additional mathematics project work 1/2013

8/22/2019 Additional Mathematics Project Work 1/2013

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PART 1

a) Write a history on logarithm.NAPIERS LOGARITHMS

The invention and origins of Logarithm was credited to John Napier, who was born in 1550 toSir Archibald and his first wife, Janet Bothwell. John Napier was an amateur Calvinist

theologian who predicted that the end of the world would occur in the years between 1688 and

1700, and designed weapons of mass destruction, intended for use against the enemies of hisreligion. But none of these were remembered widely. His lasting reputation is due to his third

hobby: mathematical computation. As a landowner, he never held a job, and his time and energy

could be entirely devoted to intellectual pursuits.

He developed an interest in reducing the labor required by the many tedious computations

that were necessary in astronomical work, involving operations with the very large values of the

trigonometric lengths given in the tables of his times. His interest in these matters might havebeen rekindled in 1590 on the occasion of a visit by Dr. John Craig, the kings physician.

On his return from Denmark, Dr. John Craig paid a visit to Napier to inform him of his

findings while visiting Uraniborg, the astronomical observatory of Tycho Brahe. Here were new,ingenious ways to perform some of those tedious computations required by astronomical

calculations, and foremost was the use ofprosthaphaeresis, which refers to the use of the

trigonometric identity:

The big problem of multiplying many-digit members in astronomical calculations was thus

reduced to the simpler one of addition and subtraction.

But what about quotients, exponentiations, and roots? Napier sought a general method to deal

with these computations, eventually found it, and gave it to the world in his bookMirificiLogarithmorum Canoni descriptio (Description of the admirable table of logarithms) of 1614.

This is a small volume of 147 pages, 90 of which are devoted to mathematical tables containing a

list of numbers, mysteriously called logarithms, whose use would facilitate all kinds of

computations, although no explanation was given in this book about how they were computed.Instead, there is a disclaimer in anAdmonition in Chapter 2 ofThe first Book, which he stated, if

his Tables were well received Napier would be happy to explain how the logarithms were

constructed; otherwise, let them go into oblivion.


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Figure 1: From Knott, Napier tercentenary memorial volume, Plate IX facing page 181.

It happened that the Descriptio was a huge editorial success: a book well received and

frequently used by scientists all over the world. Then an explanatory book became necessary but,although it was probably written before the Descriptio, it was published only posthumously

(Napier died in 1617, one year after Shakespeare and Cervantes) under the title Mirifici

Logarithmorvm Canonis constructio, when his son Robert included it with the 1619 edition oftheDescriptio. For further reading, the Constrvctiogives a glimpse into Napiers mind and his

possible sources of inspiration.

BRIGGS LOGARITHMS

The first mathematics professorship in England, a chair in geometry, was endowed by Sir

Thomas Gresham in 1596 at Gresham College in London, and Henry Briggs (1561-1631) was itsfirst occupant. He was quite impressed by the invention of logarithms. However, Briggs had

become aware of some serious shortcomings in the logarithm scheme published by Napier.


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Both Napier and Briggs had been separately thinking of ways to solve these deficiencies, and

each of them made some proposals to accomplish that. These involved, in either case, the

construction of a new set of logarithms from scratch. But they differed about the key propertieson which the new logarithms should be based.

At the time that Briggs was about to embark on the elaboration of his tables (Napiers health

was failing in his 65th

year, so it was up to Briggs to start a new series of computations) therewere two main methods to calculate logarithms, and both of them had been published in the

Appendix to Napiers Constructio. However, we do not know whether these are due to Napier or

are a product of his collaboration with Briggs. The first method was based on the extraction of


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fifth roots and the second on the extraction of square roots. Napier felt that though this method

[of the fifth root] is considerably more difficult, it is correspondingly more exact. Briggs chose

the method of the square root, and started evaluating (by hand, of course, which may take severalhours) the square root of 10.

Briggs values can be seen on the previous page, reproduced from Chapter 6, page 10, of thesecond edition ofArithmetica logarithmica. Note that he did not use the decimal point, butinserted commas to help with counting spaces. On the logarithm side he omitted many zeros,

which can be confusing. It must be pointed out that Briggs made a mistake in his computation of . These digits are wrong from the twentieth on. This mistake trickles down through hisentire table, but because the wrong digits are so far on the right, the error becomes smaller as it

propagates, and his last two entries are almost in complete agreement with values obtained today

using a computer.

