Mathematics and its importance
Mathematics is a fundamental part of human thought and logic,
and integral to attempts at understanding the world and ourselves. Mathematics
provides an effective way of building mental discipline and encourages logical
reasoning and mental rigor. In addition, mathematical knowledge plays a crucial
role in understanding the contents of other school subjects such as science,
social studies, and even music and art.
Firstly, we ask the question: why does mathematics hold such an
important and unique place among other subjects? That is, what is the
significance of mathematics in the overall school curriculum? As a point of
departure we offer a few thoughts on why mathematics should be treated as an
important subject in overall curriculum.
 Mathematics has a transversal nature. If we reflect on the
history of curriculum in general, then mathematics (geometry and algebra) were
two of the seven liberal arts in Greek as well as in medieval times. This
historical role supports the notion that mathematics has provided the mental
discipline required for other disciplines.
 Mathematical literacy is a crucial attribute of individuals
living more effective lives as constructive, concerned and reflective citizens.
Mathematical literacy is taken to include basic computational skills,
quantitative reasoning, spatial ability etc.
 Mathematics is applied in various fields and disciplines,
i.e., mathematical concepts and procedures are used to solve problems in
science, engineering, economics. (For example, the understanding of complex
numbers is a prerequisite to learn many concepts in electronics.) The
complexity of those problems often requires relatively sophisticated
mathematical concepts and procedures when compared to the mathematical literacy
aforementioned.
Mathematics
and architecture
Mathematics and architecture have
always been close, not only because architecture depends on developments in mathematics,
but also their shared search for order and beauty, the former in nature and the
latter in construction. Mathematics is indispensable to the understanding of
structural concepts and calculations. It is also employed as visual ordering
element or as a means to achieve harmony with the universe. Here geometry becomes
the guiding principle.
Golden rectangle
In Greek architecture,
the Golden
mean, (also known as the Golden rectangle, Golden Section, and Golden
Ratio) served as a canon for planning. Knowledge of the golden mean goes back
at least as far as 300BC, when Euclid
described the method of geometric construction in Book 6, Proposition 30 of his
book the Elements. It corresponds to a proportion of 1: 1.618,
considered in Western
architectural theory to be very pleasing. This number is also known as Phi. Jay Hambidge
believed that the golden mean was the ratio used by Attic Greek architects in
the design of the Parthenon
and many other ancient Greek buildings, as well as sculptures, paintings, and
vases.
In Islamic architecture, a proportion of 1: √2 was often used—the
plan would be a square and the elevation would be obtained by projecting from
the diagonal of the plan. The dimensions of the various horizontal components
of the elevation such as mouldings and cornices too were obtained from the
diagonals of the various projections and recesses in plan.
Ancient architecture such as that of the Egyptians and Indians employed planning principles and proportions that rooted the buildings to the cosmos, considering the movements of sun, stars, and other heavenly bodies. Vaastu Shastra, the ancient Indian canons of architecture and town planning employs mathematical drawings called mandalas. Extremely complex calculations are used to arrive at the dimensions of a building and its components. Some of these calculations form part of astrology and astronomy whereas others are based on considerations of aesthetics such as rhythm.
Ancient architecture such as that of the Egyptians and Indians employed planning principles and proportions that rooted the buildings to the cosmos, considering the movements of sun, stars, and other heavenly bodies. Vaastu Shastra, the ancient Indian canons of architecture and town planning employs mathematical drawings called mandalas. Extremely complex calculations are used to arrive at the dimensions of a building and its components. Some of these calculations form part of astrology and astronomy whereas others are based on considerations of aesthetics such as rhythm.
Renaissance
architecture used symmetry
as a guiding principle. The works of Andrea Palladio
serve as good examples. Later High Renaissance or
Baroque used curved
and dramatically twisted shapes in as varied contexts such as rooms, columns,
staircases and squares. St. Peter's Square in Rome, fronting the St. Peter's Basilica, is an approximately keyholeshaped
(albeit with nonparallel sides) exterior space bounded by columns giving a
very dynamic visual experience.
The term Cartesian planning given to the planning of
