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  • Bessel's equation | planetmath.org
    k p subscript r 1 r 2 k neq p r 1 Therefore r 1 r 2 2 p k subscript r 1 subscript r 2 2 p k r 1 r 2 2p neq k where k k k is a positive integer Thus when p p p is not an integer and not an integer added by 1 2 1 2 frac 1 2 we get the second particular solution gotten of 5 by replacing p p p by p p p y 2 a 0 x p 1 x 2 2 2 p 2 x 4 2 4 2 p 2 2 p 4 x 6 2 4 6 2 p 2 2 p 4 2 p 6 fragments subscript y 2 assign subscript a 0 superscript x p fragments normal 1 superscript x 2 2 2 p 2 superscript x 4 normal 2 4 2 p 2 2 p 4 superscript x 6 normal 2 4 6 2 p 2 2 p 4 2 p 6 normal normal displaystyle y 2 a 0 x p left 1 frac x 2 2 2p 2 frac x 4 2 cdot 4 2p 2 2p 4 frac x 6 2 cdot 4 cdot 6 2p 2 2p 4 2p 6 ldots right 6 The power series of 5 and 6 converge for all values of x x x and are linearly independent the ratio y 1 y 2 subscript y 1 subscript y 2 y 1 y 2 tends to 0 as x normal x x to infty With the appointed value a 0 1 2 p Γ p 1 subscript a 0 1 superscript 2 p normal Γ p 1 a 0 frac 1 2 p Gamma p 1 the solution y 1 subscript y 1 y 1 is called the Bessel function of the first kind and of order p p p and denoted by J p subscript J p J p The similar definition is set for the first kind Bessel function of an arbitrary order p ℝ p ℝ p in mathbb R and ℂ ℂ mathbb C For p ℤ p ℤ p notin mathbb Z the general solution of the Bessel s differential equation is thus y C 1 J p x C 2 J p x assign y subscript C 1 subscript J p x subscript C 2 subscript J p x y C 1 J p x C 2 J p x where J p x y 2 subscript J p x subscript y 2 J p x y 2 with a 0 1 2 p Γ p 1 subscript a 0 1 superscript 2 p normal Γ p 1 a 0 frac 1 2 p Gamma p 1 The explicit expressions for J p subscript J plus or minus p J pm p are J p x m 0 1 m m Γ m p 1 x 2 2 m p subscript J plus or minus p x superscript subscript m 0 superscript 1 m m normal Γ plus or minus m p 1 superscript x 2 plus or minus 2 m p displaystyle J pm p x sum m 0 infty frac 1 m m Gamma m pm p 1 left frac x 2 right 2m pm p 7 which are obtained from 5 and 6 by using the last formula for gamma function E g when p 1 2 p 1 2 p frac 1 2 the series in 5 gets the form y 1 x 1 2 2 Γ 3 2 1 x 2 2 3 x 4 2 4 3 5 x 6 2 4 6 3 5 7 2 π x x x 3 3 x 5 5 fragments subscript y 1 superscript x 1 2 2 normal Γ 3 2 fragments normal 1 superscript x 2 normal 2 3 superscript x 4 normal 2 4 3 5 superscript x 6 normal 2 4 6 3 5 7 normal normal 2 π x fragments normal x superscript x 3 3 superscript x 5 5 normal normal normal y 1 frac x frac 1 2 sqrt 2 Gamma frac 3 2 left 1 frac x 2 2 cdot 3 frac x 4 2 cdot 4 cdot 3 cdot 5 frac x 6 2 cdot 4 cdot 6 cdot 3 cdot 5 cdot 7 ldots right sqrt frac 2 pi x left x frac x 3 3 frac x 5 5 ldots right Thus we get J 1 2 x 2 π x sin x subscript J 1 2 x 2 π x x J frac 1 2 x sqrt frac 2 pi x sin x analogically 6 yields J 1 2 x 2 π x cos x subscript J 1 2 x 2 π x x J frac 1 2 x sqrt frac 2 pi x cos x and the general solution of the equation 1 for p 1 2 p 1 2 p frac 1 2 is y C 1 J 1 2 x C 2 J 1 2 x assign y subscript C 1 subscript J 1 2 x subscript C 2 subscript J 1 2 x y C 1 J frac 1 2 x C 2 J frac 1 2 x In the case that p p p is a non negative integer n n n the case of 7 gives the solution J n x m 0 1 m m m n x 2 2 m n subscript J n x superscript subscript m 0 superscript 1 m m m n superscript x 2 2 m n J n x sum m 0 infty frac 1 m m m n left frac x 2 right 2m n but for p n p n p n the expression of J n x subscript J n x J n x is 1 n J n x superscript 1 n subscript J n x 1 n J n x i e linearly dependent on J

