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  • 7/27/2019 Add Math Selangor 4

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    pg. 1

    PART 1

    a) Write a history on logarithm.

    History of Logarithms

    From Napier to Euler

    The method of logarithms was publicly

    propounded byJohn Napierin 1614, in a

    book titled Mirifici Logarithmorum Canonis

    Descriptio(Description of the Wonderful

    Rule of Logarithms).Joost

    Brgiindependently invented logarithms

    but published six years after Napier.

    Johannes Kepler, who used logarithm

    tables extensively to compile

    his Ephemeris and therefore dedicated it

    to Napier, remarked:

    ...the accent in calculation led Justus

    Byrgius [Joost Brgi] on the way to these

    very logarithms many years before

    Napier's system appeared; but ...instead of rearing up his child for the public benefit he

    deserted it in the birth.

    Johannes Kepler, Rudolphine Tables (1627)

    By repeated subtractions Napier calculated (1 107)L forL ranging from 1 to 100. The

    result forL=100 is approximately0.99999 = 1 105. Napier then calculated the

    products of these numbers with 107(1 105)L forL from 1 to 50, and did similarly

    with0.9998 (1 105)20 and 0.9 0.99520. These computations, which occupied 20

    years, allowed him to give, for any numberNfrom 5 to 10 million, the numberL that

    solves the equation

    Napier first called L an "artificial number", but later introduced the word "logarithm"to

    mean a number that indicates a ratio: (logos) meaning proportion,

    and (arithmos) meaning number. In modern notation, the relation to natural

    logarithms is

    T

    http://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/John_Napier
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    where the very close approximation corresponds to the observation that

    The invention was quickly and widely met with acclaim. The works

    ofBonaventura Cavalieri(Italy),Edmund Wingate(France), Xue Fengzuo(China), andJohannes Kepler's Chilias logarithmorum (Germany) helped spread

    the concept further.

    In 1647Grgoire de Saint-Vincentrelated logarithms to the quadrature of the

    hyperbola, by pointing out that the area f(t) under the hyperbola fromx=

    1 tox= tsatisfies

    The natural logarithm was first described byNicholas Mercatorin his

    work Logarithmotechnia published in 1668, although the mathematics teacher JohnSpeidell had already in 1619 compiled a table on the natural logarithm. Around

    1730,Leonhard Eulerdefined the exponential function and the natural logarithm by

    Euler also showed that the two functions are inverse to one another.

    http://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Bonaventura_Cavalieri
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    pg. 4

    maximum of the likelihood function occurs at the same parameter-value as a maximum of the

    logarithm of the likelihood (the "log likelihood"), because the logarithm is an increasing

    function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods

    forindependentrandom variables.

    Benford's lawdescribes the occurrence of digits in manydata sets, such as heights of buildings.According to Benford's law, the probability that the first decimal-digit of an item in the data

    sample is d(from 1 to 9) equals log10(d+ 1) log10(d), regardless of the unit of measurement.

    Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc.

    Auditors examine deviations from Benford's law to detect fraudulent accounting.

    2. Fractals

    The Sierpinski triangle (at the right) is constructed by repeatedly replacingequilateral

    trianglesby three smaller ones.

    Logarithms occur in definitions of thedimensionoffractals. Fractals are geometric objects that

    areself-similar: small parts reproduce, at least roughly, the entire global structure. TheSierpinski

    triangle(pictured) can be covered by three copies of itself, each having sides half the original

    length. This makes theHausdorff dimensionof this structure log(3)/log(2) 1.58. Another

    logarithm-based notion of dimension is obtained bycounting the number of boxesneeded to

    cover the fractal in question.

    http://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Benford%27s_lawhttp://en.wikipedia.org/wiki/Benford%27s_lawhttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/File:Sierpinski_dimension.svghttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Benford%27s_lawhttp://en.wikipedia.org/wiki/Independence_(probability)
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    pg. 5

    PART 2

    The volume, V, in cm3, of a solid sphere and its diameter, D, in cm, are related by the

    equation , where m and n are constants.

    Find the value of m and n by conducting the activities below.

    I. Choose 6 different spheres with diameters between 1cm to 8cm. The diameterof the 6 spheres using a pair of vernier calipers.

    II. Find the volume of each sphere using water displacement method.III. Tabulate the values of diameter, D, in cm and its corresponding volume, V, cm3.

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    pg. 6

    Find the volume of sphere using water displacement method.

    A method of finding the volume of a sphere with minimal calculations is to use the WaterDisplacement Method:

    1. Fill a beaker or graduated cylinder with enough water to completely immerse thesphere in.

    2. Record the baseline initial measurement3. Drop the sphere in4. Record final measurement5. Subtract the initial volume from the final volume ~ this is the volume of the

    sphere!

    Value of diameter,D and Volume

    Diameter,D ( Volume, V (

    D1 = 1.0 V1= 0.5

    D2 =2.8 V2= 11.5

    D3 =4.0 V3= 34.0

    D4 =5.2 V4= 74.0

    D5 =6.6 V5= 151.0

    D6 =7.8 V6= 250.0

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    pg. 7

    Diameter,D ( Volume, V (

    D1 = 1.0 V1= 0.5

    D2 =2.8 V2= 11.5

    D3 =4.0 V3= 34.0

    D4 =5.2 V4= 74.0D5 =6.6 V5= 151.0

    D6 =7.8 V6= 250.0

    We can solve by simultaneous method

    Substitute the values in the equation

    We obtain,

    ----------(1)