square root of 2

7/27/2019 Square Root of 2

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Square root of 2

The square root of 2, often known as root 2 orradical 2 and written

as , is the positivealgebraic numberthat, when multiplied by itself,

gives the number2. It is more precisely called the principal squareroot of 2, to distinguish it from the negative number with the sameproperty.

Geometrically thesquare rootof 2 is the length of a diagonal acrossasquare with sides of one unit of length; this follows fromthePythagorean theorem. It was probably the first number known tobeirrational. Its numerical value truncated to 65decimal placesis:

1.41421356237309504880168872420969807856967187537694

807317667973799... (sequenceA002193inOEIS).

The square root of 2.

The quick approximation 99/70 ( 1.41429) for the square root of

two is frequently used. Despite having adenominatorof only 70, itdiffers from the correct value by less than 1/10,000 (approx. 7.2 10

-5).

The square root of 2 is equal to the length of thehypotenuseof

aright trianglewith legs of length 1.
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History

TheBabylonianclay tabletYBC 7289(c. 18001600 BC) gives anapproximation of in foursexagesimalfigures, which is aboutsixdecimalfigures:[1]

Another early close approximation is given inancient Indianmathematicaltexts, theSulbasutras(c. 800200 BC) as follows: Increase the length [of theside] by its third and this third by its own fourth less the thirty-fourth part ofthat fourth.[2]That is,

This ancient Indian approximation is the seventh in a sequence ofincreasingly accurate approximations based on the sequence ofPellnumbers, that can be derived from thecontinued fractionexpansion

of . Despite having a smaller denominator, it is only slightly lessaccurate than the Babylonian approximation.

Pythagoreansdiscovered that the diagonal of a square isincommensurable with its side, or in modern language, that the squareroot of two isirrational. Little is known with certainty about the time orcircumstances of this discovery, but the name ofHippasusof

Metapontum is often mentioned. For a while, the Pythagoreans treated asan official secret the discovery that the square root of two is irrational,and, according to legend, Hippasus was murdered for divulgingit.[3][4][5]The square root of two is occasionally called "Pythagoras'number" or "Pythagoras' Constant", for exampleConway & Guy (1996).
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Babylonian clay tablet YBC 7289 with annotations.

Computation algorithms

There are a number of algorithms for approximating , which in expressions

as a ratio of integers or as a decimal can only be approximated. The mostcommon algorithm for this, one used as a basis in many computers andcalculators, is the Babylonian method[7]of computing square roots, which is oneof manymethods of computing square roots. It goes as follows:

First, pick a guess, ; the value of the guess affects only how manyiterations are required to reach an approximation of a certain accuracy. Then,using that guess, iterate through the followingrecursivecomputation:

The more iterations through the algorithm (that is, the more computationsperformed and the greater "n"), the better approximation of the square rootof 2 is achieved. Each iteration approximately doubles the number of correctdigits. Starting with a0 = 1 the next approximations are

3/2 = 1.5

17/12 = 1.416...

577/408 = 1.414215...

665857/470832 = 1.4142135623746....

The value of was calculated to 137,438,953,444 decimal placesbyYasumasa Kanada's team in 1997. In February 2006 the record for the

calculation of was eclipsed with the use of a home computer. ShigeruKondo calculated 200,000,000,000 decimal places in slightly over 13 daysand 14 hours using a 3.6 GHz PC with 16GiBof memory.Amongmathematical constants with computationally challenging decimal
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expansions, onlyhas been calculated more precisely.Such computationsaim to empirically check whether such numbers arenormal.

Proofs of irrationality

A short proof of the irrationality of can be obtained from therational roottheorem, that is, if is amonic polynomialwith integer coefficients, then

anyrationalrootof is necessarily an integer. Applying this to the

polynomial , it follows that is either an integer or irrational.

Because is not an integer (2 is not a perfect square), must therefore beirrational. This proof can be generalized to show that any root of any naturalnumber which is not the square of a natural number is irrational.

Seequadratic irrationalorinfinite descent#Irrationality of k if it is not anintegerfor a proof that the square root of any non-square natural number isirrational.

Proof by infinite descent

One proof of the number's irrationality is the following proof byinfinite descent.It is also aproof by contradiction, also known as an indirect proof, in that theproposition is proved by assuming that the opposite of the proposition is trueand showing that this assumption is false, thereby implying that the propositionmust be true.

1. Assume that is a rational number, meaning that there exists aninteger and an integer in general such that .

2. Then can be written as anirreducible fraction such that andarecoprimeintegers.

3. It follows that and . ( )4. Therefore is even because it is equal to . ( is necessarily even

because it is 2 times another whole number and even numbers aremultiples of 2.)

5. It follows that a must be even (as squares of odd integers are never

even).6. Because a is even, there exists an integerkthat fulfills: .

7. Substituting from step 6 fora in the second equation of step

3: is equivalent to , which is equivalent to .
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8. Because is divisible by two and therefore even, andbecause , it follows that is also even which means that b iseven.

