The algebraic method of deriving general solutions to problems is one of the most rewarding parts of mathematics as it allows knowledge to be built on by thinking outside the box. Calculus and algebra often allow me to be innovative with mathematical ideas and draw on my knowledge in order to solve problems and overcome challenges.
One of my favourite examples of mathematical equations being manipulated for a solution is in the derivation of more complex equations from known ones. I found it particularly interesting to derivate the force and angular acceleration equations into the natural frequency of a pendulum system, as well as a spring-mass system.
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Another major reason why I love algebraic problem solving is because it is applicable to such a broad range of topics. Another example of algebraic topics that I find extremely interesting matrices ranging all the way from finding the inverse of a singular matrix with a singular unknown, to using matrices to represent interceptions of planes. In the book “Physics of the Future” by Michio Kaku, one of the subchapters talks about the idea of Moore's Laws, and how they have been predicting the exponential increase in the number of transistors in silicon-based computers for the last fifty years. Moore's law suggests that every two years the number of transistors will increase by a factor of two, and the price of each transistor will halve at the same rate, maintaining a constant price to an exponential growth rate. The major issue is that as transistors get smaller and smaller, there will eventually be a physical limit, leading to the end of Moore’s Law which will create a stagnation on processing power, unless a new procedure is developed and mastered before then. This is extremely exciting and motivating as it requires a new era of trial and research which has already started, with quantum and neuromorphic computing currently being the two leading ideas.
This last summer, I got the exciting opportunity to work with a local engineering company initially for work experience, and was later offered a summer job in machining, assembly, and some low-level design of pieces using CAD software. One of the tasks I was charged with was the assembly of a batch of mechanical door sets, within a time constraint. The set was a junction of a bearing system glued to the outer shell of the lock, with an internal spring system in order to create slight resistance while turning the lock. The system was challenging due to malfunctions on the adhesion of the bearing to the outer shell, but I was able to rapidly find a solution and test all parts for proper function before the final deadline, which was both challenging and rewarding. At a recent “Ingenia live!” event organised by the Royal Academy of Engineering I heard of technological advancements that are being made by governments and private companies on space exploration, highlighting challenges (such as the prohibition of biological materials in the construction of planetary rovers due to their invasiveness to the planet) and benefits (such as satellite-based internet connection). The event was extremely informative due to the expertise of the panel, which gave me a much more realistic insight into the profession.
I was also part of a STEM based program called the Brilliant Club in which a small group were introduced to a mentor (a university student doing his masters in the specific program). As part of this problem, I was required to write an academic-style paper in which the information had to be expressed with clarity and with clear and accurate citations.