In high school some consider mathematics a mundane subject, wondering how these different counting methods will help later in life. Math has many practical applications, having a fundamental role in novel advancements in various fields. Even for students who major in different disciplines mathematics is essential. This article will cover a few mathematical theorems and frameworks taught in high school, and some of their practical use in the modern day.

Logarithms

Logarithms are recognized as one of the most critical innovations in mathematics and scientific computation. A logarithm indicates how often a number has to be multiplied by itself to get another number. Consider how many multiples of two fits into eight. That is three, the solution to the equation log2(8) [i].

Practically, logarithms are instrumental in solving exponential functions, serving as the inverse of exponents [ii]. They help students grasp the relation between exponential processes and improve the graphical understanding of functions, introducing asymptotes, domains, range, and transformations[iii]. Graphical understanding prepares students for careers in economics, science, engineering and biology among others, where visual analysis and correct interpretations are essential.

Figure 1: Illustration of Exponential and Logarithmic functions 

Graphs can model diverse practical phenomena, such as interest and investment behaviour, bacterial population growth, or voltage in batteries. 

The most powerful feature of logarithms are their properties however. These properties simplify multiplication and move powers in front of the equation, reducing the complexity of mathematical formulas [iv]. This simplification shortens both manual calculations and reduces computational power.

Figure 2: A few key logarithmic properties (Logarithms – Product Rule, sd)

Efficient computational power is essential in different modern technologies. As such, logarithms play an important role in cryptocurrencies and blockchain technology. It can help secure transactions by using logarithmic problems for encryption and digital signatures[v]. 

Logarithms are also crucial for online search engines like Google. Search engines analyses numerous components to assign a score to each web page. Some web-developers try to artificially boost those scores. So, an additional weight is added to the web pages. Using logarithmic formulas the weight of each additional score point is diminished, mitigating artificial score inflation and providing an unbiased ranking at the top results[vi]. 

Furthermore, logarithms are used sort through billions of pages in a faction of seconds as well, going back to the computational advantages of logarithmic properties [vii].

Boolean

If you chose computer science as elective in high school, or even university you likely know of Boole or the Boolean. This refers to a branch of algebra utilizing a two number system, zero and one. When representing false and true respectively Boolean algebra represents logical reasoning, and provides a framework to simplify complex logical expressions[viii]. A simple application of Boolean algebra can be found in light bulb problems and circuit switch selection. While the framework simplifies and illustrates choice, adding more options increases the amount of logical expressions needed to represent the choices, and quickly grows the complexity of the problem again.

Figure 3: Simple Illustration of a logical expression and its growing complexity 

This simple yet exhaustive logic system is the foundation of modern computing, as computers think in zeroes and ones.

Logical expression can be found in many other places as well. Social media platforms and mobile apps need decision-making algorithms based on user interactions. User filters and behaviour determine what information is presented,  but that simply boils down to whether or not an option is selected, or whether or not interest was shown in a topic.

Boolean algebra also safeguards accounts of users. A simple logical expression can check if login credentials are correct. 

Another environment expressed by logical expressions are video games. Rules can be boiled down to strings of logical expressions . This extends to the behaviour of characters, and the way players interact with the world. Logical expressions determine when events trigger, or when a score updates. 

Calculus

Newton and Leibniz are well known names, inventing modern calculus. Calculus introduced tools to analyse limits and infinite series[ix]. These tools are used to examine problems that involve division by zero as well as other paradoxes. The mathematical framework calculus provides enables a method to study the rate of change over time and space[x]. In high school the framework is frequently used to observe the rate of change in functions, or to quantify the space around functions. 

Figure 4: Through integral calculus you can calculate area S

Functions can be used to analyse motion, or to find the optimal point for a process [xi]. It helps predict behaviour in systems [xii]. Analysing practical problems through functions supports students to develop abstract thinking, promoting cognitive development. 

Calculus has broad utilization in various fields, such as economics, engineering, science, biology, chemistry and computer science. This versatility signifies that the concepts taught in high school can be extrapolated to techniques essential to higher education, which are in turn applied to real life situations.

Engineers use calculus to improve batteries in electric vehicles. They analyse battery performance under different conditions and focus on power output, or voltage variation. This analysis identifies where the batteries and electric vehicles as a whole can be improved[xiii]. 

Another application is the optimization of charging batteries. Calculus helps analyse energy transfer rates, minimizing battery recharge time, and reduces energy consumption during a drive.

Calculus optimization techniques can also be applied to sports analytics. Through the calculus framework coaches can monitor physical exertion and fatigue, or player movement during games[xiv].  Advanced calculus can create prediction models to forecast game outcomes, or optimize training regimes using the collected data[xv].

Similarly calculus can also be used for financial modelling and risk assessment. It supports investment strategies, and can examine the market environment for optimal business strategies [xvi].

Macroeconomic analysts use calculus to assess the impact of new economic policies on the economy over different timelines. Business owners employ these models to establish company strategies. 

Prediction models can also be used to examine the validity of a stock by analysing its worth its risk, determining how much should be invested in a company, if at all [xvii].

Reflecting on the topics explored it is clear that mathematics is not a mundane subject after all. The practical applications are numerous and fundamental in laying the groundwork for students who try to pursue higher education and scientific advancements, and is important for all students to develop their logical reasoning and cognitive thinking. Mathematics fosters a skill set essential for the future, where it will still be a support system to the many novel advancements yet to come.

[i] al., J. H. (2015). Getal & Ruimte VWO B Deel 3. Noordhoff Uitgevers

[ii] Ibid 

[iii] Ibid

[iv] Ibid

[v] Oolawalebright. (2024). How to do cryptography in blockchain. Opgehaald van educative: https://www.educative.io/answers/how-to-do-cryptography-in-blockchain

[vi] Varangouli, E. (2023, June 6). Everything You Need to Know About Google PageRank. Opgehaald van Semrush Blog: https://www.semrush.com/blog/pagerank/

[vii] Ali Mohammad Zareh Bidoki, N. Y. (2007). DistanceRank: An intelligent ranking algorithm. Tehran: Elsevier. Opgehaald van https://pws.yazd.ac.ir/zareh/courses/webir/papers/DistanceRank.pdf

[viii] Monk, J. D. (2018, July 11). The Mathematics of Boolean Algebra. Opgehaald van Stanford Encyclopedia of Philosophy: https://plato.stanford.edu/entries/boolalg-math/

[ix] al., J. H. (2015). Getal & Ruimte VWO B Deel 3. Noordhoff Uitgevers

[x] Ibid 

[xi] Ibid 

[xii] James Michael Stewart, e. a. (2020). Calculus. Cengage Learning.

[xiii] EV design – battery calculation. (sd). Opgehaald van x-engineer: https://x-engineer.org/ev-design-battery-calculation/

[xiv] D, S. K. (2017, December 5). How to apply calculus in sports to improve performance? Opgehaald van Digitash: https://digitash.com/engineering/mathematics/how-to-apply-calculus-in-sports/

[xv] Saini, R. (2023, October 13). How Does Maths Make Sports Better For Players? Opgehaald van Careers360: https://www.careers360.com/premium/explore-the-magic-of-mathematics-in-sports-analytics

[xvi] Heij Christiaan, e. a. (2004). Econometric Methods With Applications in Business and Economics. Athenaeum Uitgeverij.

[xvii] Ibid