9. By steps 5 and 8 a and b are both even, which contradicts that is

irreducible as stated in step 2.Q.E.D.

Because there is a contradiction, the assumption (1) that is a rationalnumber must be false. By thelaw of excluded middle, the opposite is

proven: is irrational.

This proof was hinted at byAristotle, in hisAnalytica Priora, I.23.[10]Itappeared first as a full proof inEuclid'sElements, as proposition 117 ofBook X. However, since the early 19th century historians agree that thisproof is aninterpolationand not attributable to Euclid.

Proof by unique factorization

An alternative proof uses the same approach with thefundamental theorem ofarithmeticwhich says every integer greater than 1 has a unique factorizationinto powers of primes.

1. Assume that is a rational number. Then there are

integers a and b such that a iscoprimeto b and . In other

words, can be written as anirreducible fraction.

2. The value ofb cannot be 1 as there is no integera the square of which is2.

3. There must be a primep which divides b and which does not divide a,otherwise the fraction would not be irreducible.

4. The square ofa can be factored as the product of the primes intowhich a is factored but with each power doubled.

5. Therefore by unique factorization the primep which divides b, and also itssquare, cannot divide the square ofa.

6. Therefore the square of an irreducible fraction cannot be reduced to an

integer.7. Therefore the square root of 2 cannot be a rational number.

This proof can be generalized to show that if an integer is not an exact kthpower of another integer then its kth root is irrational. The articlequadraticirrationalgives a proof of the same result but not using the fundamentaltheorem of arithmetic.
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Proof by infinite descent, not involving factoring

The followingreductio ad absurdumargument showing the irrationality of is

less well-known. It uses the additional information so

that .[12]

1. Assume that is a rational number. This would mean that there exist

positive integers m and n with such that .

Then and .

2. We may assume that n is the smallest integer so that is an integer.That is, that the fraction m/n is in lowest terms.

3. Then

4. Because , it follows that .

5. So the fraction m/n for , which according to (2) is already inlowestterms, is represented by (3) in strictly lower terms. This is a contradiction,

so the assumption that is rational must be false.

Geometric proof

Anotherreductio ad absurdumshowing that is irrational is less well-

known.

[13]

It is also an example of proof byinfinite descent. It makes use ofclassiccompass and straightedgeconstruction, proving the theorem by amethod similar to that employed by ancient Greek geometers. It is essentiallythe previous proof viewed geometrically.

LetABCbe a right isosceles triangle with hypotenuse length m and legs n. By

thePythagorean theorem, . Suppose m and n areintegers.Let m:n be aratiogiven in itslowest terms.

Draw the arcs BD and CEwith centreA. Join DE. It followsthatAB =AD,AC=AEand the BACand DAEcoincide. Therefore the

trianglesABCandADEarecongruentbySAS.

Because EBFis a right angle and BEFis half a right angle, BEFis also aright isosceles triangle. Hence BE= mn implies BF= mn. Bysymmetry, DF= mn, and FDCis also a right isosceles triangle. It also followsthat FC= n (mn) = 2nm.
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Hence we have an even smaller right isosceles triangle, with hypotenuse length2nm and legs mn. These values are integers even smallerthan mand n and in the same ratio, contradicting the hypothesis that m:n is in

lowest terms. Therefore m and n cannot be both integers, hence is

irrational.

Analytic proof

Lemma: let and such thatfor all and

Then is irrational.

Proof: suppose = a/b with a, bN+.

For sufficiently big n,

then


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but is an integer, absurd, then is irrational.

is irrational.

Proof: let and

for all .

Byinduction,

for all . For ,

and if is true forn then is true for . In fact

By application of the lemma, isirrational.

Constructive proof

In a constructive approach, one distinguishes between on the one hand notbeing rational, and on the other hand being irrational (i.e., being quantifiablyapart from every rational), the latter being a stronger property. Givenintegers a and b, because thevaluation(i.e., highest power of 2 dividing a

number) of 2b2 is odd, while the valuation ofa2 is even, they must be distinctintegers, so that, applying the law oftrichotomyin the context of an

effectivelycomputable predicateover , we obtain . Then[14]
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the latter inequality being true because we assume (otherwisethe quantitative apartness can be trivially established). This gives a lower

bound of for the difference , yielding a direct proof of

irrationality not relying on thelaw of excluded middle; seeErrettBishop(1985, p. 18). This proof constructively exhibits a discrepancy

between and any rational.

Properties of the square root of two

One-half of , also 1 divided by the square root of 2, approximately 0.7071067811 86548, is a common quantity in geometry andtrigonometrybecausetheunit vectorthat makes a 45 angle with the axes in a plane has thecoordinates

This number satisfies

One interesting property of the square root of 2 is as follows:

since This is related to the propertyofsilver ratios.

The square root of 2 can also be expressed in terms of the copies oftheimaginary unitiusing only thesquare rootandarithmeticoperations:

The square root of 2 is also the only real number other than 1whose infinitetetrateis equal to its square.

