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An overview of mathematics; its branches and learning

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- 1. A Simple Guide to Mathematics Compiled By Nalaka Gamage Aurora Computer Studies (auoracs.lk) Aurora Computer Studies (www.auroracs.lk) 1
- 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 Aurora Computer Studies (www.auroracs.lk)
- 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. 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. 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. 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. 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. 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. 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. 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. 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. Theorems • Once a conjecture has been proven, it becomes a theorem • Example: o Pythagoras theorem Aurora Computer Studies (www.auroracs.lk) 13
- 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. 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. 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. Geometry • Concerned with questions of shape, size, relative position of figures, and the properties of space. Aurora Computer Studies (www.auroracs.lk) 17
- 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. 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. 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. 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. 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. 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. 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. 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. 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. "We are what we repeatedly do. Excellence, then, is not an act, but a habit." ~ Aristotle. Aurora Computer Studies (www.auroracs.lk) 27
- 28. Thank You ! Presented by Nalaka Gamage Aurora Computer Studies (auoracs.lk) Aurora Computer Studies (www.auroracs.lk) 28

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