# Pursuit of the Unknown: 17 Equations that Changed the World

From time immemorial, gifted people have produced mathematical masterpieces that have changed the world. In this article, we look into the genius behind equations.

These equations has not only unraveled mysteries of the universe but also revolutionised various fields, demonstrating the extraordinary power of mathematical thought in shaping the world as we know it.

## 1. The Pythagorean Theorem

At its core, this theorem explores the relationship between the sides of a right triangle and has had a profound impact on:

• Geometry,
• Physics, and
• Various scientific disciplines.

Attributed to the ancient Greek mathematician Pythagoras, the theorem states that:

in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In simple terms, it’s represented as a^2 + b^2 = c^2, where ‘a’ and ‘b’ are the lengths of the triangle’s legs, and ‘c’ is the length of the hypotenuse.

In Albert Einstein’s theory of general relativity, the concept of spacetime curvature relies on mathematical models involving tensors and differential equations, some of which have their roots in the Pythagorean Theorem.

The theorem’s applicability extends to wave equations, such as the Fourier transform.

French mathematician Baron Jean-Baptiste Joseph Fourier used this transformative equation to break down complex data into combinations of simple waves, making it easier for researchers to analyze:

• Sound waves,
• Electromagnetic fields, and even
• Quantum mechanics.

## 2. The square root of -1

The square root of -1, represented as √(-1) or simply ‘i,’ is a fundamental part of complex numbers.

While you can’t have a real number that is the square root of a negative, introducing ‘i’ opened up new avenues in mathematical equations that changed the world.

It allows mathematicians and scientists to solve problems previously thought to be unsolvable.

For instance, in electrical engineering, complex numbers are used to analyze alternating current circuits, making it possible to understand and manipulate the behavior of electric currents and voltages in a more elegant way.

Moreover, ‘i’ plays a crucial role in the field of signal processing. Complex numbers and ‘i’ are indispensable tools for breaking down and analyzing these waves:

• Sound waves,
• Electromagnetic waves, or even the
• Fourier transform

## 3. Calculus

Calculus was born from the brilliant minds of mathematicians like Sir Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. It is about understanding how things change. It deals with concepts like:

• Limits,
• Derivatives, and
• Integrals,

enabling us to analyse the behaviour of variables over time and space.

One of the equations that changed the world is the fundamental theorem of calculus. This theorem provides a bridge between two seemingly distinct aspects of calculus:

• Differentiation: finding rates of change
• Integration: finding accumulated quantities.

It’s like a mathematical connector that has revolutionised fields such as physics, economics, and engineering.

In economics, calculus plays a pivotal role in modelling supply and demand curves, optimising production processes, and analysing financial markets. It allows us to make informed decisions in the complex world of economics.

## 4. Logarithms

Logarithms are a powerful mathematical tool that allows us to convert complex multiplicative processes into simpler additive ones.

They provide a means to compress vast ranges of numbers into more manageable scales. This mathematical breakthrough has played a pivotal role in various scientific fields.

One of the prime examples of logarithms in action is the Richter scale, which measures the intensity of earthquakes. The scale, named after its creator Charles F. Richter, is logarithmic in nature.

Each whole number increase on the Richter scale corresponds to a tenfold increase in the amplitude of seismic waves.

This means that a 6.0 magnitude earthquake is ten times more powerful than a 5.0 magnitude earthquake.

Logarithms make it easier to express the vast differences in earthquake energy in a comprehensible way.

Logarithms is also instrumental in the world of finance. The Black-Scholes equation, which revolutionized options pricing, relies on logarithmic transformations to calculate the theoretical value of financial derivatives.

## 5. Law of Gravity

The law of gravity, famously formulated by Sir Isaac Newton in the 17th century, is a mathematical equation that describes the force of attraction between two objects with mass.

It’s often summarised as “what goes up, must come down,” but its implications are far more profound.

Newton’s equation for gravity, known as the law of universal gravitation, allowed scientists and mathematicians to calculate the motion of celestial bodies with remarkable precision.