The square root of 2 appears inVite's formulafor:
http://en.wikipedia.org/wiki/Law_of_excluded_middlehttp://en.wikipedia.org/wiki/Law_of_excluded_middlehttp://en.wikipedia.org/wiki/Law_of_excluded_middlehttp://en.wikipedia.org/wiki/Errett_Bishophttp://en.wikipedia.org/wiki/Errett_Bishophttp://en.wikipedia.org/wiki/Errett_Bishophttp://en.wikipedia.org/wiki/Errett_Bishophttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Arithmetic_operationshttp://en.wikipedia.org/wiki/Arithmetic_operationshttp://en.wikipedia.org/wiki/Arithmetic_operationshttp://en.wikipedia.org/wiki/Arithmetic_operationshttp://en.wikipedia.org/wiki/Tetrationhttp://en.wikipedia.org/wiki/Tetrationhttp://en.wikipedia.org/wiki/Tetrationhttp://en.wikipedia.org/wiki/Vieta_formulahttp://en.wikipedia.org/wiki/Vieta_formulahttp://en.wikipedia.org/wiki/Vieta_formulahttp://en.wikipedia.org/wiki/Vieta_formulahttp://en.wikipedia.org/wiki/Tetrationhttp://en.wikipedia.org/wiki/Arithmetic_operationshttp://en.wikipedia.org/wiki/Arithmetic_operationshttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Errett_Bishophttp://en.wikipedia.org/wiki/Errett_Bishophttp://en.wikipedia.org/wiki/Law_of_excluded_middle


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form square roots and only one minus sign. Similar in

appearance but with a finite number of terms, the square rootof 2 appears in various trigonometric constants:


11/22

It is not known whether is anormal number, a stronger property thanirrationality, but statistical analyses of itsbinary expansionare consistent withthe hypothesis that it is normal to base two.

Series and product representations

The identity , along with the infinite product

representations for the sine and cosine, leads to products such as

and

or equivalently,

The number can also be expressed by taking theTaylor seriesof a

trigonometric function. For example, the series for gives

The Taylor series of with and using thedouble

factorial gives

The convergence of this series can be accelerated with anEulertransform, producing
http://en.wikipedia.org/wiki/Normal_numberhttp://en.wikipedia.org/wiki/Normal_numberhttp://en.wikipedia.org/wiki/Normal_numberhttp://en.wikipedia.org/wiki/Binary_expansionhttp://en.wikipedia.org/wiki/Binary_expansionhttp://en.wikipedia.org/wiki/Binary_expansionhttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Double_factorialhttp://en.wikipedia.org/wiki/Double_factorialhttp://en.wikipedia.org/wiki/Double_factorialhttp://en.wikipedia.org/wiki/Double_factorialhttp://en.wikipedia.org/wiki/Euler_transformhttp://en.wikipedia.org/wiki/Euler_transformhttp://en.wikipedia.org/wiki/Euler_transformhttp://en.wikipedia.org/wiki/Euler_transformhttp://en.wikipedia.org/wiki/Euler_transformhttp://en.wikipedia.org/wiki/Euler_transformhttp://en.wikipedia.org/wiki/Double_factorialhttp://en.wikipedia.org/wiki/Double_factorialhttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Binary_expansionhttp://en.wikipedia.org/wiki/Normal_number


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It is not known whether can be represented with aBBP-type

formula. BBP-type formulas are known for and ,however.

Continued fraction representation

The square root of two has the followingcontinued fractionrepresentation:

Theconvergentsformed by truncating this representation form a sequenceof fractions that approximate the square root of two to increasing accuracy,and that are described by thePell numbers(known as side and diameternumbers to the ancient Greeks because of their use in approximating theratio between the sides and diagonal of a square). The first convergents are:

1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408. The convergent p/qdiffers from the square root of 2 by almost exactly and then the nextconvergent is (p + 2q)/(p + q).

Square root of 3

The square root of 3 is the positivereal numberthat, when multiplied by itself,gives the number3. It is more precisely called theprincipal square root of 3, todistinguish it from the negative number with the same property. It is denoted by

The first sixty significant digits of itsdecimal expansionare:
http://en.wikipedia.org/wiki/BBP-type_formulahttp://en.wikipedia.org/wiki/BBP-type_formulahttp://en.wikipedia.org/wiki/BBP-type_formulahttp://en.wikipedia.org/wiki/BBP-type_formulahttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://en.wikipedia.org/wiki/Pell_numberhttp://en.wikipedia.org/wiki/Pell_numberhttp://en.wikipedia.org/wiki/Pell_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/3_(number)http://en.wikipedia.org/wiki/3_(number)http://en.wikipedia.org/wiki/3_(number)http://en.wikipedia.org/wiki/Decimal_expansionhttp://en.wikipedia.org/wiki/Decimal_expansionhttp://en.wikipedia.org/wiki/Decimal_expansionhttp://en.wikipedia.org/wiki/Decimal_expansionhttp://en.wikipedia.org/wiki/3_(number)http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Pell_numberhttp://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/BBP-type_formulahttp://en.wikipedia.org/wiki/BBP-type_formula


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1.73205080756887729352744634150587236694280525381038062805580... (sequenceA002194inOEIS)

The rounded value of 1.732 is correct to within 0.01% of the actual value.

