at addition, subtraction, multiplication, and division of integers, decimals, and fractions.
• Ordering Rational Numbers - Math Lesson Plan, Thematic Unit, Activity, Worksheet, or Teaching Idea 26 January 2010 1:52 UTC www.lessonplanspage.com [Source type: FILTERED WITH BAYES]

^ Non-integer Powers and Exponents How do you find x^n, where n can be an integer, a fraction, a decimal, or an irrational number?
• Fascinating irrational numbers: Pi and square roots 26 January 2010 1:52 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

^ Because the decimal representation of a rational fraction (like 22/7) is either finite or has a recurring sequence of digits, the dance corresponding to a rational number will either be finite or have a recurring sequence: it will be a sequence dance .
• Gender & Choreography: an Irrational Tango 26 January 2010 1:52 UTC linus.it.uts.edu.au [Source type: FILTERED WITH BAYES]

For example:
$A=0.7\,162\,162\,162\,\dots.$
Here the length of the repitend is 3. We multiply by 103:
$1000A=7\,16.2\,162\,162\,\dots.$
.Note that since we multiplied by 10 to the power of the length of the repeating part, we shifted the digits to the left of the decimal point by exactly that many positions.^ Since the words line and point are really undefined, then they may be interpreted in many different ways.
• Number and Geometry 26 January 2010 1:52 UTC www.highhopes.com [Source type: FILTERED WITH BAYES]

^ A. 3_ 4_ 5 Pythagorean triplet in pi and 1_1_2 in identical positions to the decimal point.
• http://www.artmusicdance.com/vaspi/questions.htm 26 January 2010 1:52 UTC www.artmusicdance.com [Source type: Original source]

^ Or, as noted before: since i is a 90 degree turn, the sqrt(i) is something which, when repeated twice, gives a 90 degree turn; namely, a 45 degree turn (or a 235 degree turn).
• Metaphor for imaginary numbers? - Straight Dope Message Board 26 January 2010 1:52 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

Therefore, the tail end of 1000A matches the tail end of A exactly. Here, both 1000A and A have repeating 162 at the end.
Therefore, when we subtract A from both sides, the tail end of 1000A cancels out of the tail end of A:
$999A=715.5\,.$
Then
$A=\frac{715.5}{999} = \frac{7155}{9990} = \frac{135 imes 53}{135 imes 74} = \frac{53}{74},$
.(135 is the greatest common divisor of 7155 and 9990).^ The Euclidean algorithm Greatest common divisors.
• Math 126 Number Theory 26 January 2010 1:52 UTC babbage.clarku.edu [Source type: Reference]
• Math 126 Number Theory 26 January 2010 1:52 UTC aleph0.clarku.edu [Source type: Reference]

^ Use greatest common divisors to reduce fractions and ratios n:m to an equivalent form in which gcd ( n, m ) = 1.
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]

^ Explain the meaning of the greatest common divisor (greatest common factor) and the least common multiple and use them in operations with fractions.
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]

Alternatively, since 0.5 = 1/2, one can clear fractions by multiplying the numerator and denominator by 2:
$A=\frac{715.5}{999} = \frac{2 imes 715.5}{2 imes 999} = \frac{1431}{1998} = \frac{27 imes 53}{27 imes 74} = \frac{53}{74}$
.(27 is the greatest common divisor of 1431 and 1998).^ The Euclidean algorithm Greatest common divisors.
• Math 126 Number Theory 26 January 2010 1:52 UTC babbage.clarku.edu [Source type: Reference]
• Math 126 Number Theory 26 January 2010 1:52 UTC aleph0.clarku.edu [Source type: Reference]

^ Use greatest common divisors to reduce fractions and ratios n:m to an equivalent form in which gcd ( n, m ) = 1.
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]

^ Explain the meaning of the greatest common divisor (greatest common factor) and the least common multiple and use them in operations with fractions.
• Mathematics Benchmarks, Grades K-12 26 January 2010 1:52 UTC www.utdanacenter.org [Source type: Reference]

.The bottom line, 53/74 is a quotient of integers and therefore a rational number.^ For example, integers are not dense; rational numbers are.
• Number Definitions 26 January 2010 1:52 UTC www.learner.org [Source type: Reference]
• Number Definitions 26 January 2010 1:52 UTC learner2.learner.org [Source type: Reference]

