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A Simple Guide to Mathematics

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An overview of mathematics; its branches and learning

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A Simple Guide to Mathematics

  1. 1. A Simple Guide to Mathematics Compiled By Nalaka Gamage Aurora Computer Studies (auoracs.lk) Aurora Computer Studies (www.auroracs.lk) 1
  2. 2. Agenda • Give you the big picture of mathematics • Why do we learn mathematics? • Branches of mathematics • Myths & challenges in learning • New trends in learning Aurora Computer Studies (www.auroracs.lk) 2
  3. 3. 3 Aurora Computer Studies (www.auroracs.lk)
  4. 4. 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 Aurora Computer Studies (www.auroracs.lk) 4
  5. 5. 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 Aurora Computer Studies (www.auroracs.lk) 5
  6. 6. 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. Aurora Computer Studies (www.auroracs.lk) 6
  7. 7. 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. Aurora Computer Studies (www.auroracs.lk) 7 Ref: http://philosophy.hku.hk/think/logic/whatislogic.php
  8. 8. 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. Aurora Computer Studies (www.auroracs.lk) 8 Ref: http://philosophy.hku.hk/think/logic/whatislogic.php
  9. 9. 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. Aurora Computer Studies (www.auroracs.lk) 9
  10. 10. 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 Aurora Computer Studies (www.auroracs.lk) 10
  11. 11. 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 Aurora Computer Studies (www.auroracs.lk) 11
  12. 12. 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. Aurora Computer Studies (www.auroracs.lk) 12
  13. 13. Theorems • Once a conjecture has been proven, it becomes a theorem • Example: o Pythagoras theorem Aurora Computer Studies (www.auroracs.lk) 13
  14. 14. 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 Aurora Computer Studies (www.auroracs.lk) 14
  15. 15. 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 Aurora Computer Studies (www.auroracs.lk) 15
  16. 16. 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 Aurora Computer Studies (www.auroracs.lk) 16
  17. 17. Geometry • Concerned with questions of shape, size, relative position of figures, and the properties of space. Aurora Computer Studies (www.auroracs.lk) 17
  18. 18. 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 Aurora Computer Studies (www.auroracs.lk) 18
  19. 19. 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 Aurora Computer Studies (www.auroracs.lk) 19
  20. 20. 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. Aurora Computer Studies (www.auroracs.lk) 20
  21. 21. 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 21
  22. 22. 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 Aurora Computer Studies (www.auroracs.lk) 22
  23. 23. Additional skills tested on math exams • Problem Solving • Facing Challenges • Efficiency of thinking • Strategy Formation and Planning • Strategy Execution o Concentration Aurora Computer Studies (www.auroracs.lk) 23
  24. 24. 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. Aurora Computer Studies (www.auroracs.lk) 24
  25. 25. Is mathematics hard? (cont..) “It is not that mathematics is hard, just the examinations are made challenging and interesting...” Aurora Computer Studies (www.auroracs.lk) 25
  26. 26. 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 Aurora Computer Studies (www.auroracs.lk) 26
  27. 27. "We are what we repeatedly do. Excellence, then, is not an act, but a habit." ~ Aristotle. Aurora Computer Studies (www.auroracs.lk) 27
  28. 28. Thank You ! Presented by Nalaka Gamage Aurora Computer Studies (auoracs.lk) Aurora Computer Studies (www.auroracs.lk) 28

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