THE LOGARITM ACCORDING TO EULER

The present-day notion of logarithms was made by Leonhard Euler (1707-1783) of Basel,who connected them to the exponential function in the 18th

century. The son of a preacher anddestined to enter the ministry, his ability in mathematics soon convinced his father to let him

switch careers, and he went on to become the most prolific mathematics writer of all time. In

1727, the year of Newtons death, he was invited to join the newly founded Academy of SaintPetersburg, in Russia, and soon became producing first rate research. It was the next year, in a

manuscript on the firing of cannon, that he introduced a soon to become famous number as

follows: Write for the number whose logarithm is unity. E. but he did not give a reason for thischoice of letter. By that time, he had already defined the exponential and logarithmic functions,

but the mathematical community at large had to wait until Euler was ready to publish. In 1741 he

accepted a position at the Academy of Berlin, where he would remain for twenty-five years, and

in 1744 he wrote his enormously influential treatise Introductio in Analysin Infinitorum.Published in Lausanne in 1748, it became the standard work on analysis during the second half

of the eighteenth century.

The success of the Introductio rests on the amount and importance of the mathematical

discoveries that Euler included in it, making it one of the most significant mathematics books of

all times. Its readers might have been bewildered about the fact that there is one logarithm foreach base a, but Euler easily showed that all logarithms ofy are multiples of each other.


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b) Find and explain the applications of logarithm in two different fields of study.Explanation of each application should include the following :

i) The field of study chosen.ii) Examples of problem solving related to the field of study.

Application #1 :: pH Measurement

pH is a measurement of acidity and is surprisingly a common measurement. For example, in

the chemical industry, the acidity of the reagents in many types of reactor has to be controlled toenable optimum reaction conditions. In addition, in the water industry, the acidity of freshwater

for consumption and of effluent for discharge has to be controlled carefully to satisfy legislative

requirements.

pH is an electro-chemical measurement, invariably made by means of the so-called glass

electrode. It is a notoriously difficult measurement to make because of factors such as drift and

fouling. Understanding the significance of measurements requires an appreciation of electro-chemical equilibria. And using pH for control purposes is problematic because of the inherent

non-linearities and time delays.

The formal definition of pH is given as:

Where [ ] denotes concentration of ions in aqueous solution with units of g ions/L. In the case of

hydrogen, whose atomic and ionic weights are the same, [] has units of g/L or . Thelogarithmic scale means that pH increases by one unit for each decrease by a factor of 10 in

[].Pure water dissociates very weakly to produce hydrogen and hydroxyl ions according to

At equilibrium at approximately 25, ther etrat are uh that

The dissociation must produce equal concentrations of

and

ions, so

Since pure water is neutral, by definition, it follows that for neutrality:

This gives rise to the familiar pH scale of 0-14, symmetrical about pH 7, of which 0-7

corresponds to acidic solutions and 7-14 to alkaline solutions.


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To evaluate the pH of alkaline solutions, it is usual to substitute for H+ in the above equations:

Example :: Neutralisation Control

Neutralisation is the process whereby acid and base reagents are mixed to produce a productof specified pH. In the context of waste water and effluent treatment the objective is to adjust the

pH to a value of 7 although, in practice, any value in the range of 6-8 is good enough. In many

chemical reactions the pH has to be controlled at a value other than 7, which could be anywhere

in the range of 0-14. Neutralisation is always carried out in aqueous solutions, pH is ameaningless quantity otherwise. Note that a base that is soluble in water is usually referred to as

an alkali.

pH is without doubt the most difficult of common process variables to control. For example,

the measurement is electrochemical, made with a glass and reference electrode pair as shownbelow, and is prone to contamination, hysteresis and drift. The signal produced, being

logarithmic, is highly non-linear. The process being controlled invariably ahs a wide range ofboth concentration and flow rate. The rangeability of flow gives rise to variable residence times.

To achieve satisfactory control, all of these issues have to be addressed.

Figure 2 : pH Sensing Instrument

Application #2 :: Frequency Response

Frequency response is an important means of analysis and design for control systems in the

frequency domain. In essence, it concerns the behavior of systems that are forced by sinusoidal


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inputs. A system may be forced sinusoidally and, by measuring its output signal is as specified.

Frequency response is of particular importance for design purposes form a stability point of

view.

Consider the first order system G(s) (a.k.a Transfer Function) as shown in Figure 3.

Figure 3: First order system.

If is a sine wave of amplitude A and frequency :

Then the response is given by:

By splitting this into its partial fractions, inverse transforming and some non-trivial trigonometric

manipulation, the solution may be found. Once the exponential transient has decayed away, the

response becomes:

Which is also a sine wave as depicted in Figure 4.