    Original URL path: http://www.planetmath.org/besselsequation (2016-04-25)
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  • bijection between closed and open interval | planetmath.org
    2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A33C10 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A33C10 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit bijection between closed and open interval For mapping the end points of the closed unit interval 0 1 0 1 0 1 and its inner points bijectively onto the corresponding open unit interval 0 1 0 1 0 1 one has to discern suitable denumerable subsets in both sets 0 1 0 1 1 2 1 3 1 4 S 0 1 0 1 1 2 1 3 1 4 normal S displaystyle 0 1 0 1 1 2 1 3 1 4 ldots cup S 0 1 1 2 1 3 1 4 S 0 1 1 2 1 3 1 4 normal S displaystyle 0 1 1 2 1 3 1 4 ldots cup S where S 0 1 0 1 1 2 1 3 1 4 assign S 0 1 0 1 1 2 1 3 1 4 normal S 0 1 smallsetminus 0 1 1 2 1 3 1 4 ldots Then the mapping f f f from 0 1 0 1 0 1 to 0 1 0 1 0 1 defined by f x 1 2 for x 0 1 n 2 for x 1 n n 1 2 3 x for x S assign f x 12forx01 n 2

    Original URL path: http://www.planetmath.org/bijectionbetweenclosedandopeninterval (2016-04-25)
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  • bijection between unit interval and unit square | planetmath.org
    2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A30 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A30 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit The real numbers in the open unit interval I 0 1 I 0 1 I 0 1 can be uniquely represented by their decimal expansions when these must not end in an infinite string of 9 s Correspondingly the elements of the open unit square I I I I I times I are represented by the pairs of such decimal expansions Let P 0 x 1 x 2 x 3 0 y 1 y 2 y 3 assign P 0 subscript x 1 subscript x 2 subscript x 3 normal 0 subscript y 1 subscript y 2 subscript y 3 normal P 0 x 1 x 2 x 3 ldots 0 y 1 y 2 y 3 ldots be such a pair representing an arbitrary point in I I I I I times I and let p 0 x 1 y 1 x 2 y 2 x 3 y 3 assign p 0 subscript x 1 subscript y 1 subscript x 2 subscript y 2 subscript x 3 subscript y 3 normal p 0 x 1 y 1 x 2 y 2 x 3 y 3 ldots Then it s apparent that P p maps to P p displaystyle P mapsto p 1 is an injective mapping from I I I I I times I to I I I Thus I I I I I I I times I leq I But since I I I I I times I contains more than one horizontal open segment equally long as I I I and

