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A Monthly Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories.
International Research Journal of Mathematics, Engineering and IT (IRJMEIT)
34 | P a g e
EXACT DEFINITION OF MATHEMATICS
Dharmendra Kumar Yadav
Assistant Professor, Department of Mathematics
Shivaji College, University of Delhi, Raja Garden, Delhi-27
ABSTRACT
The aim of the article is to propound a simplest and exact definition of mathematics in a single
sentence. It is observed that all mathematical and non-mathematical subjects whether science,
arts, language or commerce, follow the same steps and roots to develop, they all consist of three
parts: assumptions, properties and applications. These three terms make the exact definition of
Mathematics, which can be applied to all subjects also. Therefore all subjects can be brought
under the same umbrella of definition consisting of these three terms. Following this
mathematics has been defined as the study of assumptions, its properties and applications. Then
different branches of mathematics have been discussed. A short paragraph has been devoted to
technical teachers and students on engineering mathematics. In last how should we teach
mathematics has been emphasized? A special focus on the type of assignment has been
mentioned. This article will be useful for mathematics teachers and its learners, if it is discussed
in the first few lectures of undergraduate and post graduate level as well as will be more fruitful
for technical students as they can understand and apply it better than non-technical students.
Key Words: Axiom, Theorem, Properties, Conjecture.
AMS Subject Classification: 97D30, 00A05, 00A06, 03E65, 01A80, 97D20
Introduction
Carl Friedrich Gauss referred mathematics as the queen of science but unfortunately students
fear from this queen, although the subject is very essential to the growth of many other
disciplines. The science of mathematics depends on the mental ability. It is the means to develop
the thinking power and reasoning intelligence, which sharps the mind and makes it creative. The
development of human beings and their culture depend on the development of mathematics. This
is why, it is known as the base of human civilization. It is also the language of all material
science and the centre of all engineering branches which revolve around it. Therefore it is the
past, present and future of all sciences. Narlikar has focused on the importance of mathematics
by mentioning that in 1957 when the Soviet Union launched the first satellite Sputnik, the United
States realized that to match it, the teaching of mathematics had to receive boost. After that many
major steps have been taken to improve the quality education of mathematics not only in USA
but in the world too.
International Research Journal of Mathematics, Engineering and IT
Vol. 4, Issue 1, January 2017 Impact Factor- 5.489
ISSN: (2349-0322)
© Associated Asia Research Foundation (AARF)
Website: www.aarf.asia Email : editor@aarf.asia , editoraarf@gmail.com
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Previous Attempt to Define Mathematics
Although the research in mathematics has covered a milestone, a major drawback has been seen
in the literature of mathematics that it could not be defined properly, so that all mathematical
subjects can be combined in short and in a single sentence. No consensus on the definition of
mathematics prevails, even among professionals. According to Wikipedia, mathematics has no
generally accepted definition and there is not even consensus on whether mathematics is an art
or science. Gunter M. Ziegler mentions that in German Wikipedia the definition is interesting in
a different way: it stresses that there is no definition of mathematics, or at least no commonly
accepted one. Even the famous book by Richard Courant and Herbert Robbins entitled “What
is Mathematics?” (and subtitled “An Elementary Approach to Ideas and Methods”) does not give
a satisfactory answer. He claims that it is impossible to give a good definition in a sentence or
two. A great many professional mathematicians take no interest in a definition of mathematics, or
consider it undefinable.
Traditionally it is defined as the scientific study of quantities, including their relationship,
operations and measurements expressed by numbers and symbols. In mathematics dictionary by
James & James it has been defined as the science of logical study of numbers, shape,
arrangement, quantity, measure and many related concepts. Today it is usually described as a
science that investigates abstract structures that it created itself for their properties and patterns”.
According to Wikipedia, „Mathematics is the study of quantity, structure, space. Mathematics
seeks out patterns and uses them to formulate new conjectures. Aristotle has defined
mathematics as „The science of quantity?‟. Benjamin Pierce defined it as „Mathematics is the
science that draws necessary conclusions‟. Haskell Curry defined mathematics simply as “the
science of formal systems”. Albert Einstein stated that “as far as the laws of mathematics refer to
reality, they are not certain; and as far as they are certain, they do not refer to reality”. More
recently, Marcus du Sautoy has called mathematics “the Queen of Science …… the main
driving force behind scientific discovery”. Thus although all most all great mathematicians stated
something for it, no generally accepted definition could be produced. A little attempt has been
done in this article to define mathematics in a single sentence and exact form, which will be
accepted for centuries without any counter example.
Basic Terms of Mathematics
In every mathematical subject we find some general terms like axioms, properties, theorem, etc.