cities using a grid
plan, shows the close association between architecture and geometry.
Ancient Greek cities such as Olynthus
had such a pattern superimposed on rugged terrain giving rise to dramatic
visual qualities, though proving difficult to negotiate heights. Modern town
planning used the grid pattern extensively, and according to some, resulting in
monotony and traffic problems.
The Role of Mathematics in Physics
It took a long time in the history
of humankind before it occurred to anyone that mathematics is useful  even
vital  in the understanding of nature. Western thought was dominated from
antiquity to the Renaissance, turn by turn, by Plato and Aristotle. Plato
taught that reality consists of idealized "forms", and our world was
a flawed, inadequate shadow of reality  hardly worthy of passing notice, let
alone study. Aristotle thought that the intricacies of nature could never be
described by the abstract simplicity of mathematics.^{1} Galileo
recognized and used the power of mathematics in his study of nature, and with
his discovery modern science was born.
A role that mathematics plays in
physics not mentioned in the text is that mathematics is a really great way to
get a very concise statement that would take a lot of words in English. For
example, Newton's Second Law
can be stated as follows:
The magnitude of the
acceleration of an object is directly proportional to the net force applied to
the object, and inversely proportional to the object's mass. The direction of
the acceleration is the same as the direction of the net force.
Exactly what all of this means is
not important (at the moment)  what is important is that the statement above
can be expressed mathematically as:
The point is that to a physicist,
both statements say exactly the same thing. The symbolism of mathematics
can replace a lot of words with just a few symbols.
Many beginning physicists get the
notion that equations in physics are just something to "plug the numbers
into and get the answer"  which is one reason that numerical calculation
is not emphasized in this physics course. Physicists think differently 
equations tell them how concepts are linked together.
For instance, this equation arises in the
study of kinematics:
The symbol on the left side of the
equation represents the concept "average velocity". Since there are
two symbols (forgetting the division sign, and the counts as one
symbol) on the right side, to a physicist, the equation says (among other
things) that the average velocity of an object depends on two (and only two)
other concepts  the object's displacement (), and the time it
has been moving (t). Thus equations tell scientists how concepts are related to
one another.
Once an idea is expressed in
mathematical form, you can use the rules (axioms, theorems, etc.) of
mathematics to change it into other statements. If the original statement is
correct, and you follow the rules faithfully, your final statement will also be
correct. This is what you do when you "solve" a mathematics problem.
From a scientific point of view,
however, if you start with one statement about nature, and end up with another
statement about nature, what you have been doing is thinking about nature.
Mathematics mechanizes thinking. That's why you use it to solve problems! You
could (possibly) figure it out without the help of mathematics, but mathematics
makes it so much easier because all you have to do is follow the rules!
Mathematics
in Chemistry
Chemistry
is an exact science since it relies on quantitative models that can be
described and applied by using the mathematical language. For instance, the
theory of chemical bonding and molecular structure, rates and equilibria of
chemical reactions, molecular thermodynamics, relationships involving energy,
structure and reactivity, modeling of solvation, are swarming with problems
whose solutions require sophisticated mathematical techniques. Mathematics also plays a central role in
many areas of "applied" chemistry
and chemical engineering.
Important
examples include atmospheric chemistry,
biochemistry, and the broad field of
computer simulations. The development of faster and more accurate spectroscopic
techniques, the design of molecular devices, biomolecular
computers,
and of new empirical methods to predict reliable chemical data, and the
conception of more efficient chemical reactors are just a few of a vast number
of other topics that have strong links to applied mathematics.
A
closer interaction between chemists and mathematicians may therefore lead to
significant progress in many key problems in chemistry.
The proposed workshop will foster that interaction since it will identify a
number of important research issues which will benefit from a joint effort.
For many universities the days
when admission to a Chemistry, Chemical Engineering, Materials Science or even
Physics course could require the equivalent of Alevels in Chemistry, Physics
and Mathematics are probably over for ever.
