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Topic: Irrational number

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 Number - Wikipedia, the free encyclopedia
The first existence proofs 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.
This number is denoted by i, a symbol assigned by Leonhard Euler.
The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields. /wiki/Number   (3655 words)

 PlanetMath: irrational
An irrational number is a real number which cannot be represented as a ratio of two integers.
The sum, difference, product, and quotient (when defined) of two numbers, one rational and another irrational, is irrational.
This is version 6 of irrational, born on 2001-11-04, modified 2005-04-10. /encyclopedia/IrrationalNumber.html   (132 words)

 irrational number
There are two types of irrational number: algebraic numbers, such as the square root of 2, which are the roots of algebraic equations, and the transcendental numbers, such as pi and e, which aren't.
The vast majority of real numbers are irrational, so that if you were to pick a single point on the real number line at random the chances are overwhelmingly high that it would be irrational.
The decimal expansion of an irrational numbers doesn't come to an end or repeat itself (in equal length blocks), though it may have a pattern such as 0.101001000100001... /encyclopedia/I/irrational_number.html   (344 words)

 High-Tech Dictionary Definition
A real number which is not a ratio of two integers; for example, pi and the square root of two.An irrational number can be expressed as an infinite decimal in which not set of consecutive digits repeats itself infinitely. /resources/dictionary/definition.html?lookup=2541   (39 words)

 What's a number?
Thus Cantor studied such decimal expansions and observed that periodic expansions correspond to rational numbers whereas the non-periodic ones correspond (by his definition) to the irrational numbers.
The theory of irrational numbers belongs to Calculus.
The rest of the complex numbers could also be defined by adding this new number i to the set of reals and postulating that usual arithmetic operations (addition, subtraction, multiplication) apply to the expanded set and all the laws known to hold for these operations hold for the new set as well. /do_you_know/numbers.shtml   (3529 words)

A number that cannot be expressed as the ratio of two integers. /school/glossary/defs/irrational_number.html   (12 words)

 Composite number for pi satisfies Euler's equation.
is a transcendental number, i.e., an irrational number that is not, and cannot be, the root of an algebraic equation having rational coefficients.
Transcendental number — An irrational number that cannot be expressed as the root of an algebraic equation having rational coefficients.
is an irrational number that cannot be expressed as the root of an algebraic equation having rational coefficients, it is called a transcendental number. /r-logan/fullbook.html   (4935 words)

 Square root of 2 is irrational
It's edifying to recall an estimate of approximation of irrational numbers with rational ones.
Since the rational numbers form a dense set (i.e., in every interval no matter how small there are always rational numbers) and, since the sum of lengths of all covering intervals is found to be infinite, it would seem that, having so generously covered all rational numbers, we have automatically covered all numbers.
Consider all rational numbers in the interval from 0 to 1, excluding 0. /proofs/sq_root.shtml   (1034 words)

 Math Forum: Ask Dr. Math FAQ: Integers, Rational Numbers, Irrational Numbers
The square root of 2 is an irrational number because it can't be written as a ratio of two integers.
A real number is a number that is somewhere on a number line, so any number on a number line that isn't a rational number is irrational.
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. /dr.math/faq/faq.integers.html   (736 words)

 irrational number - a definition
An irrational number is a real number that cannot be reduced to any ratio between an integer p and a natural number q.
Irrational numbers are primarily of interest to theoreticians.
The union of the set of irrational numbers and the set of rational numbers forms the set of real numbers. /gDefinition/0,294236,sid44_gci283983,00.html   (335 words)

 Numbers and Functions as Continued Fractions - Numericana
For almost all numbers, there's no simple pattern to the sequence of partial quotients: The continued fraction representation creates a bijective relation between irrational numbers from 0 to 1 and infinite sequences of positive integers.
For negative numbers, we take the reciprocal of the opposite [which is easily dealt with, since it's positive] and obtain the result as the opposite of that (this would translate into 8 other separate cases, if we had to spell them out).
The usual probability measure (Lebesgue measure) for sets of numbers between 0 and 1 thus translates into statistical properties for their partial quotients, which were investigated by the Russian mathematician A.Ya. /~numericana/answer/fractions.htm   (3614 words)

 Rational Number System
Note: Since irrational numbers cannot be expressed as a fraction they form decimals that are neither repeating nor terminating.
The real number system is made up of rational and irrational numbers.
Any number that cannot be written as a fraction where the numerator and denominator are integers. /jreed/math9/strand1/1101.htm   (289 words)

 Irrational numbers. Evolution of the real numbers.
If we insist, however, that there be a number to represent the ratio AB : CD, then we keep the name "Square root of 2," and we call it an irrational number.
It drove home the distinction between geometry and arithmetic; between what is continuous and what is discrete; and, as we are about to see, it led to the invention of irrational numbers.
In the following Topic, we will investigate the existence of irrational numbers. /aReal/irrational-numbers.htm   (857 words)

 irrational - Wiktionary
Any numbering associating translations with definitions is unreliable or incorrect.
(mathematics) (no comparative or superlative) Of a real number, that cannot be written as the ratio of two integers.
In some cases, translations are for the wrong part of speech and so should be moved into tables for the correct part of speech. /wiki/irrational   (162 words)

