# The Catalyst Behind Novelty

Illustration by Mahela Cabrini

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.