This means there must be an
essentially remedial component of university chemistry to teach just the
Mathematics and Physics which is needed and not too much, if any more, as it is
time consuming and perhaps not what the student of Chemistry is most focused
on. There is therefore also a need for a book Physics for Chemistry.
Mathematics for Commerce
Mathematical problem solving is an
important technique for the solution of problems in any area of society. This
paper aims to develop the ideas and concepts from Mathematics in such a way
that the student develops their problem solving techniques in Mathematics and
can apply the processes to Commerce. This paper will give the student a sound
knowledge of mathematical concepts and prepare them for the demands of
100level Commerce, mathematical and statistical papers at the University of
Otago.
Learning Outcomes
 Explore
the use of formulae, relationships, equations, expressions and statistical
techniques in a variety of contexts.
 Use
number, algebra, probability, statistics and financial Mathematics in
different situations and interpret their results.
 Develop
mathematical skills in number, algebra, financial Mathematics,
probability, statistics and some curve sketching.
 Gain
and demonstrate an understanding and appreciation of problem solving techniques
in a variety of contexts.
Mathematical
economics
is
the application of mathematical methods to represent economic theories and
analyze problems posed in economics. It allows formulation and derivation of key
relationships in a theory with clarity, generality, rigor, and simplicity. By
convention, the applied methods refer to those beyond simple
geometry, such as differential and integral calculus, difference and differential equations, matrix
algebra, and mathematical programming^{[1]}^{[2]}
and other computational methods.^{[3]}
v.
Mathematics allows
economists to form meaningful, testable propositions about many wideranging
and complex subjects which could not be adequately expressed informally.
Further, the language of mathematics allows economists to make clear, specific,
positive
claims about controversial or contentious subjects that would be impossible
without mathematics.^{[4]}
Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified
mathematical relationships that clarify assumptions and implications.^{[5]}
vi.
Broad applications include:
• optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker
• static (or equilibrium) analysis in which the economic unit (such as a household) or economic system (such as a market or the economy) is modeled as not changing
• comparative statics as to a change from one equilibrium to another induced by a change in one or more factors
• dynamic analysis, tracing changes in an economic system over time, for example from economic growth.^{[1]}^{[6]}^{[7]}
• optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker
• static (or equilibrium) analysis in which the economic unit (such as a household) or economic system (such as a market or the economy) is modeled as not changing
• comparative statics as to a change from one equilibrium to another induced by a change in one or more factors
• dynamic analysis, tracing changes in an economic system over time, for example from economic growth.^{[1]}^{[6]}^{[7]}
vii.
Formal economic modeling
began in the 19th century with the use of differential calculus to represent and
explain economic behavior, such as utility
maximization, an early economic application of mathematical optimization. Economics
became more mathematical as a discipline throughout the first half of the 20th
century, but introduction of new and generalized techniques in the period
around the Second World War, as in game theory,
would greatly broaden the use of mathematical formulations in economics.^{[8]}^{[7]}
viii.
This rapid systematizing of
economics alarmed critics of the discipline as well as some noted economists. John Maynard Keynes, Robert
Heilbroner, Friedrich Hayek and others have criticized the
broad use of mathematical models for human behavior, arguing that some human
choices are irreducible to mathematics.
Math in economics can be extremely
useful. It should know! Most of my own work over the years has relied on
sometimes finicky math — It spent quite a few years of my life doing tricks
with constantelasticityofsubstitution utility functions. And the
mathematical grinding served an essential function — that of clarifying
thought. In the economic geography stuff, for example, It started with some
vague ideas; it wasn’t until I’d managed to write down full models that the
ideas came clear. After the math It was able to express most of those
ideas in plain English, but it really took the math to get there, and you still
can’t quite get it all without the equations.
Mathematics And Astrology
Mathematicians of today are very apt to declare that they do not
believe in the flimsy and hazy kind of Astrology known as Horary. They claim to