    Original URL path: http://www.planetmath.org/bijectionbetweenunitintervalandunitsquare (2016-04-25)
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  • binomial equation | planetmath.org
    0 superscript x n 1 0 x n 1 0 This name comes from the fact that the roots of the equation divide the unit circle in the complex plane into n n n equally long arcs Greek ϰ υ ϰ λ o ς ϰ normal υ ϰ λ o ς varkappa acute upsilon varkappa lambda o varsigma circle τ o μ o ς τ normal o μ o ς tau acute o mu o varsigma part Defines cyclotomic equation Related Binomial CalculatingTheNthRootsOfAComplexNumber RootOfUnity Type of Math Object Definition Major Section Reference Parent algebraic equation Groups audience Buddy List of pahio Mathematics Subject Classification 11C08 no label found 12E05 no label found Add a correction Attach a problem Ask a question Comments Binomial equation name Permalink Submitted by rm50 on Wed 01 30 2008 03 13 Jussi Do you have a reference for this term I ve never seen it used at least not for this Roger Log in to post comments Re Binomial equation name Permalink Submitted by pahio on Wed 01 30 2008 15 55 Dear Roger I know that the term binomial equation is rare in the English language at least in America but since it is quite good term I wanted to tell it in PM Unfortenately I have no algebra books in English Here some Engl references in internet http thesaurus maths org mmkb entry html action entryById id 3536 http www answers com topic binomial equation cat technology http en wikipedia org wiki Joseph Louis Lagrange In Europe the equivalents of the term are more used e g in German binomische Gleichung ref http www user tu chemnitz de syha lehre baI baI pdf in Swedish binomisk ekvation ref www matematik lu se matematiklth personal sigrid analys1 lektion19 ps Jussi Log in to post comments Re Binomial equation name Permalink Submitted by rspuzio on Wed 01 30 2008 16 34 Harry Hochstadt in his book on special functions uses the same sort of terminology when referring to Mellin s work on integral representations and differential equations for certain algebraic functions such as the general solution of the quintic Again the Finnish connection To be sure there the term used is trinomial equation but that is because the equation considered there has three terms as opposed to two While this might sound unusual to Americans here the term binomial is usually appears in the context of binomial expansions and binomial coefficients the usage is quite logical Just as the words monomial and polynomial refer to algebraic expressions with one or many terms respectively so too the terms binomial trinomial are used to refer to expressions with two and three terms respectively and the terms binomial equation and trinomial equation refer to equations gotten by setting such expressions to zero By the way Jussi could you add the form of Mellin s inversion formula for his transform to your entry on that topic I would like to attach an entry explaining Mellin s ingenious use of

    Original URL path: http://www.planetmath.org/binomialequation (2016-04-25)
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  • biquadratic equation | planetmath.org
    2Fpm 2Frdf sink 23this 3E 7B msc 3A12E05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A12E05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit biquadratic equation A biquadratic equation in a narrower sense is the special case of the quartic equation containing no odd degree terms a x 4 b x 2 c 0 a superscript x 4 b superscript x 2 c 0 displaystyle ax 4 bx 2 c 0 1 Here a a a b b b c c c are known real or complex numbers and a 0 a 0 a neq 0 For solving a biquadratic equation 1 one does not need the quartic formula since the equation may be thought a quadratic equation with respect to x 2 superscript x 2 x 2 i e a x 2 2 b x 2 c 0 a superscript superscript x 2 2 b superscript x 2 c 0 a x 2 2 bx 2 c 0 whence x 2 b b 2 4 a c 2 a superscript x 2 plus or minus b superscript b 2 4 a c 2 a x 2 frac b pm sqrt b 2 4ac 2a see quadratic formula or quadratic equation in ℂ ℂ mathbb C Taking square roots of the values of x 2 superscript x 2 x 2 see taking square root algebraically one obtains the four roots of 1 Example Solve the biquadratic equation x 4 x 2 20 0 superscript x 4 superscript x 2 20 0 displaystyle x 4 x 2 20 0 2 We have x 2 1 1 2 4 1 20 2 1 1 9 2 superscript x 2 plus or minus 1 superscript 1 2 normal 4 1 20 normal 2 1 plus or minus 1 9 2 displaystyle x 2 frac 1 pm sqrt 1 2 4 cdot 1 cdot 20 2 cdot 1 frac 1 pm 9 2 3 i e x 2 4 superscript x 2 4