^ Rational numbers and integers are all algebraic.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ Develop and analyze algorithms and compute efficiently with integers and rational numbers.
• Standard 2 : Number and Operations (Mathematics) 26 January 2010 1:52 UTC www.stemresources.com [Source type: Reference]

## Miscellaneous

Here is a famous pure existence or non-constructive proof:
.There exist two irrational numbers a and b, such that ab is rational.^ Rational and Irrational numbers .
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ However, there are numbers which are neither rational or irrational (for example, infinitesimal numbers are neither rational nor irrational).
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ Assume that a rational number r exists such that r 2 = 5.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

.Indeed, if √2√2 is rational, then take a = b = √2. Otherwise, take a to be the irrational number √2√2 and b = √2. Then ab = (√2√2)√2 = √2√2·√2 = √22 = 2 which is rational.^ Irrational numbers disturbed early mathematicians, and they developed creative methods of getting as close as they needed to the true value of the irrational quantity using rational numbers.
• Our Playground: The Real Numbers and Their Development - Wikiversity 26 January 2010 1:52 UTC en.wikiversity.org [Source type: FILTERED WITH BAYES]

^ To see this, take two rational numbers (in decimal representation) that are very close together.
• Number: What Is "This Many?" -- Platonic Realms MiniText 26 January 2010 1:52 UTC www.mathacademy.com [Source type: FILTERED WITH BAYES]
• Number: What Is "This Many?" -- Platonic Realms MiniText 26 January 2010 1:52 UTC www.mathacademy.com [Source type: FILTERED WITH BAYES]

^ By the way, there is no shortage of irrational numbers; there are an infinite number of irrational numbers between any two rational numbers .

√2√2 is transcendental because of the Gelfond–Schneider theorem.

## Open questions

It is not known whether π + e or π − e is irrational or not. .In fact, there is no pair of non-zero integers m and n for which it is known whether mπ + ne is irrational or not.^ Each sheep is paired with a distinct pebble and there are no left overs on either side.
• Counting Infinity : Gustavo Duarte 26 January 2010 1:52 UTC duartes.org [Source type: FILTERED WITH BAYES]

^ We need to use the rule method because there is no way to list rational numbers that suggests the complete list in an unambiguous way.
• Number: What Is "This Many?" -- Platonic Realms MiniText 26 January 2010 1:52 UTC www.mathacademy.com [Source type: FILTERED WITH BAYES]
• Number: What Is "This Many?" -- Platonic Realms MiniText 26 January 2010 1:52 UTC www.mathacademy.com [Source type: FILTERED WITH BAYES]

^ October 28, 2009 2:32 pm Link Of course, when it comes to computers, there is no such thing as an irrational number.

Moreover, it is not known whether the set {π, e} is algebraically independent over Q.
.It is not known whether 2e, πe, π√2, Catalan's constant, or the Euler-Mascheroni gamma constant γ are irrational.^ A similar example involves Euler's constant , the Euler-Mascheroni number ( g ): .
• Numbers and Functions as Continued Fractions - Numericana 26 January 2010 1:52 UTC home.att.net [Source type: Reference]

^ The Euler-Masheroni constant (Yes, it is irrational!
• xkcd • View topic - What's your favourite irrational number? 26 January 2010 1:52 UTC forums.xkcd.com [Source type: General]

## The set of all irrationals

.Since the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable.^ The subset of this set that consists of real numbers is as well countable.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ The set I of all irrational numbers is not countable.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ However, rational numbers form a countable set whereas irrational form a set which is not countable.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

.Under the usual (Euclidean) distance function d(xy) = |x − y|, the real numbers are a metric space and hence also a topological space.^ One complication, which does not exist in the case of real numbers, is that not all functions may be expanded in this way.
• Numbers and Functions as Continued Fractions - Numericana 26 January 2010 1:52 UTC home.att.net [Source type: Reference]

^ Real numbers and complex numbers both commute under multiplication.
• The origins of proof III: Proof and puzzles through the ages 26 January 2010 1:52 UTC plus.maths.org [Source type: FILTERED WITH BAYES]

Restricting the Euclidean distance function gives the irrationals the structure of a metric space. Since the subspace of irrationals is not closed, the induced metric is not complete. .However, being a G-delta set—i.e., a countable intersection of open subsets—in a complete metric space, the space of irrationals is topologically complete: that is, there is a metric on the irrationals inducing the same topology as the restriction of the Euclidean metric, but with respect to which the irrationals are complete.^ The set I of all irrational numbers is not countable.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ However, rational numbers form a countable set whereas irrational form a set which is not countable.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ UnBiasedTrading (TM): Is the stage being set for an even more irrational exuberance in the stock market this year?
• UnBiasedTrading (TM): Is the stage being set for an even more irrational exuberance in the stock market this year? Preview comments of Raymond Merriman 26 January 2010 1:52 UTC unbiasedtrading.blogspot.com [Source type: General]