believe only in what they call Nativties, yet at the same time they advocate
mathematical precision in arcs of direction from a birth data correct to the
minute. What is a correct data to the minute but an Horary data. A
"Nativity" and an "Horary Figure" mean precisely the same
thing when the data is from the exact minute of a birth, whether that birth be
the birth of a human being or an event in the life of a human being. An Horary
Figure must be exact to the minute, but a Nativity may or may not be, and still
it would be a Nativity.
Who ever saw a picture of the
Nativity of the Saviour, drawn to the moment of birth, and yet in every gallery
of paintings we may find a picture of the birth of the Saviour; and by the best
of scholars it is termed the "Nativity of the Lord Jesus Christ"
A strictly Horary Figure, of exact
moment of birth is the only thing in Astrology that calls for the finer
mathematics. Without the exact minute of a birth Tables of Houses are useless,
and it is impossible to make correct arcs of direction.
The finer mathematics and
spherical basis of Astrology, which calls for sidereal time, latitude,
longitude, ascending degree, zenith, Tables of Houses and arcs of direction,
aside from simple rules of arithmetic and approximated laws, belong almost
exclusively to the Horary Branch of Astrology, which is not at all an ancient
method, but is in reality a modern system but a few centuries old.
The term "mathematics"
means anything, from the numbering of the signs 1, 2, 3, etc., down to the most
intricate trigonomical calculation, but the pretentious mathematics forced upon
pure Astrology belongs only to the Horary system, and is contained in such
excellent works as "Dalton's Spherical Basis of Astrology," but this
work is only useful to Horary Astrology. I know that many minds will at first
disagree with me in this assertion, but they had better not argue against my
statement unless they wish to prove that they do not understand the subject.
Horary Astrology is generally
considered to be a figure drawn for the answering of immediate questions.
The selfsame mathematical law, or
rules are required for the casting of a figure for exact time of birth, as for
the exact time of asking a question. So far as the mathematics go, there is no
difference.
A large number of Astrologic
students learn that and nothing more, consequently they are always floundering
with mathematical problems, lordships and houses, and never really learn what
true Astrology is. By reason of the mathematical problems, relative to the
Horary arcs of the earth's diurnal motion, true Astrology and Horary Astrology
are indiscriminately mixed in nearly all of the books published, and strange to
say, those who decry the Horary features as hazy and flimsy are the biggest
sticklers for that grade of mathematics which are only needed for the Horary
findings.
Maths
and Agriculture
Within the broad concept of
farming, there are two very important elements: time and money. At the
root of both of these is mathematics. Mathematics has enabled farming to
be more economically efficient and has increased productivity. Farmers
use mathematics as a system of organization to effectively utilize their time
and manage their money. Farmers use numbers everyday for a variety of
tasks, from measuring and weighing, to land marking. I intend to explore
some of the ways math is used in farming.
One of the most frequently used
math concepts on the farm is the use of proportions. Units and
measurements used in farming are fairly unfamiliar to other areas. We can
use proportions to make conversions from the unfamiliar to the familiar.
Land is measured in acres, for example. It is hard to understand the size
of an acre because our minds are trained to visualize miles or
kilometers. We can use conversions to put an acre into perspective.
There are 43,560 square feet in an acre. This is slightly smaller than
the size of a Canadian football field, without the end zones. Other farm
measurement terminology include quarters and sections. A quarter is 160
acres and a section is 4 quarters. Professionals who work in grain
elevators frequently use conversions. Grain prices are often given per
ton, but producers want to know the price per bushel. Agriculture
specialists are required to do these conversions quickly and accurately.
The agriculture community uses numbers
to describe and grade seeds. Weights of seeds are generally expressed in
terms of bushels. For example, wheat is maybe 60 lbs/bu and perhaps
barley is 48 lbs/bu. Seeds are graded using numbers as well. For
example, spring wheat can be given a grade of 1, 2, 3, 4, or feed. Durum
can be given a grade of 1, 2, 3, 4, 5, or feed. Barley can only be given
a grade of 1, 2, or feed. Peas are either edibles or feed.
In order for peas to be edible, they need a grade of 2 or better. In the
case of durum, an HVK (hard vitreous kernels) measurement is one of the
determining factors of a grade. HVK is a percentage measurement of
