a simple guide to mathematics

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A Simple Guide to Mathematics

Compiled By Nalaka Gamage

Aurora Computer Studies(auoracs.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.

Ref: http://philosophy.hku.hk/think/logic/whatislogic.php

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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.

Ref: http://philosophy.hku.hk/think/logic/whatislogic.php

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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 mathematicso 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 applicationso Engineering applicationso 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/Mathematics

• http://philosophy.hku.hk/think/• https://

en.wikipedia.org/wiki/Outline_of_academic_disciplines

• http://web.cerritos.edu/dford/SitePages/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 GamageAurora Computer Studies(auoracs.lk)