These are the basics of the subjects whose meanings are given as follows:
Axioms: James & James stated that the axioms of a subject are the basic propositions from
which all other propositions can be derived. They are accepted as the starting points and are
accepted true without any proof. With the help of axioms we decide whether a given
mathematical statement is true or false. It is also known as assumptions or hypothesis or
postulates or propositions.
Therefore mathematics can be regarded as a set and study of assumptions, because it starts with
axiom. But it ends with reality. Here the statement „end with reality‟ means although we start
with assumptions, but after its application and in the final result, we reach at the real destination.
Example: One right angle is equal to 900 is an assumption. It has no proof. Although now there
are many properties from which it can be proved, but the property from which it will be proved,
will also be an assumption.
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International Research Journal of Mathematics, Engineering and IT (IRJMEIT)
36 | P a g e
Property: Any mathematical statement derived using axioms is known as a property. A straight
angle is equal to 1800 is a property, which is proved as the sum of two right angles. It can be
further divided in two parts: Theorem and Conjecture.
Theorem: A general conclusion which can be proved by the help of axioms is called a theorem.
In general it is proved logically using assumptions as true.
Example: The sum of the three angles of a triangle is 1800 is a theorem.
Conjecture: A mathematical statement which has many examples but cannot be proved or yet to
be proved is known as conjecture. A well known conjecture is Goldbach’s Conjecture which
states that „Every even integer greater than 4 can be written as sum of two odd primes‟. So far
either a proof or a counter example has not been found.
Applications: If we apply the assumptions and its properties to solve real life problems, we say
that such type of assumptions have applications.
Example: The sum of the three angles of a triangle is 1800 is an application of the assumption
that one right angle is 900 and a straight angle is 1800.
Thus we see that in mathematical subject we have three main terms: assumptions, properties and
applications. So we can say that every mathematical subject is composed of three terms:
assumptions, properties and applications. Thus we can define ‘Mathematics is the study of
assumptions, its related properties and applications‟. In fact every subject is the set of
assumptions, its properties and applications as has been explained in the analysis below:
Analysis of Some Properties
Analysis 1: We remember that the sum of the three angles of a triangle is 1800 as shown in fig.-
1. Fig.-5 shows that one right angle is equal to 900. Fig.-3 and Fig.-4 show that two right angles
make one straight angle which is equal to 1800. Fig.-2 shows that the sum of three angles of a
triangle is equal to 1800, which is generally derived from the fact that when three angles <A, <B
and <C are placed on a straight line with their vertices coinciding at one point, makes a straight
angle which is equal to 1800.
Fig.-1
A B C
Fig.-2
Fig.-3
Fig.-4 Fig.-5
1800
<A+<B+<C=1800
A
B
C
900
900
900
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We know that in geometry angles are measured in terms of right angle. In British System, a
right angle is divided into 90 equal parts called Degrees, so a right angle makes 900. In French
System, a right angle is divided into 100 equal parts called Grades, so a right angle makes 100g.
Whereas in Circular Measure, a right angle is equal to
2
. Thus the sum of the three angles of
a triangle is 1800 in British system, 200g in French system and 𝜋 radian in Circular measure
system.
Therefore we first supposed that a right angle is equal to 900. Then we developed the definition
of a linear pair of angles, for it the sum of two angles is 1800 and then developed a property that
a straight angle is equal to 1800. By the help of this property, we proved that the sum of the three
angles of a triangle is 1800. What we followed here is that, first we assumed, then developed a
property and finally applied it.
Analysis 2: A spherical balloon is pumped at the rate of 10 cubic inches per minute, find the rate
of increase of its radius when its radius is 15 inches.
Let y be the volume and x the radius of the balloon at any time t. Then
10
dt
dy
cubic inches per
minute and we have to find
dt
dx
when x=15 inches. Since the balloon is spherical,
3
3
4ry
.
90
1
154
10
4
10
4
4222
2 xx
dt
dy
dt
dx
dt
dx
x
dt
dy
Hence rate of increase of the radius when radius is 15 inches is
90
1
inch/minute. For the
function y=f(x), the differential coefficient of y with respect to x has been denoted and defined
by
x
xfxxf
x
dx
dy
)()(
0
lim
provided that the limit exists. It has been called the measurement of rate of change in y with
respect to x. After that the derivative of
n
xy
is
1
n
nx
dx
dy
, was found with many other
elementary functions. Then it was applied in many such problems as discussed above. So it also
followed the three main steps: assumption, properties and applications. Similarly many more
examples in mathematics can be analyzed.
Analysis 3: Let us consider the shadow formation on a screen by a point source of light. A
source of light at S falls on the opaque body AB and makes a shadow A‟B‟ on the screen.
Let us consider the following:
dsh: diameter of the shadow;
db: diameter of the body
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International Research Journal of Mathematics, Engineering and IT (IRJMEIT)
38 | P a g e
S: source of light;
AB: Opaque Body;
A‟B‟: shadow of the body on screen
lb: distance between source & opaque body;
lsh: distance between opaque body & screen.