A close fraction is (1.732142857...).

Archimedesreported (1351/780)2 > 3 > (265/153)2, accurate to1/608400 (6-places) and 2/23409 (4-places), respectively.

Thesquare rootof 3 is anirrational number. It is also knownas Theodorus' constant, named afterTheodorus of Cyrene.

It can be expressed as thecontinued fraction[1;1,2,1,2,1,2,1,...](sequenceA040001inOEIS), expanded on the right.

It can also be expressed bygeneralized continued fractionssuch as

which is [1;1, 2,1, 2,1, 2,1, ...] evaluated at every second term.

List of numbers

Irrational and suspected irrational numbers

(3) 23

5 Se

Binary 1.1011101101100111101...

Decimal 1.7320508075688772935...

Hexadecimal 1.BB67AE8584CAA73B...

Continued fraction
http://oeis.org/A002194http://oeis.org/A002194http://oeis.org/A002194http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Theodorus_of_Cyrenehttp://en.wikipedia.org/wiki/Theodorus_of_Cyrenehttp://en.wikipedia.org/wiki/Theodorus_of_Cyrenehttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://oeis.org/A040001http://oeis.org/A040001http://oeis.org/A040001http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/Generalized_continued_fractionhttp://en.wikipedia.org/wiki/Generalized_continued_fractionhttp://en.wikipedia.org/wiki/Generalized_continued_fractionhttp://en.wikipedia.org/wiki/List_of_numbershttp://en.wikipedia.org/wiki/List_of_numbershttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constanthttp://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constanthttp://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constanthttp://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constanthttp://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_5http://en.wikipedia.org/wiki/Square_root_of_5http://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Plastic_numberhttp://en.wikipedia.org/wiki/Plastic_numberhttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Decimalhttp://en.wikipedia.org/wiki/Hexadecimalhttp://en.wikipedia.org/wiki/Hexadecimalhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Hexadecimalhttp://en.wikipedia.org/wiki/Decimalhttp://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Plastic_numberhttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Square_root_of_5http://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constanthttp://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constanthttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/List_of_numbershttp://en.wikipedia.org/wiki/Generalized_continued_fractionhttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://oeis.org/A040001http://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Theodorus_of_Cyrenehttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://oeis.org/A002194


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Proof of irrationality

This irrationality proof for the square root of 3 usesFermat's method ofinfinitedescent:

Suppose that 3 is rational, and express it in lowest possible terms (i.e., as a

fully reduced fraction) as for natural numbers m and n. Then 3 can be

expressed in lower terms as , which is a contradiction.[1](The twofractional expressions are equal because equating them, cross-multiplying, and

canceling like additive terms gives and hence , which is true

by the premise. The second fractional expression for3 is in lower terms since,comparing denominators, since since since .

And both the numerator and the denominator of the second fractional

expression are positive since and .)

Geometry and trigonometry

If anequilateral trianglewith sides of length 1 is cut into two equal halves, bybisecting an internal angle across to make a right angle with one side, the rightangle triangle'shypotenuseis length one and the sides are of length 1/2

and 3/2. From this the trigonometric function tangent of 60 degrees equals3,and the sine of 60 and the cosine of 30 both equal half of3.

The square root of 3 also appears in algebraic expressions for variousothertrigonometric constants, including[2]the sines of 3, 12, 15, 21, 24, 33,39, 48, 51, 57, 66, 69, 75, 78, 84, and 87.

It is the distance between parallel sides of a regularhexagonwith sides oflength 1. On the complex plane, this distance is expressed

as i3 mentionedbelow.

It is the length of thespace diagonalof a unitcube.The shapeVesica piscishas a major axis: minor axis ratio equal to the squareroot of three, this can be shown by constructing two equilateral triangles withinit.
http://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Square_root_of_3#cite_note-Grant-1http://en.wikipedia.org/wiki/Square_root_of_3#cite_note-Grant-1http://en.wikipedia.org/wiki/Square_root_of_3#cite_note-Grant-1http://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Exact_trigonometric_constantshttp://en.wikipedia.org/wiki/Exact_trigonometric_constantshttp://en.wikipedia.org/wiki/Exact_trigonometric_constantshttp://en.wikipedia.org/wiki/Square_root_of_3#cite_note-2http://en.wikipedia.org/wiki/Square_root_of_3#cite_note-2http://en.wikipedia.org/wiki/Hexagonhttp://en.wikipedia.org/wiki/Hexagonhttp://en.wikipedia.org/wiki/Hexagonhttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Square_root_of_3#Square_root_of_.E2.88.923http://en.wikipedia.org/wiki/Square_root_of_3#Square_root_of_.E2.88.923http://en.wikipedia.org/wiki/Space_diagonalhttp://en.wikipedia.org/wiki/Space_diagonalhttp://en.wikipedia.org/wiki/Space_diagonalhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Vesica_piscishttp://en.wikipedia.org/wiki/Vesica_piscishttp://en.wikipedia.org/wiki/Vesica_piscishttp://en.wikipedia.org/wiki/Vesica_piscishttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Space_diagonalhttp://en.wikipedia.org/wiki/Square_root_of_3#Square_root_of_.E2.88.923http://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Hexagonhttp://en.wikipedia.org/wiki/Square_root_of_3#cite_note-2http://en.wikipedia.org/wiki/Exact_trigonometric_constantshttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Square_root_of_3#cite_note-Grant-1http://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Pierre_de_Fermat


