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The 12 Equations That Explain Reality

There are some equations that are just magical. Follow the symbols where they lead and light becomes an electromagnetic wave, mass turns out to be a form of energy, spacetime bends, and a new particle is predicted where none had been suspected. You don’t need to be a mathematician to marvel that a few marks on a page can uncover hidden features of reality.

The Great Courses has released a new course, “12 Equations That Explain Reality”, which introduces you to a dozen such masterpieces that capture with breathtaking economy the way the world works. Your guide is award-winning physicist and popular podcaster Sean Carroll, Homewood Professor of Natural Philosophy at Johns Hopkins University.

The chapters in this course cover:

01: Why Reality Obeys the Pythagorean Theorem

Equations summarize patterns in the world. Begin the course with the Pythagorean theorem, an equation that relates the lengths of the three sides of a right triangle. Nothing sounds simpler, but the theorem brings up deep questions in mathematics and philosophy: the idea of symbolic mathematics, the existence of irrational numbers, and the process of proving a theorem with axioms and logic.

02: Newton's Second Law of Motion

Newton's Second Law--F = ma, or force equals mass times acceleration--is arguably the most important equation in physics. Explore how it made motion predictable and embodied a revolutionary insight: Objects continue moving at constant speed in a straight line unless some outside influence changes their motion, overturning the Aristotelian belief that continued motion requires a continuous mover.

03: Calculus: The Derivative and the Integral

The study of motion came to life with calculus, developed independently in the late 1600s by Isaac Newton and Gottfried Wilhelm Leibniz. Its key idea is the derivative--the instantaneous rate of change--calculated by measuring change over intervals that shrink toward a single instant. Learn the fundamental definition of the derivative, which expresses this idea in a compact equation.

04: Newton's Inverse Square Law and Gravity

Learn how Newton made yet another key contribution with his inverse-square law of universal gravitation, which revealed that the force making objects fall to Earth also governs the Moon and planets, weakening with the square of distance. Combined with his Second Law, it enabled him to calculate planetary orbits. Dissect this relatively simple equation and draw out its far-reaching implications.

05: Euler's Formula: The Most Beautiful Equation

Venture into the realm of complex numbers, formed from real and imaginary parts. Meet i, the square root of -1, and e, a constant fundamental to calculus. Explore the trigonometric functions sine and cosine, along with radians, the natural measure of angles. Then, witness how Euler's formula unites them in Euler's identity, eiπ +1 = 0, often called the most beautiful expression in mathematics.

06: Maxwell's Equations for Electromagnetism

Explore Maxwell's equations for electromagnetism, which unified electricity and magnetism through the concept of fields. Learn how to read the four equations, and see how they predict waves of oscillating electric and magnetic fields traveling at a constant speed, c. When that speed proved to be the speed of light, Maxwell reached a startling conclusion: Light itself is an electromagnetic wave.

07: Probability and Bayes's Theorem

Probability pervades science, from interpreting data to evaluating hypotheses. Bayes's theorem calculates how new evidence changes the probability that a proposition is true, allowing scientists to revise their confidence in a theory as data accumulates. Even people who have never heard of Bayes use a similar approach informally when weighing the credibility of sources and claims.

08: Entropy: The Second Law of Thermodynamics

Investigate an equation that has profound implications for the arrow of time and the earliest era of the universe. The Second Law of Thermodynamics simply says that energy naturally spreads out and does not spontaneously gather itself back together. Discover how this principle emerged from 19th-century studies of steam engines and was ultimately explained through the statistical behavior of atoms.

09: Einstein, Special Relativity, and Space-Time

Analyze the most famous equation of all, E = mc2, Einstein's formula connecting mass and energy. See how it emerged from his revolutionary 1905 work on special relativity, which resolved contradictions in electrodynamics and led to a new conception of space and time as a four-dimensional continuum: spacetime. Explore why c, the speed of light, is central to both relativity and this equation.

10: Einstein, General Relativity, and Gravity

Special relativity provided a framework that Einstein and others applied to different phenomena in physics. Gravity proved exceptionally difficult. Trace the insights that led Einstein to his greatest triumph: a general theory of relativity, expressed in an equation that links matter and energy to the curvature of spacetime and predicts black holes, gravitational waves, and an expanding universe.

11: Schrödinger's Equation for Quantum Mechanics

Enter the baffling realm of quantum mechanics, where Schrödinger's equation provides a startlingly accurate description of matter at the atomic scale. Learn how it governs the wave function, which encodes possible measurement outcomes and their probabilities. Also, see how Heisenberg's radically different approach yielded the same predictions and led to his celebrated uncertainty principle.

12: Dirac's Equation for the Electron Field

For the final equation, probe Dirac's equation for the electron, a crucial step toward quantum field theory, in which particles are understood as manifestations of fields. See how it predicted the positron, revealing an entire realm of antiparticles now routinely observed in experiments. That surprising outcome affirms the physicists' motto: "Our equations are smarter than we are."

For those interested in this topic, this course is a very good introduction to these equations.

It should also be remembered that mathematics is a tool to help prove scientific theories, and by itself is not a proof of them. Scientific theories are proven by the evidence from observations and experiments, with mathematics used to explain these observations and experiments.

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Please note that my Science Articles disclaimer applies to this article, as it does to all of my science articles. These science articles are for the purposes of general scientific information that is understandable to a layperson, so that they can grasp the concepts of the topic being discussed. As such, they are neither as accurate nor as thorough as academics, scientists, engineers, and science writers strive to achieve. My hope is that these science articles will be informative and interesting, and that they will pique the reader's interest in exploring these topics in further detail via the hyperlinks that I have embedded into these articles.