Figure 4: Steady state response of first order system to sinusoidal input.

In Figure 4, the amplitude of is which comprises the amplitude A of the input amplifiedby the steady state gain K and attenuated by the factor

.


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The product of gain and attenuation factor

is known as the amplitude ratio. The frequency of is the same as that of but is shifted in phase by an amount where:

t a Since the oscillatory behavior is associated with roots of the form , this fact may beexploited in determining the frequency response of a system, by substituting and formthe complex conjugate giving:

This may be depicted in Argand diagram form as shown in Figure 5.

It may also be expressed in polar co-ordinates as follows:

where the modulus is the length of the vector, is given by

( ) ( ) and the argument, which is the angle of the vector, is given by:

t a ta


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The derived equation shows the complexity to plot in Cartesian space.

Bode Plots

Because of the complexity, it is convenient to present information about frequency in graphical

form known as Bode Plots. A Bode diagram consist of two graphs; One I s a plot of thelogarithm of the magnitude of a sinusoidal transfer function; the other is a plot of the phaseangle; both are plotted against the frequency on a logarithmic scale, as shown in Figure 6.

The standard representation of the logarithmic magnitude of is given by

where the base of the logarithm is 10. The unit used in this representation of the magnitude is the

decibel, usually abbreviated dB. In the logarithmic representation, the curve are drawn on semi-

log paper, using the log scale for frequency and the linear scale for either magnitude (but in

decibels) or phase angle (in degrees). The frequency range of interest determines the number oflogarithmic cycles required on the abscissa.

Furthermore, a simple method for sketching an approximate log-magnitude curve is available. Itis based on asymptotic approximations. Such approximation by straight-line asymptotes is

sufficient if only rough information on the frequency-response characteristics is need. Should the

exact curve be desired, corrections can be made easily to these basic asymptotic plots. Expandingthe low frequency range by use of a logarithmic scale for the frequency is highly advantageous,

since characteristics at low frequencies are most important in practical systems.

Example :: Electronic Filters


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The concept of filters has been an integral part of the evolution of electrical engineering from

the beginning. Several technological achievements would not have been possible withoutelectrical filters. Because of this prominent role of filters, much effort has been expended on the

theory, design and construction of filters.

A filter is a circuit that I s designed to pass signals with desired frequencies and reject orattenuate others. As a frequency-selective device, a filter can be used to limit the frequency

spectrum of a signal to some specified band of frequencies. Filters are the circuit used in radio

and TV receivers to allow us to select one desired signal out oof a multitude of broadcast signalsin the environment.

A filter is a passive filterif it consists of only passive elements resistors R, inductorL, and

capacitorC. It is said to be an active filterif it consists of active elements (such as transistors andoperational amplifiers) in addition to passive elementsR, L and C. There are four types of filters

whether passive or active.

1. A lowpass filter passes low frequencies and stops high frequencies, as shown inFigure 7a & 7b.

2. A highpass filter passes high frequencies and rejects low frequencies, as shown isFigure 8a & 8b.


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3. A bandpass filter passes frequencies within a frequency band and blocks or attenuatesfrequencies outside the band, as shown in Figure 9a & 9b.

4. A bandstop filter oases frequencies outside a frequency band and blocks or attenuatesfrequencies within the band, as shown in Figure 10a & 10b.


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PART 2

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

= mDn, where m and n are constants.

You can find the value ofm and ofnby conducting the activities below.

a i) Choose six different spheres with diameters between 1 cm to 8 cm.

Measure the diameters of the six spheres using a pair ofvernier callipers.

SPHERE DIAMETER

(CM)

1 1.0

2 2.4

3 3.84 5.2

5 6.6

6 8.0

ii) Find the volume of each sphere without using the formula of volume.

(You can use the apparatus in the science lab to help you)

Archimedes principle states that the upward buoyant force exerted on a body immersed in a

fluid is equal to the weight of the fluid the body displaces. In other words, an immersed object is

buoyed up by a force equal to the weight of the fluid it actually displaces.

Using the method of water displacement and knowing the water density is , the volumeof a solid sphere is equivalent to the weight of the water displaced (in grams), and thus it can bemeasured.

SPHERE VOLUME

()1 0.5236

2 7.2382

3 28.7310

4 73.6223

5 150.53296 268.0832

iii) Tabulate the values of the diameter,D, in cm, and its corresponding volume, V, in cm3.


