Arithmetic is the very first topic covered in most Secondary One mathematics. Students learn about factors, primes, prime factorisation, highest common factor and lowest common multiple. One application of prime factorisation is finding square roots and cube roots of whole numbers. For example, to find the square root of 2025, we express 2025 as product of primes 34 x 52, from which we obtain the answer 32 x 5. On the other hand, the square root of 200 is not a whole number because 200 = 23 x 52 and the power of 2, which is 3, is odd. In contrast, we cannot say that the square root of 324 = 27 x 12 is not a whole number because the power of 27 is odd. Of course, we argue that we cannot make this conclusion because 27 x 12 is not a prime factorisation of 324. So what is so special about prime numbers? Why do we have to write a whole number N in prime factorisation form to determine conclusively whether the square root of N is a whole number or not? In this course, we shall discuss the mathematical principle behind this, and also explain why the algorithm for finding HCF and LCM using prime factorisation works.
We shall also touch on representation of real numbers in decimals. What is the difference between rational and irrational numbers written in decimals? How do we know whether a fraction m/n written in decimal is terminating or non-terminating (2/5 = 0.4 is terminating, 2/3 = 0.666… is non-terminating)? These questions can be answered using elementary number theory.
We also discuss some real-life applications of number theory; for example, how is the official reference of NRIC No., or the ISBN of books, obtained? We shall share some interesting mathematical puzzles involving number theory. For example, can you get four gallons of water using five and three gallon jugs with no markings? (In the movie “Die Hard 3”, John McClain (played by Bruce Willis) and Zenus (played by Samuel L. Jackson) have to solve this puzzle posed by the villain Peter Krieg (played by Jeremy Irons) in order to defuse a bomb. Had the villain asked them to get four gallons using three and six gallons jugs, would they be able to solve it?)
Learning Dimension: Deepening Knowledge and Practice in Academic Curriculum