The distances between different objects have been indicated in the following figure:
Opaque Body Screen
A’ A’
A
d
Source S C’ dsh d
lb B B’
B’
lsh
Now since triangles SAB and SA‟B‟ are similar, we have
sh
b
shb
b
d
d
ll
l
BA
AB
SC
SC
'''
b
sh
bsh l
l
dd 1
From this we conclude that the size of the shadow is always greater than the size of the opaque
body or object. It will be equal to the size of the object if the screen is in contact with the opaque
body. As the screen is moved away from the opaque body the shadow size increases. To find the
size of the shadow we used the concepts of similar triangles, which is itself an assumption and its
properties.
Analysis 4: In language we first learn its alphabets, then words formed using alphabets and
finally we make sentences using many words. For example, in English language, we first study
from A to Z. Then we study words like Ram, Eat, Apple, etc having special meanings followed
by grammar as properties. Finally we make sentence Ram eats an apple. So every language
follows the three steps of development: assumption, properties and application. In English 26
alphabets are assumption, words and grammar are both assumptions and properties, where as a
complete sentence is the application.
The same procedures follow in other languages like Hindi, Urdu, Sanskrit, etc. Similarly in
Commerce, we assume the definitions of GDP, NDP, Income, Development, Growth, etc. and
then apply it to study about the status of the society or nations.
dsh
db
C
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From above examples, analysis and the basic terms of mathematical subjects, we can define
mathematics in the shortest and compact form as „Mathematics is the study of assumptions, its
properties and applications‟, which can be taken as the exact definition of mathematics. In fact
„all mathematical and non-mathematical subjects are the set and study of assumptions, its
properties and applications‟, whether it is science, arts, commerce, literature, etc.
Branches of Mathematics
As far as the branch of mathematics is concerned, it is divided into four fields: Arithmetic,
Algebra, Analysis and Geometry. In arithmetic we learn about numbers and basic arithmetical
operations. When we apply arithmetic in solving real life problems, we get equations and thus
lead to Algebra. When the basic properties of arithmetic and algebra fail, we need analysis. In
analysis we generally study about limits, continuity, etc.
In Geometry we study about shapes and size of the figures. It is divided into two parts: Plane (or
Euclidean) Geometry and Spherical (or Non-Euclidean) Geometry. Spherical geometry is
further divided into two parts: Elliptical (or Riemann) Geometry and Non-elliptical (or
Hyperbolic) Geometry.
The basic difference among the above is that, in plane geometry, the sum of the three angles of a
triangle if 1800 where as in spherical geometry it is not equal to 1800, but either greater than or
less than of it. In elliptical geometry it is more than 1800 but in hyperbolic geometry it is less
than 1800.
The shapes of a triangle in above three geometries are as follows:
Similarly other properties and figures can be studied in three geometries.
Pure and Applied Mathematics
For the sake of simplicity, mathematics is divided into two branches: Pure and Applied
Mathematics.
Pure Mathematics is concerned with increasing knowledge of the subject rather than using
knowledge in practical ways, i.e. its study is theoretical. For example, trigonometry, geometry,
set theory, vector, etc. It is concerned with concepts and ideas that do not necessarily have any
immediate practical application. Its importance can be better understood by the well known
statement “A pure mathematician makes dreams even beyond the imagination of human
beings, and it is the scientists and technologists to apply them.”
Applied Mathematics is concerned with using knowledge of pure mathematics. It is not
theoretical but practical. In it we use the theories and concepts of pure mathematics. For
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International Research Journal of Mathematics, Engineering and IT (IRJMEIT)
40 | P a g e
example, mechanics, dynamics, statics, physics, etc. It is concerned with the use of mathematical
theories and principles as tools to solve problems in any field, whether it is research or problems
in actuarial science, economics, financial analysis, market research analysis, meteorology,
oceanography, aviation, aerodynamics, robotics, population studies, commercial surveys,
physics, chemistry, biology, social sciences, earth sciences, industrial research and development.
The study of problems in applied mathematics leads to new developments in pure mathematics
and theories developed in pure mathematics often find applications later. Work in applied
mathematics requires a theoretical background, which enables the mathematician to understand
the physical dimensions and technicalities of the problem. The study of pure and applied
mathematics is interdependent. Applied mathematics is like a flowing river having pure
mathematics on its two banks playing like domain on the boundaries. The two branches are so
interrelated and mixed that no sharp (or dividing) line can be drawn between them.