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SPHERE DIAMETER

(CM)

VOLUME

()1 1.0 0.5236

2 2.4 7.2382

3 3.8 28.7310

4 5.2 73.62235 6.6 150.5329

6 8.0 268.0832

b Find the value of m and ofn using logarithms with any two sets of values

obtained in the table above.

SPHERE DIAMETER

(CM) VOLUME

()1 1.0 0.0000 -0.5236 0.5236

2 2.4 0.3802 0.8596 7.2382

3 3.8 0.5798 1.4584 28.73104 5.2 0.7160 1.8670 73.6223

5 6.6 0.8195 2.1776 150.5329

6 8.0 0.9031 2.4283 268.0832

Given that the volume, V, of a solid sphere is

(1)Taking logarithm on both sides yield

which can be expanded to

(2)

Using logarithms with any two sets of values , and , , we have

(3)

(4)From Eq. (3) and (4), it is obvious that can be eliminated by taking (3)(4)

which can be simplified by the Laws of Logarithms


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Dividing both sides with , we can determine value ofn

(5a)Although it is not necessary to find n in the reduced form of

*+ (5b)Let , and ,, we have

With , the value ofm can be determined by manipulating Eq. (1)


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PART 3

(A) In our daily life, the relation between two variables is not always in a linear form. Forexample, the relation between the volume, V, and the diameter,D, in Part 2 above. Plot Vagainst

D using suitable scales.

(B) When the graph VagainstD is drawn, the value ofm and ofn are not easily determinedfrom the graph. If the non-linear relation is changed to a linear form, a line of best fit can be

drawn and the values of the constants and other information can be obtained easily.

a) Reduce the equation V = mDn to a linear form.

in which the reduced equation is a linear form of ,where ;M = n ; ; and .

b) Using the data from Part 2, plot the graph and draw the line of best fit.


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c) From the graph, findi) the value ofm and ofn, thus express Vin terms ofD,

To obtain the value ofm we must know the y-intercept, -0.281, because

The value ofn can be determined by calculating the slope of the graph, that is

Thus volume, V, of a solid sphere can be expressed in terms ofD as (6)

ii) the volume of the sphere when the diameter is 5 cm, and

With this value 0.699 on the x-axis, we can look up on the linear graph and interpolate the

corresponding value 1.816 on they-axis.


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iii)

the radius of the sphere when the volume is 180 cm

3

. With this value 2.25527 on the y-axis, we can look up on the linear graph and interpolate thecorresponding value 0.845 on thex-axis.

FURTHER EXPLORATION

a) Compare the equation obtained in Part 3 (B) c (i) with the formula of volumeof sphere. Hence, find the value of .

The formula for volume of sphere is given by

Comparing the formula and Eq. (6), we have

()

b) Suggest another method to find the value of .


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One simple method is to determine the circumference C, of a circle with diameterD. The

circumference of a circle is the length around it and the associate formula is given by To measure the circumference C, of a circle with diameterD effectively, the shadow of a solid

sphere can be projected on a screen using a bright light source as shown below. Then, thediameter of the casted shadow can be scaled linearly according to the actual diameter of the solid

sphere, because the formula shows the linear relationship.

REFLECTION

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

practise? Express your feelings and opinions creatively through the usage ofsymbols, drawings, lyrics of a song or a poem.


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INTRODUCTION

One of the mathematical concepts which we must be familiar with is logarithms. Before the

days of scientific calculators, logarithms were used to multiply or divide extreme numbers using

mathematical tables. For these calculations, ten was the most common base to use. Logarithm tothe base of ten is also called the common logarithm. Other bases such as two, five and eight canalso be used. The ancient Babylonians had used bases up to 60.

Logarithms have many applications in various fields of studies. In the early 17th

century itwas rapidly adopted by navigators, scientists, engineers and astronomers to perform

computations more easily.


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REFERENCES

[1] J. Napier, Mirifici logarithmorum canonis descriptio, Edinburgh (1614).

[2] J. Napier, H. Briggs, Mirifici logarithmorum canonis constructi, London (1619).[3] H. Briggs, Arithmetica logarithmica, London (1624).

[4] L. Euler, Introductio in analysin infinitorum, (1748).[5] John Blanton, English translation toIntroduction to Analysis of the Infinite, Springer-Verlag (1988).

[6] E.A Gonzlez-Velasco, Logarithms, in Journey through Mathematics, pp 78-147, Springer

Science+Business Media (2011).

additional mathematics project work 1/2013

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