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The square root of 3 is equal to the length between parallel sides of a regular

hexagon with sides of length 1

Square root of 3Multiplicationof3 toimaginary unitgives a square root of3, animaginarynumber. More exactly,

(seesquare root of negative numbers).

It is anEisenstein integer. Namely, it is expressed as the difference betweentwo non-realcubic roots of 1(which are Eisenstein integers).

Square root of 5

The square root of 5 is the positivereal numberthat, when multiplied by itself,gives the prime number5. It is more precisely called the principal square rootof 5, to distinguish it from the negative number with the same property. Thisnumber appears in the fractional expression for thegolden ratio. It can bedenoted insurdform as:

It is anirrationalalgebraic number.[1]The first sixty significant digits ofitsdecimal expansionare:

2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152

57242 7089... (sequenceA002163inOEIS).

which can be rounded down to 2.236 to within 99.99% accuracy. As ofApril 1994, its numerical value in decimal had been computed to at leastone million digits.[
http://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/%E2%88%923_(number)http://en.wikipedia.org/wiki/%E2%88%923_(number)http://en.wikipedia.org/wiki/%E2%88%923_(number)http://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/Square_root_of_negative_numbershttp://en.wikipedia.org/wiki/Square_root_of_negative_numbershttp://en.wikipedia.org/wiki/Square_root_of_negative_numbershttp://en.wikipedia.org/wiki/Eisenstein_integerhttp://en.wikipedia.org/wiki/Eisenstein_integerhttp://en.wikipedia.org/wiki/Eisenstein_integerhttp://en.wikipedia.org/wiki/Root_of_unityhttp://en.wikipedia.org/wiki/Root_of_unityhttp://en.wikipedia.org/wiki/Root_of_unityhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/5_(number)http://en.wikipedia.org/wiki/5_(number)http://en.wikipedia.org/wiki/5_(number)http://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Nth_roothttp://en.wikipedia.org/wiki/Nth_roothttp://en.wikipedia.org/wiki/Nth_roothttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Algebraic_numberhttp://en.wikipedia.org/wiki/Algebraic_numberhttp://en.wikipedia.org/wiki/Square_root_of_5#cite_note-1http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-1http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-1http://en.wikipedia.org/wiki/Decimal_expansionhttp://en.wikipedia.org/wiki/Decimal_expansionhttp://en.wikipedia.org/wiki/Decimal_expansionhttp://oeis.org/A002163http://oeis.org/A002163http://oeis.org/A002163http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/Square_root_of_5#cite_note-2http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-2http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-2http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-2http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://oeis.org/A002163http://en.wikipedia.org/wiki/Decimal_expansionhttp://en.wikipedia.org/wiki/Square_root_of_5#cite_note-1http://en.wikipedia.org/wiki/Algebraic_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Nth_roothttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/5_(number)http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Root_of_unityhttp://en.wikipedia.org/wiki/Eisenstein_integerhttp://en.wikipedia.org/wiki/Square_root_of_negative_numbershttp://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/%E2%88%923_(number)http://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Multiplication


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List of numbers

Irrational and suspected irrational numbers

(3) 2 3 5 S e

Binary 10.0011110001101111...

Decimal 2.23606797749978969...

Hexadecimal 2.3C6EF372FE94F82C...

Continued fraction

Proof of irrationality

This irrationality proof for the square root of 5 usesFermat's method ofinfinitedescent:

Suppose that 5 is rational, and express it in lowest possible terms (i.e., as a

fully reduced fraction) as for natural numbers m and n. Then 5 can be

expressed in lower terms as , which is a contradiction.[3](The twofractional expressions are equal because equating them, cross-multiplying, and

canceling like additive terms gives and hence , which is trueby the premise. The second fractional expression for 5 is in lower terms since,comparing denominators, since since

since . And both the numerator and the denominator of the second

fractional expression are positive since and .)