Engineering Mathematics
It is a part of applied mathematics. Kreyszig has mentioned that for the sake of deciding the
depth of studying mathematics, while teaching mathematics to the engineering students, we
should limit our knowledge of mathematics to the extent as far as the applications of the subject
are concerned. Technical students need solid knowledge of basic principles, methods, and
results, and a clear perception of what engineering mathematics is all about, in all three phases of
solving problems related to real and physical world: Modeling, Solving and Interpreting.
How To Teach Mathematics?
Ronning has pointed out that mathematics is becoming more and more important in study. More
and more decisions are made and actions are being taken on the basis of mathematical models.
So the problem arises that, what should we emphasize when we teach mathematics? What kind
of understanding do we want the students to develop? What kind of mathematics, and how much,
do all students need to know?
A simple answer of the question is that every chapter must be divided into three parts:
assumptions, properties and applications. When we start teaching, we must mention that what are
the basic assumptions in the chapter keeping in view that definition of a term is itself an
assumption. What can we obtain from the assumptions and in last how and where can we apply
these concepts? Even students may be allowed to remember the definition as their own
assumptions and then try to find out some properties related to them and the previous
mathematical or non-mathematical knowledge they have. In this way the learner will enjoy the
subject and they will improve their ability of mathematical power.
As far as the engineering mathematics is concerned, Kreyszig states that it would make no sense
to overload students with all kinds of little things that might be of occasional use. Instead it is
important that students become familiar with ways to think mathematically, recognize the need
for applying mathematical methods to engineering problems, realize that mathematics is a
systematic science built on relatively few basic concepts and involving powerful unifying
principles, and get a firm grasp for the interrelation between theory, computing and experiment.
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What should be the Format of Assignment?
Assignments play an important role in learning. In general it has been found that the teacher
gives some examples as assignments to solve. In this manner students generally do not care
about basic assumptions. They apply formula blindly and miss the basic concepts. They miss the
game of enjoyment that how properties are developed using assumptions, which finally fails our
aim of increasing mathematical ability in students. In fact the proper format of assignments must
follow the three main terms of the subject: assumption, properties and applications.
In other words, students must be directed to do assignment in three steps: first they must define
the basic terms of the chapter followed by the properties with proof. They must make a table of
the formulae and then solve at least five examples on each formulae or properties. In this way
students learn the definition, formulae and understand the basic structure of the chapter, which
makes them perfect in application. Thus our motives become more and more successful in
increasing the interest of mathematics among students.
Conclusion:
Finally we conclude the exact definition of mathematics as the study of assumptions, its
properties and applications. In teaching we must maintain the order of assumptions, properties,
and applications. This order must be maintained in assignments to get the desired aim of teaching
mathematics. Similarly all non-mathematical subjects can be defined as the set of assumptions,
its properties and applications.
Acknowledgement
Author acknowledges that the example on the size of the shadow has been taken from a lecture
delivered by Late Dr. P. Singh, Professor of Physics, BIT, Patna and the figure of triangles in
three geometries has been copied from Google. Some of the definitions have also been taken
directly from the sources mentioned in references.
References
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Chapter-1, Souvenir Press, 2008
Burton D. M., Elementary Number Theory, 7th Indian Edition, McGraw Hill Education
(India) Pvt Ltd, New Delhi, 2012, pp. 50-53
Courant R., Robbins H., What is Mathematics? An elementary approach to Ideas and
Methods, 2nd Edition, Oxford University Press, New York, 1996
Gakkhad S. C., Teaching of Mathematic, N. M. Prakashan, Chandigarh, 1991
James & James, Mathematics Dictionary, 4th Edition, CBS Publishers & Distributors,
India, 2001, pp.23, 169-170, 239, 387
Kennedy L. M. & Tipps S., Guiding Children‟s Learning of Mathematics, 8th Edition,
Wadsworth Publishing Company An International Thomson Publishing Company, 1997
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pp. 1-17
Narlikar J. V., Mathematics: The Queen of Sciences, Education Article, Manorama Year
Book 2013, pp. 480-484
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Nelson D., The Penguin Dictionary of Mathematics, Penguin Books, 3rd Edition, 2003,
pp. 21, 78, 147, 185-186, 271, 399
Ronning F., Developing Knowledge in Mathematics by Generalising and Abstracting,
Mathematics Newsletter, Ramanujan Mathematical Society, Vol.17 (4), March 2008,
pp.109-118
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Research (Preface: What is Mathematics? by Gunter M. Ziegler), Springer, XIV, pp.220,
2011
Sinha K. C., A Text Book of Calculus, Students‟ Friends, Patna, 1994, pp. 236-237
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Moscow, 1987
Ziegler G. M. & Loos A., Teaching and Learning “What is Mathematics”,
www.google.com
https://en.wikipedia.org/wiki/Hyperbolic_geometry
https://en.wikipedia.org/wiki/Elliptic_geometry
https://en.wikipedia.org/wiki/Definitions_of_mathematics
