Counting on applications

Diagram 1: Various Waveforms

The advancement of technology depends very much on mathematical inputs and solutions.

IT may not be so straightforward to explain the applications of mathematics to a non-science background layman because the mathematical theory of an engineering product is quite complicated.

For example, the mathematical theory behind a computer and a smart phone is Boolean algebra and state machines. It will take time and knowledge to understand the details of the theory.

Interestingly, countries such as USA, Korea, Japan and China are able to develop advanced engineering products, and one of the main reasons is that they have many experts with strong mathematical knowledge.

The USA has the largest number of mathematicians in the world. Knowledge and understanding in Mathematics is key to many applications in life especially in the field of engineering, information technology, software development and many more.

Fast Fourier transform, wavelet transform, error detection and correction techniques are all about mathematics. They are behind modern digital audio, picture and video compression technology. We are enjoying MP3, JPEG, and MP4 all because of these signal processing mathematical knowledge.

See Diagram 1

Two mathematical subjects – Probability Theory and Statistics – are used in many places including the insurance industry, health and medical industries and marketing survey industries. The statistical analysis of public health data is the foundation for national health management.

National health institutes will normally carry out a survey on the lifestyle of the general public to learn how our lifestyles are affecting our health or to learn the distributions of certain diseases.

With the outcomes from the data analysis, our government can implement national health budget and control by allocating the right amount of budget in support of public health.

In medical industries, sophisticated design and experimental techniques from statistics are required to make sure the new medicines are effective with minimal experiments.

To understand how a medicine works, for example in the medication for suppressing HIV viruses, sometimes very complicated dynamical system modelling involving differential equations may be used.

Even graph theory, a mathematical subject considered to be “pure”, has been applied to kidney matchmaking and have saved many lives.

Diagram 2: Basic components of Hodgkin–Huxley-type models.

See Diagram 2

Mathematics is not only employed in engineering and medical industries, but also in environmental science to study the distribution of species, the changes in climates and modelling the propagation of pollution.

With more and more human activities, pollution and deforestation, statistical analysis shows that many subspecies have gone extinct and many species are on the brink of extinction.

Scientists are trying to build more accurate models to understand the dangers of pollution, global warming and the deforestation to human population growth and economic growth.

So far, there are very few results because the mathematical models are either too complicated or too simple to provide proper predictions.

The current engineering practice does not take environmental issues and sustainability into account as yet.

The higher the integration of engineering components in an engineering product, the lower the re-usability of the product.

For example, the modern smart phone is so compact that any damage in a particular part makes the whole product unusable, causing it to be thrown away.

However, there are many heavy metal elements in a smart phone, so, unless it is disposed of properly, it may become a pollution nightmare in the future.

New mathematics that allow easy fault detection and correction needs to be developed and mathematicians, physicists, chemists need to work together to find better ways to process these waste products.

Environmental issues belong to fundamental science problems that can only be resolved by going back to first principles, where mathematics plays a fundamental role.

Diagram 3: The mathematics behind computer aided design is computational geometry.

See Diagram 3

Knowledge of high school mathematics is insufficient to support the above applications. Much more advanced mathematics is required. The foundation to advanced mathematics is university mathematics. It has inherited the culture of mathematics, founded in ancient Greece, with logic as the foundation. Since then, mathematics has become a subject where fascinating theorems are derived with a set of axioms and rules.

A major achievement in mathematics was the book titled The Elements, written by Euclid.

This has been the mathematical textbook until the 17th century, when Newton and Leibniz developed the fundamental theorem of calculus.

Newton and Leibniz used the idea of “infinitesimal small” to define differentiation, however they were not able to distinguish “infinitesimal small” and zero properly and faced criticism for the lack of rigour.

The problem with the lack of rigour is that awkward results will arise.

It was until Cauchy introduced the notion of “limit” that a foundation of calculus was developed and since then students are forced to work with the notion of limits and applying proper proving methods at university mathematics.

Through this training, students familiarise themselves with the mathematical objects and logical rules in scientific disciplines.

There is an opinion that university mathematics focuses too much on theory rather than practical knowledge.

This is a very wrong perception because university subjects are a good balance of theoretical knowledge and practical knowledge.

The theoretical knowledge is for students to build higher order thinking skills such as abstraction and to be able to work with abstract mathematical models.

We always say that to build a skyscraper, we need a solid foundation. Similarly, university mathematics is the solid foundation for advanced mathematics, which is core to many of the modern technologies.

The role of mathematics in the university is to provide the toughest training to students both in understanding the importance of logical thinking and try to relate basic science with mathematics.

Students who have gone through such training can be said to equip themselves with the ability to adapt to any new technologies, which have complicated mathematics behind them.

The founders of Google are PhD candidates who researched on the dominant eigenvector problem of a probability matrix.

They succeeded in employing this mathematical model to ranking search results and created one of the most important IT companies.

However, they have never stopped improving their search engine with more sophisticated mathematical models.

Other companies such as Netflix are employing advanced data mining techniques which require advanced knowledge of probability theory and statistics to analyse the huge amount of user data they have on their servers to understand the user preference and to advertise based on users’ preference.

So, it is not surprising that countries best in university mathematics were the best in science and engineering.

In Malaysia, we actually need more people to study the fundamental sciences such as mathematics, physics and chemistry, essential knowledge in a modern information technology society.

To advance our technologies, students who enjoy mathematics should preferably study university mathematics, to learn to master modern technologies and create future technologies.

The writer Dr Liew How Hui is assistant professor with UTAR’s Department of Mathematical and Actuarial Sciences. This article is one in a series of STEM articles from UTAR.