A Simple Guide to Mathematics

A Simple Guide to
Mathematics
Compiled By
Nalaka Gamage
Aurora Computer Studies
(auoracs.lk)
(www.auroracs.lk)
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Agenda
• Give you the big picture of
mathematics
• Why do we learn mathematics?
• Branches of mathematics
• Myths & challenges in learning
• New trends in learning
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What is mathematics?
• Mathematics (from Greek máthēma,
“knowledge, study, learning”) is the study
of topics such as quantity (numbers),
structure, space, and change. (Ref:
Wikipedia)
• Mathematics is part of formal science
academic field
• Mathematics is closely related to logic
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Natural science Vs. formal science
• Natural science is a branch of science
concerned with the description, prediction,
and understanding of natural phenomena,
based on observational and empirical
evidence.
• Natural science can be broken into two
main branches:
o Life science (or biological science)
• Biology, Medicine..
o Physical science
• physics, chemistry, astronomy and Earth science
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Natural science Vs. formal
science (cont.)
• Formal sciences are disciplines concerned with
formal systems, such as logic, mathematics,
statistics, theoretical computer science,
information theory, game theory, systems theory,
decision theory, and portions of linguistics and
economics.
• Whereas the natural sciences seek to
characterize physical systems, the formal
sciences are concerned with characterizing
abstract structures described by sign systems.
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Natural science Vs. logic
• Science does involve logic, since we
need logic to test our theories and check
whether they are consistent.
• But for a theory to be a scientific one, it
must either describe actual empirical
facts about the world or the laws of
nature in our universe.
• Logic is more abstract or formal form of
scientific facts.
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Ref: http://philosophy.hku.hk/think/logic/whatislogic.php

Mathematics Vs. logic
• The relationship between mathematics
and logic is rather complicated
philosophical issue.
• To give a very short answer, the content
of mathematics is restricted to abstract
objects like numbers, functions, triangles,
etc.
• But the principles and laws of logic are
very general and are not restricted to
such concepts.
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Ref: http://philosophy.hku.hk/think/logic/whatislogic.php

How mathematics is evolved ?
• A postulate (also sometimes called an
axiom) is a statement that is agreed by
everyone to be correct.
• Mathematicians seek out patterns and
use them to formulate new conjectures.
• Once a conjecture has been proven, it
becomes a theorem.
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Postulate (Axiom)
• A postulate (also sometimes called an
axiom) is a statement that is agreed
by everyone to be correct.
• A common sense truth
• Example:
o We can draw only one straight line through two points
in the space
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Conjectures
• A conjecture is like an hypothesis to a scientist
• Scientists write hypotheses and test them to see
if they are true
• Similarly, in mathematics, a conjecture is just an
initial conclusion that you formed based on what
you see and already know, but for which no
proof has been found.
• In rare cases a conjecture may later be found to
be invalid
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Conjectures –An example
• Fermat's Last Theorem states that no
three positive integers a, b, and c can
satisfy the equation an + bn = cn for
any integer value of n greater than
two.
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Theorems
• Once a conjecture has been proven, it
becomes a theorem
• Example:
o Pythagoras theorem
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Branches of mathematics
• Arithmetic
• Algebra
• Geometry
• Logic
• Statistics & probability
• Calculus
• Applied mathematics
o Business & decision theory related applied mathematics
o Engineering and science related applied mathematics
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Arithmetic
• Mathematics related to numbers
• As a kid, you first expose to
mathematics through arithmetic
• Basic life skill and usages in all areas
including business, engineering and
computing
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Algebra
• Algebra is the study of mathematical
symbols (unknowns) and the rules for
manipulating these symbols
• Has usages from simple day to day
life to engineering and business
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Geometry
• Concerned with questions of shape,
size, relative position of figures, and
the properties of space.
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Logic
• Logic is the use and study of valid
reasoning.
• The study of logic features most
prominently in the subjects of
philosophy, mathematics, and
computer science.
• Sets are used to understand
and explain logic
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Statistics & probability
• Statistics is the study of the collection,
analysis, interpretation, presentation, and
organization of data
• Probability is the measure of the
likeliness that an event will occur
• Both areas play a key role in planning
designing strategic decision making
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Calculus
• Calculus is the mathematical study of
change
• The fundamental concept behind calculus is
“Limits”
o In mathematics, a limit is the value that a function or sequence
"approaches" as the input or index approaches some value.
• Today, calculus has widespread uses in
science, engineering and economics and
can solve many problems that algebra
alone cannot.
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Calculus (cont.)
• It has two major branches,
o Differential calculus
• Concerning rates of change and slopes of curves
o Integral calculus
• Concerning accumulation of quantities and the areas under
and between curves
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Skills developed by learning &
practicing mathematics
• Logical Thinking
• Identify Patterns (A key aspect of intelligence)
• Analytical Skills
• Abstraction
• Formal Representation (important in effective
communication)
• Problem solving
• Applications
o Business applications
o Engineering applications
o Day to day applications
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Additional skills tested on
math exams
• Problem Solving
• Facing Challenges
• Efficiency of thinking
• Strategy Formation and Planning
• Strategy Execution
o Concentration
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Is mathematics hard?
• Excuse me! Then how comes most 100/100 in exams
are for mathematics ?
• Mathematics is one of the easy subject to learn
• Remember mathematics papers sometimes test the
additional skills (mentioned in the previous slide)
• That is why most of the maths papers are made
challenging & complex.
• It is not because maths is hard or examiner hates you.
• He just gives you complex problems to make you
tougher, more sharp and skilled which is one of the
key to your success in real world.
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Is mathematics hard? (cont..)
“It is not that mathematics is hard, just
the examinations are made challenging
and interesting...”
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Useful Resources
• https://en.wikipedia.org/wiki/Mathema
tics
• http://philosophy.hku.hk/think/
• https://en.wikipedia.org/wiki/Outline_o
f_academic_disciplines
• http://web.cerritos.edu/dford/SitePage
s/FALL2011/dgAllConjectures.pdf
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"We are what we repeatedly
do. Excellence, then, is not an
act, but a habit."
~ Aristotle.
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Thank You !
Presented by
Nalaka Gamage
(auoracs.lk)
(www.auroracs.lk)
28

A Simple Guide to Mathematics

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A Simple Guide to Mathematics