Let’s take a look at some important ideas from the world of math that have changed the way we understand things.

Mathematical equations are like special lenses that allow us to see the world in a different light. They help us understand reality and discover new perspectives on things we may not have noticed before.

It’s fascinating how advancements in math often coincide with our deeper understanding of the universe. Let’s explore nine historical equations that have completely transformed our view of everything, from the tiniest particles to the expansive cosmos.

Pythagorean theorem

One of the fundamental trigonometric principles taught in schools concerns the relationship between the sides of a right triangle. It states that the square of the length of each of the shorter sides, when added together, equals the square of the length of the longest side. Expressed as a^2 + b^2 = c^2, this concept has been recognized for over 3,700 years, originating from the ancient Babylonians.

The Greek mathematician Pythagoras is traditionally attributed with formalizing the version of the equation we use today, as noted by the University of St. Andrews in Scotland. Besides its applications in construction, navigation, and mapmaking, the Pythagorean theorem played a crucial role in shaping our understanding of numbers.

In the fifth century B.C., mathematician Hippasus of Metapontum made a significant observation. He realized that an isosceles right triangle with base sides of 1 unit in length would yield a hypotenuse equal to the square root of 2, an irrational number.

This discovery was groundbreaking, as it introduced the concept of numbers that extend infinitely after the decimal point without repeating. Legend has it that Hippasus faced dire consequences for his revelation, allegedly being thrown into the sea by followers of Pythagoras who were troubled by the existence of such numbers, as documented by the University of Cambridge.

F = ma and the law of gravity

British luminary Sir Isaac Newton is celebrated for numerous groundbreaking discoveries, including his second law of motion. This law asserts that force equals the mass of an object multiplied by its acceleration, expressed as F = ma.

Building upon this principle, Newton formulated his law of universal gravitation in 1687, which states that the force between two objects is determined by their masses and the distance between them, represented by the equation F = G (m1 * m2) / r^2.

Here, m1 and m2 denote the masses of the objects, r signifies the distance separating them, and G is a crucial constant derived through experimentation.

These concepts have profoundly influenced our comprehension of various physical phenomena, such as planetary motion within the solar system and the principles governing space travel using rockets.

The wave equation

In the 18th century, scientists embraced Newton’s innovative laws to explore the world around them. In 1743, French polymath Jean-Baptiste le Rond d’Alembert formulated an equation to describe the vibrations of an oscillating string or the motion of a wave, as documented in a 2020 paper in the journal Advances in Historical Studies. This equation can be expressed as follows:

1/v^2 * ∂^2y/∂t^2= ∂^2y/∂x^2

In this equation, v represents the velocity of a wave, while the other components describe the wave’s displacement in a single direction. When applied across two or more dimensions, the wave equation enables scientists to forecast the movement of water, seismic waves, and sound waves.

Moreover, it serves as the foundation for equations like the Schrödinger equation in quantum physics, which forms the basis of numerous modern computer-based devices.

Fourier’s equations

Even if you’re not familiar with the name, the contributions of French baron Jean-Baptiste Joseph Fourier have impacted your life. His mathematical equations, formulated in 1822, have enabled researchers to deconstruct complex data into simpler wave combinations, making analysis much more manageable.

Initially met with skepticism, Fourier’s transformative idea challenged the belief that intricate systems couldn’t be simplified elegantly. Today, Fourier transforms are indispensable tools in various scientific fields, including data processing, image analysis, optics, communication, astronomy, and engineering.

Maxwell’s equations

In the 1800s, the study of electricity and magnetism was in its infancy, as scholars sought to grasp and utilize these mysterious forces. Scottish scientist James Clerk Maxwell made significant strides in our comprehension of both phenomena in 1864 when he unveiled a series of 20 equations outlining the workings and connections between electricity and magnetism.

These equations were later condensed to four, now fundamental to the curriculum of first-year physics students and serving as the foundation for virtually all electronic technologies in our modern world.

E = mc^2

Albert Einstein’s iconic equation, E = mc^2, stands as a cornerstone in the realm of transformative equations. Introduced in 1905 as part of his groundbreaking theory of special relativity, it unveiled the intrinsic unity between matter and energy. In this equation, E symbolizes energy, m denotes mass, and c represents the constant speed of light.

While the profound implications embedded within this seemingly simple formula remain challenging for many to grasp, its significance is undeniable. Without E = mc^2, our comprehension of celestial bodies, the mechanics of the universe, and the construction of colossal particle accelerators like the Large Hadron Collider would be severely limited.

Friedmann’s equations

It may seem audacious to attempt to encapsulate the vast expanse of the cosmos within a set of equations, yet Russian physicist Alexander Friedmann did just that in the 1920s. Building upon Einstein’s theories of relativity, Friedmann demonstrated that the dynamics of an expanding universe could be comprehensively described from the moment of the Big Bang onward through two equations.

These equations amalgamate all the crucial elements of the cosmos, encompassing its curvature, its mass and energy content, its rate of expansion, and various fundamental constants such as the speed of light, the gravitational constant, and the Hubble constant, which signifies the universe’s accelerating expansion.

Einstein famously resisted the notion of an expanding or contracting universe, which his theory of general relativity implied would occur due to gravitational effects. He attempted to introduce a variable represented by the Greek letter lambda to counteract gravity and maintain a static cosmos.

Although he later deemed it his greatest error, decades later, the concept resurfaced in the form of the enigmatic substance known as dark energy, which propels the universe’s accelerated expansion.

Shannon’s information equation

Claude Shannon, an American mathematician and engineer, paved the way for the ubiquitous 0s and 1s of computer bits with his groundbreaking work in 1948. In a seminal paper, Shannon introduced an equation (C = B * 2log(1+S/N)) delineating the maximum efficiency of information transmission.

Here, C represents the achievable capacity of an information channel, B denotes its bandwidth, and S and N signify the average signal and noise power, respectively. The formula’s output is measured in bits per second. Shannon credited mathematician John W. Tukey with coining the term “bit” as a shorthand for “binary digit.”

May’s logistic map

The concept that simple inputs can yield remarkably complex outcomes gained prominence in the mid-20th century with the rise of chaos theory. In 1976, Australian physicist, mathematician, and ecologist Robert May published a groundbreaking paper titled “Simple mathematical models with very complicated dynamics” in the journal Nature.

The paper introduced the equation xn+1 = k * xn(1 – xn), illustrating how systems with just a few components and feedback loops could generate random and unpredictable behavior.

In the equation xn+1 = k * xn(1 – xn), xn represents a quantity in a system at the present time that interacts with itself through (1 – xn). K stands as a constant, while xn+1 signifies the system at the subsequent moment. Despite its simplicity, varying k values can yield vastly different outcomes, including complex and chaotic behaviors.

May’s equation has found applications in elucidating population dynamics in ecological systems and generating random numbers for computer programming.