.One can see this without knowing the aforementioned fact about G-delta sets: the continued fraction expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable.^ The set Q of all rational numbers is equivalent to the set N of all integers.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

^ If integers are an infinite number, then the irrational number must also be an infinte set.
• How many irrational numbers are there? 26 January 2010 1:52 UTC askville.amazon.com [Source type: General]

^ Rational numbers and integers are all algebraic.
• What's a number? from Interactive Mathematics Miscellany and Puzzles 26 January 2010 1:52 UTC www.cut-the-knot.org [Source type: Reference]

Furthermore, the set of all irrationals is a disconnected metric space.

## References

1. ^ Cantor, Georg (1955, 1915). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover. ISBN 978-0486600451.
2. ^ The 15 Most Famous Transcendental Numbers. by Clifford A. Pickover. URL retrieved 24 October 2007.
3. ^ http://www.mathsisfun.com/irrational-numbers.html; URL retrieved 24 October 2007.
4. ^ URL retrieved 26 October 2007.
5. ^ T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 411–2, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1402002602 .
6. ^ Ed. Dold-Samplonius et al, "2000 Years Transmission of Mathematical Ideas", pp. 31-44
7. ^ Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics.
8. ^ James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number". The Two-Year College Mathematics Journal. .
9. ^ Kline, M. (1990). Mathematical Thought from Ancient to Modern Times, Vol. 1. New York: Oxford University Press. (Original work published 1972). p.33.
10. ^ Kline 1990, p. 32.
11. ^ Robert L. McCabe (1976). "Theodorus' Irrationality Proofs". Mathematics Magazine. .
12. ^ Charles H. Edwards (1982). The historical development of the calculus. Springer.
13. ^ Kline 1990, p.48.
14. ^ Kline 1990, p.49.
15. ^  ..
16. ^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences 500: 253–277 [254], doi:10.1111/j.1749-6632.1987.tb37206.x .
17. ^ a b Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences 500: 253–277 [259], doi:10.1111/j.1749-6632.1987.tb37206.x
18. ^ Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1402002602 .
19. ^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences 500: 253–277 [260], doi:10.1111/j.1749-6632.1987.tb37206.x .
20. ^ Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences 500: 253–277 [261], doi:10.1111/j.1749-6632.1987.tb37206.x .
21. ^ Prof. Ahmed Djebbar (June 2008). "Mathematics in the Medieval Maghrib: General Survey on Mathematical Activities in North Africa". FSTC Limited. Retrieved 2008-07-19.
22. ^ Katz, V. J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine (Mathematical Association of America) 68 (3): 163–74.
23. ^ Salvatore Pincherle (1880). "Saggio di una introduzione alla teorica delle funzioni analitiche secondo i principi del prof. Weierstrass". Giornale di Matematiche.
24. ^ J. H. Lambert (1761). "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques". Histoire de l'Académie Royale des Sciences et des Belles-Lettres der Berlin: 265–276.

• Adrien-Marie Legendre, Éléments de Géometrie, Note IV, (1802), Paris
• Rolf Wallisser, "On Lambert's proof of the irrationality of π", in Algebraic Number Theory and Diophantine Analysis, Franz Halter-Koch and Robert F. Tichy, (2000), Walter de Gruyer

# Simple English

An irrational number is defined to be any number that is the part of the real number system that cannot be written as a complete ratio of two numbers.

File:Square root of 2
√2 is irrational

Irrational numbers often occur in geometry. For instance, if we have a square which has sides of 1 meter, the distance between opposite corners is the square root of two. This is an irrational number. In decimal for it is written as 1.414213... Mathematicians have proved that the square root of every natural number is either an integer or an irrational number.

One well known irrational number is pi. This is the circumference of a circle divided by its diameter. This number is the same for every circle. The number pi is approximately 3.1415926359... .

An irrational number cannot be fully written down in decimal form. It would have an infinite number of digits after the decimal point. These digits would also not repeat.

# Citable sentences

Up to date as of December 29, 2010

Here are sentences from other pages on Irrational number, which are similar to those in the above article.