Continued fraction

It can be expressed as thecontinued fraction[2; 4, 4, 4, 4, 4...](sequenceA040002inOEIS). The sequence ofbest rational approximationsis:
http://en.wikipedia.org/wiki/List_of_numbershttp://en.wikipedia.org/wiki/List_of_numbershttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constanthttp://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constanthttp://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constanthttp://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constanthttp://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Plastic_numberhttp://en.wikipedia.org/wiki/Plastic_numberhttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Decimalhttp://en.wikipedia.org/wiki/Hexadecimalhttp://en.wikipedia.org/wiki/Hexadecimalhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Fermathttp://en.wikipedia.org/wiki/Fermathttp://en.wikipedia.org/wiki/Fermathttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Square_root_of_5#cite_note-Grant-3http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-Grant-3http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-Grant-3http://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://oeis.org/A040002http://oeis.org/A040002http://oeis.org/A040002http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximationshttp://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximationshttp://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximationshttp://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximationshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://oeis.org/A040002http://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Square_root_of_5#cite_note-Grant-3http://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Infinite_descenthttp://en.wikipedia.org/wiki/Fermathttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Hexadecimalhttp://en.wikipedia.org/wiki/Decimalhttp://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Silver_ratiohttp://en.wikipedia.org/wiki/Plastic_numberhttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constanthttp://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constanthttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/List_of_numbers


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Convergentsof the continued fraction are colored; their numerators are 2, 9,38, 161, ... (sequenceA001077inOEIS), and their denominators are 1, 4,

17, 72, ... (sequenceA001076inOEIS). The other (non-colored) termsaresemiconvergents.

Babylonian method

When is computed with theBabylonian method, starting with r0 = 2 andusing rn+1 = (rn + 5/rn) / 2, the nth approximant rn is equal to the 2

n-th convergentof the convergent sequence:

Relation to the golden ratio and Fibonacci numbers

Thisgolden ratio is thearithmetic meanof1and the square root of5.[4]Thealgebraicrelationship between the square root of 5, the golden ratio

and theconjugate of the golden ratio( ) are expressed in thefollowing formulae:

(See section below for their geometrical interpretation as decompositionsof a root-5 rectangle.)

The square root of 5 then naturally figures in the closed form expressionfor theFibonacci numbers, a formula which is usually written in terms of

the golden ratio:

The quotient of 5 and (or the product of 5 and ), and its reciprocal,provide an interesting pattern of continued fractions and are related tothe ratios between the Fibonacci numbers and theLucas numbers:
http://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://oeis.org/A001077http://oeis.org/A001077http://oeis.org/A001077http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://oeis.org/A001076http://oeis.org/A001076http://oeis.org/A001076http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://en.wikipedia.org/wiki/Continued_fraction#Semiconvergentshttp://en.wikipedia.org/wiki/Continued_fraction#Semiconvergentshttp://en.wikipedia.org/wiki/Continued_fraction#Semiconvergentshttp://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_methodhttp://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_methodhttp://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_methodhttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Arithmetic_meanhttp://en.wikipedia.org/wiki/Arithmetic_meanhttp://en.wikipedia.org/wiki/Arithmetic_meanhttp://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-4http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-4http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-4http://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Golden_ratio#Golden_ratio_conjugatehttp://en.wikipedia.org/wiki/Golden_ratio#Golden_ratio_conjugatehttp://en.wikipedia.org/wiki/Golden_ratio#Golden_ratio_conjugatehttp://en.wikipedia.org/wiki/Fibonacci_numberhttp://en.wikipedia.org/wiki/Fibonacci_numberhttp://en.wikipedia.org/wiki/Fibonacci_numberhttp://en.wikipedia.org/wiki/Lucas_numberhttp://en.wikipedia.org/wiki/Lucas_numberhttp://en.wikipedia.org/wiki/Lucas_numberhttp://en.wikipedia.org/wiki/Lucas_numberhttp://en.wikipedia.org/wiki/Fibonacci_numberhttp://en.wikipedia.org/wiki/Golden_ratio#Golden_ratio_conjugatehttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Square_root_of_5#cite_note-4http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/Arithmetic_meanhttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_methodhttp://en.wikipedia.org/wiki/Continued_fraction#Semiconvergentshttp://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://oeis.org/A001076http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttp://oeis.org/A001077http://en.wikipedia.org/wiki/Convergent_(continued_fraction)


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The series of convergents to these values feature the series ofFibonacci numbers and the series ofLucas numbersas numeratorsand denominators, and viceversa, respectively:

The 52 diagonal of a half square forms the basis for the geometricalconstruction of agolden rectangle.