 RJN's More Digits of Irrational Numbers Page
They are not copyrighted and we do not think it is legally justifiable to copyright such a basic thing as the digits of a commonly used irrational number.
The first 5 million digits of the number e (currently unchecked).
The first 10 million digits of the number e (currently unchecked). /htmltest/rjn_dig.html   (289 words)

 Rational and irrational numbers - Topics in precalculus
The existence of these irrationals was first realized by Pythagoras in the 6th century B.C. He called them "unnameable" or "speechless" numbers.
But when an irrational number is expressed as a decimal, then, clearly, the decimal cannot terminate -- for if it did, the number would be rational!
They are the numbers of arithmetic: The whole numbers, fractions, mixed numbers, and decimals; together with their negative images. /aPreCalc/rational-irrational-numbers.htm   (744 words)

 The Prime Glossary: irrational number
Almost all real numbers are irrational; so if you were to pick a real number "at random," then the "probability" that it is irrational is one.
A real number is an irrational number if it is not a rational number.
The decimal expansion of irrational numbers do not repeat (in equal length blocks), though they can have a simple pattern such as /glossary/page.php?sort=IrrationalNumber   (102 words)

 Peter Kaminski: Irrational? Transcendental!
An irrational number is one that cannot be expressed as the ratio of two -integers-.
(A number that is is called "algebraic.") For example, the square roots of 2 and 3 are irrational, but they are not transcendental, because they are solutions to the equations x^2 - 2 = 0 and x^2 - 3 = 0, respectively.
The number e can certainly be expressed as the ratio of two numbers, but at least one of these numbers must not be an -integer-. /archives/000378.html   (1083 words) - mathematical creations
I was under the impression that an irrational number is a number with no repeating intergers or repeating patterns How does that fit into your definition of a rational number.
Itseems to me that a rational number added to a cardical number is still irrational; multiplying an irrational number by a rational number is irrational.
But since pi has an infinite number of digits without a repeating pattern, regardless of where you move the decimal point, it will still be irrational. /showthread.php?t=43886   (573 words)

 Rational and Irrational Numbers
Although irrational numbers are not often used in daily life, they do exist on the number line.
All numbers that are not rational are considered irrational.
An irrational number can be written as a decimal, but not as a fraction. /ipka/A0876704.html   (229 words)

These numbers are formed by a number multiplied by itself (1, 4, 9, 16...).
There are a variety of numbers which you are required to know about.
These are whole positive or negative numbers (-3, -2, -1, 0, 1, 2, 3). /maths/number.htm   (77 words)

 The Square Root of 2 and Other Irrational Numbers
The rational numbers and the irrationals are interwoven on the number line, so that between any two rationals (no matter how close they are) there is an irrational, and between any two irrationals (no matter how close they are) there is a rational.
Consequently, if the set of irrational numbers were countable, we would have an argument that the set of real numbers is countable-- since the set of real numbers would then be a union of two countable sets.
must be irrational, we will use the method of proof by contradiction introduced in the vignette on prime numbers. /math/vignettes/sqrt2.htm   (724 words)

 Golden Ratio
Most people are familiar with the number Pi, since it is one of the most ubiquitous irrational numbers known to man. But, there is another irrational number that has the same propensity for popping up and is not as well known as Pi.
Another property of the Fibonnaci numbers is that no two consecutive numbers in the sequence have a common prime factor.
This wonderful number is Phi, and it has a tendency to turn up in a great number of places, a few of which will be discussed in this essay. /emt669/Student.Folders/Frietag.Mark/Homepage/Goldenratio/goldenratio.html   (2304 words)

 Rational Numbers
In decimal form, irrational numbers do not repeat in a pattern or terminate.
is actually a non-ending decimal and is an irrational number.
A rational number is a number that can be expressed as a fraction or ratio (rational). /Regents/math/rational/Lrat.htm   (154 words)

 irrational, irrational number
a number which cannot be written as a fraction where the numerator and denominator are both integers and where the denominator does not equal zero; a nonrepeating or nonterminating decimal number.
Counting or natural numbers are at the very bottom, the smallest set.
You are hereby granted permission to make ONE printed copy of this page and its picture(s) for your PERSONAL and not-for-profit use. /math/spoken/here/1words/i/i26.htm   (85 words)

 Number Types Lesson
On the other hand, all those numbers that can be written as non-repeating, non-terminating decimals are non-rational, so they are called the "irrationals".
The commonest question I hear regarding number types is something along the lines of "Is a real number irrational, or is an irrational number real, or neither...
In decimal form, a number is either non-terminating and non-repeating (so it's an irrational) or not (so it's a rational); there is no overlap between these two number types! /modules/numtypes.htm   (797 words)

 Quia - MATH A - Rational or Irrational?
Identify whether a given number is rational or irrational.
Quia - MATH A - Rational or Irrational?
To learn how to make your own, just like this, click here. /pop/37541.html   (38 words)

 BBspot - Irrational Building Number Causes Tragedy at Google
Unfortunately, the county’s 911 system was unable to deal with the irrational numbers and tragedy befell the search engine company.
Google numbered its building 0, 1, e and pi to satisfy the intellectual superiority that many Google employees feel.
Mountain View, CA – What once seemed a harmless, whimsical choice in building numbers, turned a simple call for help into a tragedy. /News/2005/03/google_numbers.html   (426 words)

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