From Wikipedia, the free encyclopedia
.^ 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 noninteger fractions make up the set of real numbers. Numbers 26 January 2010 1:52 UTC www.bsu.edu [Source type: Reference]
.^ 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_2Thinking CriticallyBrief ReviewSets 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]
.^ 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]
.^ 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]} .^ 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]}
.^ 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.gapsystem.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]
.^ 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
is irrational.
History
.^ 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/WAKIViewArticle.aspx?pin=xnu068900a&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.
.^ The square root of 2 is an irrational number.
^ 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]} .^ 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/WAKIViewArticle.aspx?pin=xnu068900a&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]}
.^ 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 BarNatan: Classes: 200405: 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]} .^ 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/200908/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.
.^ 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]
.^ 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:

.^ William Jones, a selftaught 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.
.^ 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]
.^ Complex numbers and geometry  the third in the series. Hanna UsckaWehlou, 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 BarNatan: Classes: 200405: 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]
.^ 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]} .^ 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]
.^ 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/WAKIViewArticle.aspx?pin=xnu068700a&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]
.^ Book X considers commensurable and incommensurable magnitudes.
^ Proportion is an equation that states that two ratios are equal, for example 2:1 = 6:3.
^ 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.
.^ 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
.^ 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
.^ 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.cuttheknot.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
AlMahani (d. 874/884) examined and classified
quadratic irrationals and cubic irrationals.
.^ 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]}
.