The most important equation…
As a scientist, I dealt with many equations. As a full-time writer, not so much. Many people know Newton’s F = ma. That equation, and its rotational form, τ = Iα, appear in Survivors of the Chaos (the first novel contained in The Chaos Chronicles Trilogy Collection). That’s about it for equations in my fiction.
Some people might think that Einstein’s equation linking the curvature tensor of space and time to the matter-energy tensor in General Relativity is the most important one in science, or Maxwell’s equation for the electromagnetic field (tensor form so it’s just one equation). The latter certainly has more application in our everyday lives than the former, if that’s important.
The most important equation, though, is exp (i π) + 1 = 0.
exp (i π) is usually written differently—e to the i times π—but I use the less common form to avoid superscripts. The functional notation exp ( . ) is shorthand for the irrational number e raised to some power. Here i is the imaginary unit that is the square root of -1; all complex numbers can be written in the form a + bi where a and b are real numbers. exp (i π) can be written that way too: cos(π) + i sin(π). The symbol π is the ratio of the circumference of the circle to its diameter, and some readers might remember that cos(π) = -1 and sin(π) = 0.
So there you have it: that most important equation just says that -1 + 1 = 0. “Big deal!” you might say. Well, it says that in a most beautiful and profound way, connecting the five most important numbers in mathematics. First 1, the first natural number, the numbers used in counting; second, 0, an important human invention signifying nothing, but also used as a placeholder, as in 1000; e and π, the two most important irrational numbers in the set of real numbers; and finally i, needed to create the complex numbers so important to scientists and engineers.
Why do these numbers link up this way? Maybe that’s not the right question. The better question: Isn’t the discovery that they do link up this way in one equation more important than any physical law ever discovered? There’s something immutable and unchangeable about this equation beyond any other in science.
Think about it. True science is empirical. Experimental data lead to “laws” like F = ma that explain many natural phenomena, but Newton’s law must be modified to explain special relativistic effects occurring at high velocities. There’s nothing empirical about exp (i π) + 1 = 0 at all. It would exist for some ETs living at the bottom of a methane ocean under huge pressures, or on a planet in the 82 Eridani star system (they’re there, and you can read about them in Sing a Zamba Galactica, novel #2 in the collection mentioned above). The symbols they might use in the equation could be different—maybe just sonograms—but that doesn’t matter.
That equation exists without any phenomena at all, and it can be considered a cornerstone of universal mathematics. It’s a mathematical tautology of the utmost importance.
Think about it this way: the equation connects natural numbers, integers, real numbers including two fundamental irrational numbers (ratios of integers are rational numbers), and complex numbers. The only thing that this equation doesn’t include are quaternions, the biggest set of numbers that includes them all that scientists sometimes use to describe three-dimensional rotations.
Now, isn’t that one beautiful equation?
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Comments are always welcome.
Mind Games. You know A. B. Carolan as the writer of The Secret Lab and The Secret of the Urns. Those novels are sci-fi mysteries for young adults (and adults who are young at heart). In Mind Games, A. B. tells a new story that’s set a bit farther into the future than his first two books. Della Dos Toros is a young girl with psi powers living in the Dark Domes of the planet Sanctuary. Her adopted father doesn’t let her use those powers, but she must do so to find his killer. This story about ESP and androids adds another action-packed novel to the ABC Sci-Fi Mystery series. Available in both print and ebook versions.
Around the world and to the stars! In libris libertas!