For instance, it enabled astronomers to predict the positions of planets in the solar system and even discover new celestial bodies like Neptune through perturbations in their orbits.

The law of gravity also played a pivotal role in the development of Albert Einstein’s theory of general relativity.

Einstein’s theory expanded our understanding of gravity by describing it as the curvature of space and time, leading to groundbreaking insights into the nature of the universe itself.

Moreover, without the law of gravity, the concept of:

• Weight,
• mass, and
• even the second law of thermodynamics

would not be as comprehensible as they are today. It’s a cornerstone of classical mechanics, essential for understanding the motion of objects on Earth and beyond.

## 6. Navier-Stokes Equations

Have you ever marveled at the graceful flow of a river, the turbulence of a storm, or the delicate flight of a butterfly?

These phenomena, governed by the intricate laws of fluid dynamics, find their mathematical expression in the Navier-Stokes equations.

Named after Claude-Louis Navier and George Gabriel Stokes, the Navier-Stokes equations are a set of partial differential equations that describe the behavior of fluid flow.

These equations take into account factors like:

• Viscosity,
• Pressure, and
• Velocity,

offering a comprehensive framework for understanding the complex motion of fluids, whether they are gases or liquids.

One of the most compelling aspects of the Navier-Stokes equations is their ability to simulate and predict a wide range of fluid dynamics scenarios.

For instance, they are crucial in weather forecasting, allowing meteorologists to model the movements of air masses and predict the path of hurricanes and storm systems.

Moreover, the Navier-Stokes equations have played a crucial role in the development of computational fluid dynamics (CFD), a branch of engineering that uses mathematical models and simulations to analyze fluid flow and heat transfer in various applications.

## 7. Euler’s Polyhedra Formula

Named after the Swiss mathematician Leonhard Euler, this formula relates the number of:

• Vertices (corners),
• Edges (sides), and
• Faces of a polyhedron (a three-dimensional geometric shape).

Euler’s Polyhedra Formula is elegantly expressed as V – E + F = 2, where V represents vertices, E represents edges, and F represents faces.

But what makes this seemingly simple equation so powerful? It turns out that Euler’s formula has far-reaching implications for understanding complex shapes and their properties.

For instance, it has been instrumental in the study of crystal structures, where the formula helps determine the number of distinct crystalline arrangements in various minerals.

Moreover, in topology, a branch of mathematics concerned with the properties of space and continuity, Euler’s formula plays a pivotal role in classifying surfaces.

It provides a framework for distinguishing between different types of surfaces based on their topological properties, allowing mathematicians and scientists to explore the characteristics of higher-dimensional spaces.

In the field of graph theory, which deals with networks and connections, Euler’s formula serves as a foundation for understanding and analyzing complex networks, from social interactions to computer networks.

## 8. Wave Equation

The wave equation is a mathematical equation that describes the behavior of waves as they propagate through space and time.

It captures the essence of wave phenomena, from the way sound travels through air to the propagation of electromagnetic waves, including radio waves and light.

One of the groundbreaking applications of the wave equation is in the field of telecommunications. It enables us to design and optimize the transmission of information through various types of waves.

For instance, it underpins the technology behind fiber-optic communication, allowing you to send messages and data across long distances at the speed of light.

In the realm of medical imaging, the wave equation plays a vital role in techniques like ultrasound and MRI, providing detailed images of the human body’s interior. Doctors and researchers rely on these technologies for diagnosis and treatment planning.

## 9. Normal distribution

You may heard of the more common name of Normal Distribution, the bell curve.

It is a mathematical concept that describes the distribution of data in a symmetrical pattern, resembling a bell.

This distribution is characterized by its mean (average) and standard deviation, which determine the curve’s shape and spread.

One of the most remarkable applications of the normal distribution is in the field of statistics. It serves as the foundation for inferential statistics, allowing us to make predictions and draw conclusions about populations based on sample data.

Whether you’re:

• Conducting opinion polls,
• Analysing test scores, or
• Studying the heights of a population

the normal distribution plays a crucial role in understanding the data.

