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ADDITONAL

MATHEMATICS

PROJECTWORK 2013

NAME : FATIN NUR SHAFIQAH BINTI MOHD FAIZALCLASS : 5 LAMBDA

SCHOOL : SMK BANDAR TUN HUESSEIN ONN 2


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6/12/2013[ADDITIONAL MATHEMATICS PROJECT PAPERLOGARITHMS 2013] June 1, 2013

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PART 1

The History

of

Logarithms


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PART 1

The History of Logarithms

Predecessors

The Babylonians sometime in 20001600 BC may have invented the quartersquare multiplication algorithm to multiply two numbers using only addition,subtraction and a table of squares. However it could not be used for divisionwithout an additional table of reciprocals. Large tables of quarter squares wereused to simplify the accurate multiplication of large numbers from 1817 onwardsuntil this was superseded by the use of computers.

Michael Stifel publishedArithmetica Integra in Nuremberg in 1544, which containsa table of integers and powers of 2 that has been considered an early version of alogarithmic table.

In the 16th and early 17th centuries an algorithm called prosthaphaeresis was usedto approximate multiplication and division. This used the trigonometric identity

or similar to convert the multiplications to additions and table lookups. Howeverlogarithms are more straightforward and require less work. It can be shown using

complex numbers that this is basically the same technique.


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From Napier to Euler

The method of logarithms was publicly propounded by John Napier in 1614, in abook titled Mirifici Logarithmorum Canonis Descriptio (Description of the WonderfulRule of Logarithms). Joost Brgi independently 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:

Johannes Kepler, Rudolphine Tables (1627)

By repeated subtractions Napier calculated (1 107)L forL ranging from 1 to 100.The result forL=100 is approximately 0.99999 = 1 105. Napier then calculated theproducts of these numbers with 107(1 105)L forL from 1 to 50, and did similarly

with 0.9998 (1 105)20 and . These computations, which occupied 20years, allowed him to give, for any numberN from 5 to 10 million numberL thatsolves the equation


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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, therelation to natural logarithms is:

where the very close approximation corresponds to the observation that

The invention was quickly and widely met with acclaim. The works of BonaventuraCavalieri (Italy), Edmund Wingate (France), Xue Fengzuo (China), and Johannes

Kepler's Chilias logarithmorum (Germany) helped spread the concept further.


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TheApplications

of

Logarithms


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The Applications of Logarithms

1. Psychology

Logarithms occur in several laws describing human perception:

Hick's law proposes a logarithmic relation between the time individuals take forchoosing an alternative and the number of choices they have.

Fitts's law predicts that the time required to rapidly move to a target area is alogarithmic function of the distance to and the size of the target.

In psychophysics, the WeberFechner law proposes a logarithmic relationshipbetween stimulusand sensationsuch as the actual vs. the perceived weight of an

item a person is carrying. (This "law", however, is less precise than more recentmodels, such as the Stevens' power law.)

Psychological studies found that individuals with little mathematics educationtendto estimate quantities logarithmically, that is they position a number on anunmarked line according to its logarithm, so that 10 is positioned as close to 100 as100 is to 1000. Increasing education shifts this to a linear estimate (positioning1000 10x as far away) in some circumstances, while logarithms are used when the

numbers to be plotted are difficult to plot linearly.

2. Music

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3.

4.

5.


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PART 2


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PART 2

As we were been told, my group and I had chosen 6 different spheres which we had made

by using plasticine. The diameters of the six spheres are ranging between 1cm to 8cm and

the measurements of the diameter are made using vernier calipers in the physics lab. Below

are the cross-sectional areas of all of the six spheres together with their diameters.

diagram 1 : cross sectional areas and diameters

S here 1 : 1.48 S here 2 : 2.12 S here 3: 2.66

S here 4 : 3.68S here 5 : 4.44

S here 6 : 5.88


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The six small spheres then are put into a Eureka can that is full with water in order to measure their

volume by using water displacement method. The value of volume obtained is tabulated together

with the diameter.

Table 1: The diameter of the spheres with the volumeThe volume, Vin cm3, of a solid sphere and its diameter, D, in cm are related by the equationV=mDn, where m and n are constants.

With 2 sets of values that we chose from the findings, the value of m and n are calculated asfollows :

1 (radians/real number)

Input

Solution

Table 2: The mathematical solution for m and n

From the solution, we know that:

a) The value of m isb) The value of n is

Sphere Diameter , D (cm ) Volume, V (cm3)

1 1.48 2.2

2 2.12 5.4

3 2.66 12.1

4 3.68 20.8

5 4.44 55.2

6 5.88 106.9


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Part 3


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PART 3

The relation between the volume, V, and the diameter, D, in Part 2, is sketched as follows:

Diagram 2: Sketched graph for relation of V=mDn

The following graph is drawn using graph paper by using a scale of 1 cm to 5 units on the y-axis and 1 cm to 0.5 units on the x-axis


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Diagram 3: Drawn graph for relation of V=mDn


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From the both sketched and drawn graphs, we know that the value of m and n are not easilyobtained from the graphs due to the non-linear relation. We would reduce the equationV=mDn to a linear form by adding log10 to the both side of the equation so that it could be

easier for us to draw a line of best fit and determine the values of m and n.

Now, we could plot a graph with the line of best fit using the new equation that is derivedfrom the equation V=mDn which is log10 V= n log10 D + log10 m

Table 3: Values of log10 D with log10 V

The graph is sketched using Microsoft Mathematics (2010):

Diagram 4: Sketched graph for relation oflog10 V= n log10 D + log10 m

Log10 D Log10 V


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A graph is drawn on the graph paper using a scale of 2 cm to 0.1 unit on the x-axis and 4 cmto 0.5 unit on the y-axis:

Diagram 5 : drawn graph for relation of log10 V= n log10 D + log10 m


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From the drawn graph, we could simply find the value of m by taking the value of y-

intercept, c in the graph,

To know the value of n, we just calculate the gradient of the straight line,

Now, we can express Vin terms of D,

log10 V = n log10 D + log10 m

y = mx + c

log10 m = y-intercept, c

log10 m =

n = gradient of straight line

n =

n =


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FURTHER EXPLORATION

By comparing the equation that we have got with the formula of volume of sphere,

, we could find the value of pi ( )

Let pi ( ) be the subject of this equation:

There is another ways that are found in order to discover the value of pi. We would like to

show one of the methods, which is by comparing the area of circle (using )

with the formula

It is known that the area of circle can be proved by following diagrams:

Diagram 6: Determining what is the area of circle

A=?


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Below is tabulation of data regarding the radii and circumferences of the cross-sectional

area of six spheres which we have measured using 20- cm string together with 1-metre ruler.

SPHERE RADIAUS (cm) CIRCUMFERENCE ( cm )

1

2

3

4

5

6

Table 4: Values of radii and the circumferences

We chose Sphere 2 as the data value of radius and circumference.


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REFLECTION

What have you learnt while conducting the project? What moral values did you practice?

Represent your feelings or opinions creatively through the usage of symbols, drawings or

lyrics in a song or a poem.

It teach me how to love logarithms Experiment conducting Sharing Hard work


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