hardness, which is examined by natural translucency, in a 25g sample. For
a number one durum, the seeds need to have 80% HVK, for a number two durum, the
seeds need to have 60% HVK, and so on. Elevators have a special scale
that gives a measurement of HVK percentage. All of these number systems
are used to categorize seeds. Theses numbers are determinants for grain
pricing and are extremely valuable for producers and consumers.
Estimation is an important concept
for farmers. Much of farming is unpredictable, due to weather reliance
and grain markets. Farmers try to estimate the yield of a certain field
of grain. To do this, farmers pick a plant and count how many seeds are
on the head. By looking at the square footage of a field and estimating
the number of heads, farmers can find an approximation of the yield. It
can be very difficult to estimate crop yields and sometimes, professional
estimates are very inaccurate.
Farmers will also estimate
elements of time. They know approximately how many hours they will need
to seed and harvest and can plan accordingly. These estimates of time are
based on crop types and machine availability, as well as human resources.
Farmers consider past trends of weather and moisture conditions to decide when
to start seeding. Furthermore, farmers can estimate the time remaining
until harvest by calculating growing degree days. This is the measurement
of heat units needed by the plant to reach its full maturity. It also
accounts for the ripening of the crop. An approximation is made of how
many days remain until the crop is ready to be harvested. Some processes
affect this calculation such as desiccating the crop, and can change the number
of growing degree days.
Farmers need to consider all
aspects of their farming operation in order to make it successful.
Farmers create mathematical systems of equations and inequalities to help them
make decisions about which crops to plant in which fields. This system of
organization is commonly referred to as linear programming. The
constraints of farming can include costs of seed, labour, time, crop insurance,
machinery, chemical / fertilizer, and so on. Livestock producers
also use linear programming when making feed for cattle. A variety of
ingredients are mixed together to make feed and producers want the most
nutritious combination of ingredients that is also cost efficient. There
are also formulas that illustrate the relationships between relative humidity,
time, and moisture content that are used by farmers to estimate drying time
before baling alfalfa.
The following problem illustrates
how a young livestock farmer may use linear equations, linear inequalities, and
mathematical organization:
Maurice, who is 14, lives on a
farm and wants to earn some money for a snowboard this fall. It costs
$450. He talks to his parents and decides he will raise some freerange
chickens to sell. His parents agree to lend him the money to do this if
he pays them back when the chickens are sold.
He makes a list of his costs and his time
requirements:
1. He needs to prepare
a warm, dry shelter for the chicks. He will need chick starter (special
food) that costs $1 per chick. It will take 2 hours of his time.
2.
His mom will give him a ride to the hatchery where he can buy the chicks for
$1.20 each. Maurice has to pay $12 for gas and it takes him 2 hours to
collect the chicks.
3.
For the next sixteen weeks, the chicks grow into adulthood. Once they are
finished the chick starter formula, they eat grass, which costs him
nothing. Maurice expects to lose 10% of his birds before they are fully
grown. He has to spend 5 hours per week doing chores like providing fresh
water, checking on them, cleaning their roosts, etc.
4.
Maurice is a bit squeamish, and he decides that he'll take the fullygrown
chickens to a commercial processor for killing and cleaning. His father has a
truck that will be able to carry the chickens to the slaughterhouse, but it
will cost $20 in fuel and take 4 hours of Maurice's time. The processor
charges $3 per bird for its services.
5.
When he gets the chickens home, Maurice has to weigh, bag and freeze
them. That takes an average of 5 minutes per bird. He estimates the
average weight will be 5.5 pounds. Fortunately, his mother is letting him
use her extra freezer for free.
6.
He decides that he can charge $3.00 per pound if his customers pick up their orders
(he won't be doing any deliveries).
Question
1: How many chickens does he need to order so that he can raise the money for
the snowboard, keeping in mind that Maurice needs to repay his parents for the
money he borrowed from them?
Question
2: If Maurice purchases that many chickens, what would his hourly wage be if he
made $450?
Solution
Question 1: Make a chart recording the
expenses, revenue, and labour values from the problem. We can define
variables for c (the number of chickens bought) and c¢ (the number of chickens
that survived the 16 weeks).
Expenses
(in dollars) 
Revenue
(in dollars) 
Labour
(in hours) 