    Original URL path: http://www.planetmath.org/biquadraticequation (2016-04-25)
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  • Bohr's theorem | planetmath.org
    2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A12D99 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Theorem Bohr 1914 If the power series n 0 a n z n superscript subscript n 0 subscript a n superscript z n displaystyle sum n 0 infty a n z n satisfies n 0 a n z n 1 superscript subscript n 0 subscript a n superscript z n 1 displaystyle left sum n 0 infty a n z n right 1 1 in the unit disk z 1 z 1 z 1 then 1 and even the inequality n 0 a n z n 1 superscript subscript n 0 subscript a n superscript z n 1 displaystyle sum n 0 infty a n z n 1 2 is true in the disk z 1 3 z 1 3 z frac 1 3 Here the radius 1 3 1 3 frac 1 3 is the best possible Proof One needs Carathéodory s inequality which says that if the real part of a holomorphic function g z n 0 b n z n assign g z superscript subscript n 0 subscript b n superscript z n g z sum n 0 infty b n z n is positive in the unit disk then b n 2 Re b 0 for n 1 2 formulae sequence subscript b n 2 Re subscript b 0 for n 1 2 normal b n leqq 2 mbox Re b 0 quad mbox for n 1 2 ldots Choosing now g z 1 e i φ f z assign g z 1 superscript e i φ f z g z 1 e i varphi f z where φ φ varphi is any real number and f z f z f z the sum function of the series in the theorem we get a n 2 Re 1 e i φ a 0 2 1 a 0 cos φ subscript a n 2 Re 1 superscript e i φ subscript a 0 2 1 subscript a 0 φ a n leqq 2 mbox Re 1 e i varphi a 0 2 1 a 0 cos varphi and especially a n 2 1 a 0 for n 1 2 formulae sequence subscript a n 2 1 subscript a 0 for n 1 2 normal a n leqq 2 1 a 0 quad mbox for n 1 2 ldots If f z a 0 not equivalent to f z subscript a 0 f

    Original URL path: http://www.planetmath.org/bohrstheorem (2016-04-25)
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  • Bolzano's theorem | planetmath.org
    2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A40A30 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A30B10 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A30B10 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home

    Original URL path: http://www.planetmath.org/bolzanostheorem (2016-04-25)
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  • Bombelli's method of computing square roots | planetmath.org
    to the left then the value of the square root changes only such that its decimal point moves one step in the same direction If the integer part of a number has one or two digits the integer part of its square root has evidently one digit Accordingly one may infer the following rule If the integer part of the radicand is cut starting from the decimal point into pieces of two digits when the leftmost piece may consist of only one digit then the number of the pieces expresses the number of digits in the integer part of the square root We now illustrate the computing of square root by using 2238 9 2238 9 sqrt 2238 9 as an example and denote its first digits by x y z x y z normal x y z ldots The integer part of 22 389 22 389 sqrt 22 389 has one digit which is x x x This is the greatest one digit integer whose square is at most 22 Hence x 4 x 4 x 4 By the above rule the integer part of 2238 9 2238 9 sqrt 2238 9 has two digits and thus equals to 10 x y 40 y 10 x y 40 y 10x y 40 y The number y y y is the greatest of the one digit integers such that the square 10 x y 2 100 x 2 2 10 x y y 2 100 x 2 y 10 2 x y superscript 10 x y 2 100 superscript x 2 normal 2 10 x y superscript y 2 100 superscript x 2 y normal 10 2 x y 10x y 2 100x 2 2 cdot 10xy y 2 100x 2 y 1 0 cdot 2x y does not exceed 2238 i e such that the product y 10 2 x y y 80 y y normal 10 2 x y y 80 y y 10 cdot 2x y y 80 y is at most the remainder 2238 100 x 2 638 2238 100 superscript x 2 638 2238 100x 2 638 We see that y 7 y 7 y 7 We can continue similarly and determine next the digit z z z The calculations may be organised right from the start as follows absent sqrt 22 22 22 38 38 38 90 90 90 4 4 4 7 7 7 3 3 3 1 1 1 7 7 7 normal ldots 16 16 16 4 4 4 6 6 6 38 38 38 8 8 8 7 7 7 6 6 6 09 09 09 7 7 7 29 29 29 90 90 90 9 9 9 4 4 4 3 3 3 28 28 28 29 29 29 3 3 3 1 1 1 61 61 61 00 00 00 9 9 9 4 4 4 6 6 6 1 1 1 94 94 94 61 61 61 1 1 1 66 66 66 39 39 39 00

    Original URL path: http://www.planetmath.org/bombellismethodofcomputingsquareroots (2016-04-25)
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