From Pythagoras to Schrödinger, equations have not only advanced scientific understanding but have also driven technological innovations that define modern life. This exploration highlights how these mathematical tools have transformed our world and continue to unlock new possibilities.

In the very beginning, let me make it categorically clear to the readers that this write-up has nothing to do with the nature of equations, as to whether they are invented or discovered, nor does it seek to undermine the role of inequalities, which, according to our well-known mathematician Prof Amin Sofi, far outweigh equations in terms of their significance and importance in mathematics. I will leave this question to the philosophers of mathematics to answer. On this matter, suffice it to quote Robert Lanza from his book Beyond Biocentrism: “One might see that the physical world is not the same as the abstract mathematics or even simple logic we might use to describe it. Logic demands symbolic thinking, whereas the actual world doesn’t have to play by these semantic rules.”

Coming back to my point, this write-up, as the title suggests, is all about equations and the role they have played in shaping not only the modern world but in making intellectual advancements possible. Before I answer this question, I would like everyone to ponder whether our world would be the same as it is today. If it has changed, which it has, what do we owe that change to? A simple look back at history will show that the change is due to the equations of physics and mathematics formulated over hundreds of years.

So, what are equations, why do we need them, and what role have they played in making the world as it is today? The Oxford Dictionary of Mathematics defines an equation as “a statement that indicates two mathematical expressions are equal.”

Ian Stewart, the British mathematics popularizer, while answering this question of why we need equations, writes in his book 17 Equations That Changed the World that “equations are the lifeblood of mathematics, science, and technology. Without them, our world would not exist in its present form. The course of human history has been redirected, time and again, by equations. Equations have hidden power. They reveal the inner secrets of nature. Equations model deep patterns in the world. The power of equations lies in the philosophically difficult correspondence between mathematics, a collected creation of the human mind, and an external physical reality.”

Equations not only offer a structured way to analyze and solve complex problems but also provide a way to communicate scientific findings. Not only do the laws of nature find their language and representation in equations, but equations can also act as tools to unify two seemingly different phenomena, forces, or theories. In the modern world, equations, because of their immense predictive power, are widely used in domains other than physics and mathematics, in fields like climate change, genetics, and epidemiology to model disease spread.

However, equations can be intimidating as well. Prof Stephen Hawking, in his masterpiece A Brief History of Time, writes that his publishers told him that every equation he included in his book might halve his sales, but despite that, they could not do away without including Einstein’s famous mass-energy equivalence relationship.

As young students, the first and foremost equation we are introduced to is Pythagoras’ theorem, which connects the three sides of a right-angled triangle. The theorem and its consequences have had a gigantic impact on human history. It literally opened up our world. The power of this theorem lies in the fact that it gave birth to a new branch of mathematics called trigonometry. This equation relates to algebra and geometry. In terms of its application, it continues to play a key role in modern-day construction, surveying, and navigation, besides having played a key role in the formulation of Einstein’s special and general relativity—the two most beautiful theories on space, time, and matter. Pythagoras’ theorem also played a pivotal role in the invention of coordinate geometry. Its extension to triangles without right angles and the triangles on a sphere allowed us to map our continents and measure our planet.

The second most beautiful equation that a student comes across is Newton’s second law of motion, generally expressed in mathematical form as F = d(mv)/dt. This law, according to many physicists like Kirchhoff, occupies a central position in Newton’s laws of motion. The discovery of this second law of motion, elegantly expressed in the above form, was a dramatic moment in the history of science. This equation is the prescription for formulating the dynamical equations of motion in inertial frames. The second law gives us a specific way of determining how velocity changes under different influences called forces. The important thing to realize is that this equation or relationship involves not only changes in the magnitude of momentum or velocity but also in their direction. The motion of pendulums, oscillators with springs and weights in them, and so on could all be analyzed completely after Newton’s law was enunciated. It beautifully explained the motion of planets, which were a complete mystery before Newton. This equation, or the basic principles of classical mechanics, possesses a beauty in that it can be derived from Schrödinger’s equation, provided the quantities it relates are understood to be averages rather than precise values.

The third class of equations that have shaped the modern world is Maxwell’s set of four equations that unify electricity and magnetism. Maxwell, through these equations, is considered to have made one of the great unifications of physics. The beauty of Maxwell’s equations is paraphrased in these words of acclaimed Nobel Prize-winning American theoretical physicist Richard Feynman: “God said, ‘Let there be light,’ and it was. Maxwell could say, ‘Let there be electricity and magnetism,’ and light is.”

The applications of Maxwell’s equations are far too many to count. From MRI scanners in hospitals to the creation of computers, from the generation of electricity to magnetic tapes, they have played a central role. One of the pioneering tasks that Maxwell’s equations accomplished was the prediction that electromagnetic waves do exist, travelling at the speed of light, so light itself is such a wave, which in turn motivated the invention of radio, radar, and wireless communication. It wouldn’t be an exaggeration to say that Maxwell’s equations didn’t just change the world—they opened a new one, the world we live in.

The fourth equation is Schrödinger’s wave equation. It explains the whole dynamics of a particle in modern physics. This equation is at the core of modern physics and quantum mechanics. It has made possible the invention of transistors, MRIs and lasers.

So, equations do have a profound role in shaping our lives. Coming up with new equations will not only lead to a better understanding of the universe we live in or reveal new secrets of nature, but they can also improve our technology and expose us to new horizons of life. Other equations will be discussed in a separate write-up.

The writer is a physics student

By Nasir Rather

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Nasir Rather