Pi......From a Student's Perspective

Examiners in the HSC Advanced Mathematics (2 unit) course have commented that many students become confused when the constant 𝜋 is used in calculations. What is it about 𝜋 that causes problems? Students from an early age learn that 𝜋 is the ratio of the circumference of a circle to its diameter and so this reinforces the geometrical interpretation of 𝜋. However, 𝜋 can be involved in problems that have no connection with circles and this is when confusion in examination answers can arise. Some students convert an answer involving 𝜋 into revolutions or degrees just because of the  connection of 𝜋 to the geometry of a circle. Pi is a number, an irrational number, and should be treated like any other number in calculations. In the 2015 Mathematics examination question 15 c involved calculating the time when the volume of water in a pool first started to decrease given the rate of change of the volume. The answer was 2𝜋 seconds and many candidates lost marks by converting this into degrees.

“The fact that 𝜋, a purely geometric ratio, could be evolved out of so many arithmetic relationships- out of infinite series, with apparently little or no relation to geometry-was a never ending source of wonder and a never ending stimulus to mathematical activity”

— Mathematics and the Imagination by Edward Kasner and James Newman, G Bell and Sons 1951, page 78

Energy and Entropy

The word energy is widely used in everyday life. "I do not have the energy to exercise today" or "the world has an energy crisis" are familiar sayings. We use the word energy so much that it is taken for granted that we know what it means (?). The first law of thermodynamics tells us that energy can neither be created nor destroyed. When hot and cold water are mixed together the heat energy lost by the hot water equals the heat energy gained by the cold water. Another word in the vocabulary of physicists and engineers is entropy. Entropy is an indicator of the disorder of a system and the second law of thermodynamics states that natural processes tend to move toward a state of greater disorder (entropy) as time passes. The entropy of a system can never decrease. Students sometimes think that entropy is mysterious since it can be created, whereas energy cannot. When hot and cold water are mixed together the entropy of the hot water decreases and the entropy of the cold water increases by a greater amount. The total entropy of the mixture increases during the process. The mixing process created the entropy. The mixture does not naturally separate back into hot and cold components as this would involve a decrease in total entropy. Entropy is nature's arrow, always increasing, pointing in the direction in which systems evolve in time.

“If, when we observed a highly irreversible process (such as cooling coffee by placing an ice cube in it), we said, “That surely increases the entropy,” we would soon be as familiar with the word entropy as we are with the word energy.”

— Fundamentals of Classical Thermodynamics by Richard E Sonntag and Gordon J Van Wylen, 3rd edition page 232