Geometry

Geometrically, the square root of 5 corresponds to thediagonalofarectanglewhose sides are of length1and2, as is evident fromthePythagorean theorem. Such a rectangle can be obtained by halvingasquare, or by placing two equal squares side by side. Together with the
http://en.wikipedia.org/wiki/Lucas_numberhttp://en.wikipedia.org/wiki/Lucas_numberhttp://en.wikipedia.org/wiki/Lucas_numberhttp://en.wikipedia.org/wiki/Golden_rectanglehttp://en.wikipedia.org/wiki/Golden_rectanglehttp://en.wikipedia.org/wiki/Golden_rectanglehttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Diagonalhttp://en.wikipedia.org/wiki/Diagonalhttp://en.wikipedia.org/wiki/Diagonalhttp://en.wikipedia.org/wiki/Rectanglehttp://en.wikipedia.org/wiki/Rectanglehttp://en.wikipedia.org/wiki/Rectanglehttp://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/2_(number)http://en.wikipedia.org/wiki/2_(number)http://en.wikipedia.org/wiki/2_(number)http://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Square_(geometry)http://en.wikipedia.org/wiki/Square_(geometry)http://en.wikipedia.org/wiki/Square_(geometry)http://en.wikipedia.org/wiki/Square_(geometry)http://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/2_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/Rectanglehttp://en.wikipedia.org/wiki/Diagonalhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Golden_rectanglehttp://en.wikipedia.org/wiki/Lucas_number


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algebraic relationship between 5 and , this forms the basis for thegeometrical construction of agolden rectanglefrom a square, and for theconstruction of a regularpentagongiven its side (since the side-to-diagonalratio in a regular pentagon is ).

Forming adihedralright anglewith the two equal squares that halve a 1:2rectangle, it can be seen that 5 corresponds also to the ratio between thelength of acubeedgeand the shortest distance from one of itsverticesto theopposite one, when traversing the cube surface (the shortest distance whentraversing through the inside of the cube corresponds to the length of the cubediagonal, which is thesquare root of threetimes the edge).

The number 5 can be algebraically and geometrically related to the squareroot of 2and thesquare root of 3, as it is the length of thehypotenuseof a righttriangle withcathetimeasuring 2 and 3 (again, the Pythagorean theorem

proves this). Right triangles of such proportions can be found inside a cube: thesides of any triangle defined by thecentrepoint of a cube, one of its vertices,and the middle point of a side located on one the faces containing that vertexand opposite to it, are in the ratio 2:3:5. This follows from the geometricalrelationships between a cube and the quantities 2 (edge-to-face-diagonal ratio,or distance between opposite edges), 3 (edge-to-cube-diagonal ratio) and 5(the relationship just mentioned above).

A rectangle with side proportions 1:5 is called aroot-five rectangle and is partof the series of root rectangles, a subset ofdynamic rectangles, which are

based on 1 (= 1), 2, 3, 4 (= 2), 5... and successively constructed using thediagonal of the previous root rectangle, starting from a square. A root-5rectangle is particularly notable in that it can be split into a square and twoequal golden rectangles (of dimensions 1), or into two golden rectangles ofdifferent sizes (of dimensions 1 and 1 ). It can also be decomposed asthe union of two equal golden rectangles (of dimensions 1 ) whoseintersection forms a square. All this is can be seen as the geometricinterpretation of the algebraic relationships between 5, and mentionedabove. The root-5 rectangle can be constructed from a 1:2 rectangle (the root-4rectangle), or directly from a square in a manner similar to the one for the

golden rectangle shown in the illustration, but extending the arc of length 52 toboth sides.
http://en.wikipedia.org/wiki/Golden_rectanglehttp://en.wikipedia.org/wiki/Golden_rectanglehttp://en.wikipedia.org/wiki/Golden_rectanglehttp://en.wikipedia.org/wiki/Pentagonhttp://en.wikipedia.org/wiki/Pentagonhttp://en.wikipedia.org/wiki/Pentagonhttp://en.wikipedia.org/wiki/Dihedral_anglehttp://en.wikipedia.org/wiki/Dihedral_anglehttp://en.wikipedia.org/wiki/Right_anglehttp://en.wikipedia.org/wiki/Right_anglehttp://en.wikipedia.org/wiki/Right_anglehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Edge_(geometry)http://en.wikipedia.org/wiki/Edge_(geometry)http://en.wikipedia.org/wiki/Edge_(geometry)http://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Square_root_of_threehttp://en.wikipedia.org/wiki/Square_root_of_threehttp://en.wikipedia.org/wiki/Square_root_of_threehttp://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Cathetushttp://en.wikipedia.org/wiki/Cathetushttp://en.wikipedia.org/wiki/Cathetushttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Dynamic_rectanglehttp://en.wikipedia.org/wiki/Dynamic_rectanglehttp://en.wikipedia.org/wiki/Dynamic_rectanglehttp://en.wikipedia.org/wiki/Dynamic_rectanglehttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Cathetushttp://en.wikipedia.org/wiki/Hypotenusehttp://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_threehttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Edge_(geometry)http://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Right_anglehttp://en.wikipedia.org/wiki/Dihedral_anglehttp://en.wikipedia.org/wiki/Pentagonhttp://en.wikipedia.org/wiki/Golden_rectangle


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Conway triangle decomposition into homothetic smaller triangles.