1.

1c

2


2.

1.20c
+ 12

2


3.

c¢ =
0.9c

80


4.

3c¢ +
20

4


5.

16.5c¢

1/12
c¢

Maurice’s total profit is his revenue minus
his expenses. We can sum up his expenses as follows:
Then, Maurice’s profit would be:
Maurice’s profit needs to be at least $450 if
he wants to purchase the snowboard. We can set up an inequality to find
the minimum number of chickens that must be purchased for Maurice to afford the
snowboard.
Maurice needs to purchase at least
49 chicks in order to afford the snowboard once he sells the chickens, fully
grown.
Question 2: We can use information from
the chart above to find the total number of hours Maurice worked.
Total time in hours = 91.78
If Maurice works a total of 91.78 hours and
makes $450, his hourly wage can be found simply by dividing.
The Impact of Mathematics on Cellular and Molecular Biology
The
application of mathematics to cellular and molecular biology is so pervasive
that it often goes unnoticed. The determination of the dynamic properties of
cells and enzymes, expressed in the form of enzyme kinetic measurements or
receptorligand binding are based on mathematical concepts that form the core
of quantitative biochemistry.
Molecular
biology itself can trace its origins to the infusion of physical scientists
into biology with the inevitable infusion of mathematical tools. The utility of
the core tools of molecular biology was validated through mathematical
analysis.
Examples
include the quantitative estimates of viral titers, measurement of
recombination and mutation rates, the statistical validation of radioactive
decay measurements, and the quantitative measurement of genome size and
informational content based on DNA (i.e., base sequence) complexity.
These
examples are cited not to document the accomplishments of mathematical
biologists but to bring focus to the fact that mathematical tools are intrinsic
to biological fields. The discussion that follows focuses more clearly on the
more sophisticated development of new mathematical concepts and statistical
models to explain the complexity of biological systems. Biological complexity
derives from the fact that biological systems are multifactored and dynamic.
Quantitative
research in these fields is based upon a wide variety of laboratory techniques,
with gel electrophoresis and enzymebased assays among the most common.
Measurements include activity, molecular weight, diameters, and sizes in bases,
and with all these an understanding of the accuracy, precision, sources of
variation, calibration, etc. In short, the quality of the measurement process
is of central significance.
The
goal of the present discussion is to provide a framework in which ongoing
research in mathematical cell and molecular biology may be logically placed,
and future opportunities can be described. This framework will provide for the
analysis of the resource needs for future development and carries implications
for current shortfalls.
One factor is that undergraduate and graduate
training in biology treats mathematics too superficially, especially in light
of its role as an underpinning for quantitative research.
Banking
Keeping Track
Mathematics and
banking are closely connected. Extensive math is involved inkeeping track of the money in a bank. Needless to
say, mathematical precision isessential. Banks handle substantial sums of
money. Any inaccuracies in math cancreate huge negative ramifications. Everyone
from the tellers at your local bankbranch to banking executives must have a
firm grasp of math skills in order to dotheir jobs. Mathematic skills are
needed to insure the money is coming from andgoing to the correct places.
Transactions
When someone
deposits or withdraws money from a bank, more math is involved tocalculate the
total in their accounts. Again, precision and accuracy is key. If a
bankemployee accidentally puts too much money into someone's account, it will
cost thebank money. If an employee puts too little into someone's account, it
can greatlydamage the bank's image. The same goes for withdrawals. The amount
taken fromthe account must match the amount the client withdrew from the
account. Accuratemath is essential to all transactions; extensiverecord keepinginsures mathematicaccuracy. The math is repeatedly
checked and rechecked.
Accumulating Interes
Savings and
checking accounts have a wide variety of different interest ratesdepending on
the account type. Interest is calculated and added to a person'saccount at
regular intervals, depending on what their account type dictates.Calculating
interest is done using percentages. In each interest period, the setpercentage
is added to a person's account.Money market accounts also accumulate interest
at set intervals. This interest rate,however, varies greatly to correspond to
the changing markets, and thusly involvesmore complex mathematics.
Loans
4.Loans involve complex
mathematics. First, the bank needs to calculate how high of a loan it should give you and how much you will be
paying each month for the loan.Calculating this requires consideration of a
wide variety of factors ranging from howlong the loan is for to how strong your
credit history is.
Mathematics and Banking