In the realm of quality control and manufacturing, the normal distribution guides processes to maintain consistent product quality.

It helps identify defects and deviations from the norm, ensuring that products meet the desired specifications. This application is evident in industries ranging from automotive to pharmaceuticals.

Furthermore, the normal distribution is the cornerstone of risk management and finance. It underpins the Black-Scholes equation, a fundamental tool in options pricing, and is essential for assessing and managing financial risks in investment portfolios.

## 10. Fourier Transform

At its core, the Fourier Transform allows us to break down complex and messy data into combinations of simple waves.

Fourier Transform extends far beyond the confines of mathematics. In the field of quantum mechanics, it bridges matter and energy, offering insights into the behavior of atoms and subatomic particles.

This tool has been instrumental in enhancing our understanding of the universe, from the characteristics of an expanding universe, as explored by Russian physicist Alexander Friedmann, to the intricate dance of planets in the solar system.

Fourier Transform’s role in transforming complex phenomena into manageable equations. This transformation is evident in fields like information theory, where it simplifies the handling of large data sets, a cornerstone of modern communication technologies.

The Fourier Transform has a significant role in medical imaging, particularly in MRI technology. By turning complex signals from the human body into clear images, it aids in the early detection and treatment of diseases.

## 11. Information Theory

Information Theory revolutionized how you perceive data transmission and processing. Coined by Claude Shannon in the mid-20th century, it tackles the management of information, specifically:

• Quantification,
• Storage, and
• Communication of information.

What makes Information Theory a standout is its practical application in modern technology. It underpins the digital revolution.

Think about your smartphone, internet, and digital television; they all rely on principles derived from Information Theory. It also paved the way for developments in quantum mechanics and general relativity.

Information Theory introduced key concepts like ‘bit’, the basic unit of information, which transformed the way data is measured and managed. It led to the development of more efficient data compression algorithms, crucial for high-speed data transmission.

The theory also plays a crucial role in cryptography, ensuring secure communication in the digital age. It’s fascinating to consider that the secure websites you visit daily use principles based on Information Theory.

## 12. Relativity

Relativity in mathematics, particularly Einstein’s theory of relativity, is a cornerstone that reshaped our understanding of space, time, and gravity.

Unlike the Euclidean geometry you learned in school, where the sides of a right triangle and the angles of a right angle are constant, relativity introduced a universe where these can bend and warp.

This theory, comprising special and general relativity, goes beyond the Pythagorean theorem or Newton’s law of universal gravitation.

It shows how massive objects like planets cause a curvature in space-time, leading to what you experience as gravity. This was a radical shift from the classical mechanics of Newton.

Einstein’s equations have changed the world, influencing everything from GPS systems in your smartphone to understanding the orbits of planets in the solar system.

The theory’s implications extend to predicting phenomena like black holes and gravitational waves, once mere theoretical concepts, now proven realities.

In the pursuit of the unknown, relativity has been a guiding light, reshaping the fundamental principles of physics and altering your perception of the universe’s very fabric.

## 13. Schrodinger’s Equation

Unlike the familiar Newtonian physics, where objects follow predictable paths, Schrödinger’s Equation delves into the probabilistic nature of particles at the quantum level.

This mathematical model provides a wave equation that describes the quantum state of a system.

The equation’s beauty lies in its ability to predict the likelihood of finding a particle in a certain position, a stark contrast to the certainty of classical mechanics.

Its impact is profound and far-reaching. Schrödinger’s Equation is crucial for the development of modern electronics. The transistors in your smartphone and computer, the backbone of digital technology, rely on principles derived from this equation.

It also plays a pivotal role in the field of chemistry, particularly in understanding chemical bonds and the behavior of electrons in atoms.

Schrödinger’s Equation also opened doors to new realms in physics, influencing theories from the microscopic world of particles to the vast expanse of the universe.

It is a cornerstone in the pursuit of the unknown, offering a mathematical basis for the complex interactions of matter and energy.