Trigonometry

Like 2 and 3, the square root of 5 appears extensively in the formulaeforexact trigonometric constants, including in the sines and cosines of everyangle whose measure in degrees is divisible by 3 but not by 15.[8]The simplestof these are

As such the computation of its value is important forgeneratingtrigonometric tables. Since 5 is geometrically linked to half-squarerectangles and to pentagons, it also appears frequently in formulaefor the geometric properties of figures derived from them, such asin the formula for the volume of adodecahedron.
http://en.wikipedia.org/wiki/Exact_trigonometric_constantshttp://en.wikipedia.org/wiki/Exact_trigonometric_constantshttp://en.wikipedia.org/wiki/Exact_trigonometric_constantshttp://en.wikipedia.org/wiki/Square_root_of_5#cite_note-8http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-8http://en.wikipedia.org/wiki/Square_root_of_5#cite_note-8http://en.wikipedia.org/wiki/Generating_trigonometric_tableshttp://en.wikipedia.org/wiki/Generating_trigonometric_tableshttp://en.wikipedia.org/wiki/Generating_trigonometric_tableshttp://en.wikipedia.org/wiki/Generating_trigonometric_tableshttp://en.wikipedia.org/wiki/Dodecahedronhttp://en.wikipedia.org/wiki/Dodecahedronhttp://en.wikipedia.org/wiki/Dodecahedronhttp://en.wikipedia.org/wiki/Dodecahedronhttp://en.wikipedia.org/wiki/Generating_trigonometric_tableshttp://en.wikipedia.org/wiki/Generating_trigonometric_tableshttp://en.wikipedia.org/wiki/Square_root_of_5#cite_note-8http://en.wikipedia.org/wiki/Exact_trigonometric_constants


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Diophantine approximations

Hurwitz's theoreminDiophantine approximationsstates that everyirrationalnumberxcan be approximated by infinitely manyrationalnumbersm/n inlowest termsin such a way that

and that 5 is best possible, in the sense that for any larger constant than5, there are some irrational numbersxfor which only finitely many suchapproximations exist.

Closely related to this is the theorem that of any threeconsecutiveconvergentspi/qi,pi+1/qi+1,pi+2/qi+2, of a number , at least one ofthe three inequalities holds:

And the 5 in the denominator is the best bound possible since theconvergents of thegolden ratiomake the difference on the left-hand sidearbitrarily close to the value on the right-hand side. In particular, onecannot obtain a tighter bound by considering sequences of four or moreconsecutive convergents.

Algebra

Thering contains numbers of the form ,where a and b areintegersand is theimaginary number . This ring is afrequently cited example of anintegral domainthat is not aunique factorizationdomain. The number 6 has two inequivalent factorizations within this ring:

Thefield , like any otherquadratic field, is anabelian extensionof therational numbers. TheKroneckerWeber theoremtherefore guarantees thatthe square root of five can be written as a rational linear combination ofroots

of unity:
http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory)http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory)http://en.wikipedia.org/wiki/Diophantine_approximationshttp://en.wikipedia.org/wiki/Diophantine_approximationshttp://en.wikipedia.org/wiki/Diophantine_approximationshttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Lowest_termshttp://en.wikipedia.org/wiki/Lowest_termshttp://en.wikipedia.org/wiki/Lowest_termshttp://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Ring_(mathematics)http://en.wikipedia.org/wiki/Ring_(mathematics)http://en.wikipedia.org/wiki/Ring_(mathematics)http://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/Integral_domainhttp://en.wikipedia.org/wiki/Integral_domainhttp://en.wikipedia.org/wiki/Integral_domainhttp://en.wikipedia.org/wiki/Unique_factorization_domainhttp://en.wikipedia.org/wiki/Unique_factorization_domainhttp://en.wikipedia.org/wiki/Unique_factorization_domainhttp://en.wikipedia.org/wiki/Unique_factorization_domainhttp://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Quadratic_fieldhttp://en.wikipedia.org/wiki/Quadratic_fieldhttp://en.wikipedia.org/wiki/Quadratic_fieldhttp://en.wikipedia.org/wiki/Abelian_extensionhttp://en.wikipedia.org/wiki/Abelian_extensionhttp://en.wikipedia.org/wiki/Abelian_extensionhttp://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber_theoremhttp://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber_theoremhttp://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber_theoremhttp://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber_theoremhttp://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber_theoremhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber_theoremhttp://en.wikipedia.org/wiki/Abelian_extensionhttp://en.wikipedia.org/wiki/Quadratic_fieldhttp://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Unique_factorization_domainhttp://en.wikipedia.org/wiki/Unique_factorization_domainhttp://en.wikipedia.org/wiki/Integral_domainhttp://en.wikipedia.org/wiki/Imaginary_numberhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Ring_(mathematics)http://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Convergent_(continued_fraction)http://en.wikipedia.org/wiki/Lowest_termshttp://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Diophantine_approximationshttp://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory)


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Identities of Ramanujan

The square root of 5 appears in various identitiesofRamanujaninvolvingcontinued fractions.

For example, this case of theRogersRamanujan continued fraction:
http://en.wikipedia.org/wiki/Srinivasa_Ramanujanhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujanhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujanhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fractionhttp://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fractionhttp://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fractionhttp://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fractionhttp://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fractionhttp://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan

square root of 2

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