5.Mathematics and banking are
tightly linked. Banking requires constant use of mathematics. Often times, complex formulas are necessary to compute
interest andloans. Not only does banking require extensive math skills, it also
requires intenseprecision and accuracy. Banking and mathematics are inexorably
intertwine.
Mathematics
and the environment
Some
people claim that the discoveries made by scientists contribute to the
destruction of the natural environment. Professor Louis Gross at the University
of Tennessee shows that the case can equally be made for the opposite. He is a
mathematical ecologist, applying advanced mathematics to the problems of
managing the natural environment to maximise the benefits to the whole natural
system.
The pressures of human life have an effect on
the rest of nature and by understanding how the relationships work, everyone
and everything might get some of what they want.
It turns
out that these problems are not trivial mathematically. The flow pattern of a
river might have a linear relationship with the rainfall in a particular place,
but what happens when the river bursts its banks? Or if it rains after a period
of drought? And how do you know what the rainfall is going to be anyway?
Not only
are many natural processes essentially stochastic they also require nonlinear
algebra to describe them. Getting meaningful results is a huge mathematical and
computational exercise. This is why Gross, like many scientists from other,
more conventional fields, has turned his attention to the mathematics of the
natural world – it has some of the most interesting mathematical problems.
Mathematical
biology has achieved a high profile through cell biology and genomics, but at
the scale of the whole ecosystem it is still in its emerging stage and the
field has many opportunities to do new things.
He graduated in mathematics, with a minor in physics. But after three years of working in radio astronomy, he decided that it was time to change tack. He says: ‘I came to the realisation that the really good people not only had good mathematical ability, but they also had good handson skills and could fix equipment.
He graduated in mathematics, with a minor in physics. But after three years of working in radio astronomy, he decided that it was time to change tack. He says: ‘I came to the realisation that the really good people not only had good mathematical ability, but they also had good handson skills and could fix equipment.
They
also had huge physical intuition and I realised I did not have that physical
intuition, which is why I decided to get into biological science. I always had
an interest in the outdoors, because I was very active in the scouting
movement, which had a strong emphasis on natural history.’
Maths
in Google Earth
In
the project, the children will be using the line tool and the path tool to
measure horizontal distances on Google Earth. This can be used to estimate the
perimeter and perhaps the area of various buildings. Then try to let them
explore the programme and discover other handy tools which is linked to maths.
Afterwards, they could set themselves a questions that they will answer
themselves.
Questions
children could consider in their on research work:
 What
are the longest and shortest airport runways in the world?
 Can
they find any runways that are parallel or perpendicular to each other?
 What
is the area and perimeter of their school or school playground/fields?
 On
average, who walks further to school, girls or boys?
 What
is the distance between various capital cities?
There
are many other questions which children might also consider, so this is also an
opportunity for them to ask and answer their own questions about shape and size
of buildings or other features anywhere on the Earth's surface!
Medicine and MathS
Both doctors and
nurses use math every day while providing health care for people around the
world. Doctors and nurses use math when they write prescriptions or
administer medication. Medical professionals use math when drawing up
statistical graphs of epidemics or success rates of treatments. Math
applies to xrays and CAT scans. Numbers provide an abundance of
information for medical professionals. It is reassuring for the general
public to know that our doctors and nurses have been properly trained by
studying mathematics and its uses for medicine.
Prescriptions and Medication
Regularly, doctors
write prescriptions to their patients for various ailments. Prescriptions
indicate a specific medication and dosage amount. Most medications have
guidelines for dosage amounts in milligrams (mg) per kilogram (kg).
Doctors need to figure out how many milligrams of medication each patient will
need, depending on their weight. If the weight of a patient is only known
in pounds, doctors need to convert that measurement to kilograms and then find
the amount of milligrams for the prescription. There is a very big
difference between mg/kg and mg/lbs, so it is imperative that doctors
understand how to accurately convert weight measurements. Doctors must
also determine how long a prescription will last. For example, if a
patient needs to take their medication, say one pill, three times a day.