## 14. Second Law of Thermodynamics

The Second Law of Thermodynamics states that the total entropy of a closed system can never decrease over time. It signifies that systems will naturally progress towards a state of maximum entropy or disorder.

This law reshaped our understanding of energy and its transformations. It’s a key player in the pursuit of the unknown, providing a fundamental understanding of the universe’s behavior, from the microscopic level of atoms to the vastness of stars.

One of the most intriguing applications of the Second Law is in the field of information theory. It relates to the concept of entropy in a system, offering insights into data transmission and encryption.

In a sense, it’s about understanding the order and disorder in the flow of information, a crucial aspect of modern digital communication.

The Second Law is also pivotal in engineering and technology, particularly in the design of engines and refrigerators.

It explains why you can’t build a perpetual motion machine, a device that would function indefinitely without an energy source. This understanding is crucial in developing efficient and sustainable technologies.

## 15. Maxwell’s Equations

Maxwell’s Equations is a set of four mathematical equations. Formulated by James Clerk Maxwell, they unified

• Electricity,
• Magnetism, and
• Optics

into a single, coherent theoretical framework.

Before Maxwell, electricity and magnetism were seen as distinct phenomena. His equations showed that electric and magnetic fields are intertwined, and light itself is an electromagnetic wave.

This revelation had profound implications, altering the course of modern physics and technology.

One of the most tangible impacts of Maxwell’s Equations is in the realm of communication technology. They laid the groundwork for the development of:

• Television, and
• Mobile communications.

Every time you use your smartphone or tune into a radio station, you’re witnessing Maxwell’s theories in action.

Maxwell’s theories also foundational in the study of quantum mechanics and general relativity, bridging the gap between classical physics and the bizarre world of quantum phenomena.

## 16. Black-Scholes Equation

Developed by economists Fisher Black and Myron Scholes, the equation provides a theoretical estimate of the price of European-style options.

Before its advent, pricing options was more art than science, fraught with guesswork and uncertainty.

This model brought a new level of precision, turning the chaotic world of options trading into a more ordered universe. This equation’s impact echoes through various facets of economics and finance,.

It lays the mathematical basis for a significant part of the modern financial industry, influencing everything from risk management strategies to the way hedge funds operate.

What’s remarkable about the Black-Scholes model is its simplicity and depth. It’s a perfect blend of advanced concepts from stochastic processes, yet its application is straightforward enough to be indispensable for traders and financial analysts.

This blend of complexity and accessibility is a trait shared by many mathematical discoveries that have profoundly shaped our understanding of the universe.

## 17. Chaos Theory

Chaos theory revolutionizes how we perceive order and predictability in seemingly random systems. Think of:

• Weather patterns,
• Stock market, or
• The orbits of planets in the solar system

Chaos Theory provides a lens to view the intricate, often unpredictable behaviors of these systems.

At the heart of Chaos Theory lies the idea that small changes can have large consequences, a concept popularly known as the ‘butterfly effect.’ This theory defies the classical laws of motion, where predictability and determinism reigned supreme.

Instead, it aligns more with the principles of quantum mechanics and general relativity, which embrace the complexity and unpredictability of the universe.

In meteorology, it helps in understanding and predicting weather patterns, which are inherently chaotic. In the realm of biology, it aids in deciphering the complex dynamics of ecosystems.

Even in the field of economics, Chaos Theory has shed light on market dynamics, which, like the weather, can be unpredictable and influenced by a multitude of factors.

## 17 Equations That Changed The World

The equations are much more than mathematical formulas; they are the keystones of our scientific and intellectual heritage. They have not only illuminated paths to previously uncharted realms of knowledge but also profoundly influenced our daily lives.

These equations stand as a testament to human ingenuity, reminding us that the pursuit of understanding is a journey that continually shapes and redefines our world.

The Author

Dr Andrew Stapleton has a Masters and PhD in Chemistry from the UK and Australia. He has many years of research experience and has worked as a Postdoctoral Fellow and Associate at a number of Universities. Although having secured funding for his own research, he left academia to help others with his YouTube channel all about the inner workings of academia and how to make it work for you.