Then one month of pills is approximately 90 pills. However, most patients
prefer two or three month prescriptions for convenience and insurance
purposes. Doctors must be able to do these calculations mentally with
speed and accuracy.
Doctors must also
consider how long the medicine will stay in the patient’s body. This will
determine how often the patient needs to take their medication in order to keep
a sufficient amount of the medicine in the body. For example, a patient
takes a pill in the morning that has 50mg of a particular medicine. When
the patient wakes up the next day, their body has washed out 40% of the
medication. This means that 20mg have been washed out and only 30mg
remain in the body. The patient continues to take their 50mg pill each
morning. This means that on the morning of day two, the patient has the
30mg left over from day one, as well as another 50mg from the morning of day
two, which is a total of 80mg. As this continues, doctors must determine
how often a patient needs to take their medication, and for how long, in order
to keep enough medicine in the patient’s body to work effectively, but without
overdosing.
The amount of medicine
in the body after taking a medication decreases by a certain percentage in a
certain time (perhaps 10% each hour, for example). This percentage decrease can
be expressed as a rational number, 1/10. This constant rational decrease
creates a geometric sequence. So, if a patient takes a pill that has
200mg of a certain drug, the decrease of medication in their body each hour can
be expressed as follows:
200, 20, 2, 1/5, 1/50, ...
As you can see, the
amount of medication in the body after 5 hours is quite small, almost
zero. The sequence of numbers shown above is geometric because there is a
common ratio between terms, in this case 1/10. This means that each hour,
the amount of medication decreases by 1/10. Doctors can use this idea to
quickly decide how often a patient needs to take their prescribed medication.
Ratios and Proportions
Nurses also use ratios
and proportions when administering medication. Nurses need to know how
much medicine a patient needs depending on their weight. Nurses
need to be able to understand the doctor’s orders. Such an order may be
given as: 25 mcg/kg/min. If the patient weighs 52kg, how many milligrams
should the patient receive in one hour? In order to do this, nurses must
convert micrograms (mcg) to milligrams (mg). If 1mcg = 0.001mg, we can
find the amount (in mg) of 25mcg by setting up a proportion.
By crossmultiplying
and dividing, we see that 25mcg = 0.025mg. If the patient weighs 52kg,
then the patient receives 0.025(52) = 1.3mg per minute. There are 60
minutes in an hour, so in one hour the patient should receive 1.3(60) =
78mg. Nurses use ratios and proportions daily, as well as converting
important units. They have special “shortcuts” they use to do this math
accurately and efficiently in a short amount of time.
Numbers give doctors
much information about a patient’s condition. White blood cell counts are
generally given as a numerical value between 4 and 10. However, a count
of 7.2 actually means that there are 7200 white blood cells in each drop of
blood (about a microlitre). In much the same way, the measure of
creatinine (a measure of kidney function) in a blood sample is given as X mg
per deciliter of blood. Doctors need to know that a measure of 1.3 could
mean some extent of kidney failure. Numbers help doctors understand a
patient’s condition. They provide measurements of health, which can be
warning signs of infection, illness, or disease.
Mathematics in Sports
Although not always realized, mathematics plays a very important role in sports. Whether discussing a players statistics, a coaches formula for drafting certain players, or even a judges score for a particular athlete, mathematics are involved. Even concepts such as the likelihood of a particular athlete or team winning, a mere case of probability, and maintain equipment are mathematical in nature.
Let's begin by looking at the throwing of a basketball. Now, we can use the equation
to help figure out the velocity at which a basketball player must throw the ball in order for it to land perfectly in the basket. When shooting a basketball you want the ball to hit the basket at as close to a right angle as possible. For this reason, most players attempt to shoot the ball at a 45^{o} angle. To find the velocity at which a player would need to throw the ball in order to make the basket we would want to find the range of the ball when it is thrown at a 45^{o} angle. The formula for the range of the ball is
But since the angle at which the ball is thrown is 45^{o}, we have Now, if a player is shooting a 3 point shot, then he is approximately 25 feet from the basket. If we look at the graph of the range function we can get an idea of how hard the player must throw the ball in order to make a 3 point shot.
So, by solving the formula knowing that the range of the shot must be 25 feet we have
So in order to make the 3 point shot, the player must throw the ball at approximately 28 feet per second, 19 mph
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