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 |    Not for the faint of art. |
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Complex Numbers A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number. The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi. Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary. Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty. ![Merit Badge in Quill Award
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| Because I select these things at random from a list, it's not all that common that I hit upon similar topics two days running. Today is one of those days, though. It would probably be more accurate to say that the equations changed our understanding of the world. The world spins on, regardless of our mathematical prowess (or lack thereof). True, we then go on to shape the world based on the new understanding, but... well, perhaps I'm getting ahead of myself here. Do you know that mathematical equations affect our day-to-day lives? Trivially true; I mentioned the world spinning on in the last paragraph, and that motion follows mathematical and physical realities. While there are many mathematical equations that have molded mathematics and human history, let’s have a look at 10 of them: You'll have to go to the link to see the actual equations, I'm afraid. Especially the later ones use symbols I can't be arsed to reproduce here. 1. The Pythagorean Theorem: Hopefully everyone knows this one. The theorem states that: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. This article is brief; don't expect definitions of things like "right triangle," "square," "legs of a triangle," or especially "hypotenuse." 2. Isaac Newton’s Law Of Universal Gravitation: I've heard it proclaimed that "Newton Was Wrong!" Well, he was, in a trivial sense, but as the article points out, this equation does just fine when dealing with things on a human scale and slightly bigger. 3. Albert Einstein’s Theory Of Relativity: Oh boy, is this one simplistic. E=mc2 is hardly the totality of the Theories of Relativity. Also, there's another term in the equation that's generally left out of its popular form; it's generally enough to know that a) energy and matter are really the same "thing" and b) the speed of light (in a vacuum) is a constant. 4. The Second Law Of Thermodynamics: Oddly, this list isn't in chronological order. We knew about this one before Einstein. The weird thing here isn't so much the equation itself, but how it relates to the passage of time. Rudolf Clausius’second law of thermodynamics states that the total entropy can never decrease over time for an isolated system, that is, a system in which neither energy nor matter can enter nor leave. I have heard this used to argue against the process of evolution. Such arguments conveniently ignore the fact that our planet isn't an isolated system; we get a constant influx of energy from a handy nearby fusion reactor. 5. Logarithm functions: As the article points out, these lost a lot of their necessity once computers entered the picture. They're still useful to learn, though, as logarithms have the other benefit of turning an exponential chart into an easier to understand linear one. 6. Maxwell’s Equations: First published between 1861 and 1862, by combining the electric and magnetic fields into a set of four equations they define the key mathematics behind radio waves of all types also called as electro-magnetic radiation by scientists and engineers. Truly one of the great triumphs of applied mathematics, though understanding the equations and their arcane symbols takes a bit of work. Hell, electromagnetism is sorcery as far as I'm concerned. But apparently even sorcery has mathematical underpinnings. Also, check out what the UK was doing while we were over here fighting a war over whether or not it should be legal to own human beings. 7. Chaos Theory: Chaos theory is a branch of mathematics focused on the behavior of dynamical systems that are highly sensitive to initial conditions. Seriously jumping through time, here. There passed almost exactly 100 years between Maxwell and the beginnings of the understanding of chaos theory. Also, this article doesn't nearly do justice to the topic; I've read entire books on the subject and I only understand the basics, myself. 8. Wave Equation: The wave equation is a linear second-order partial differential equation... Don't worry. I noped on out of there when I read that bit, too. 9. Schrödinger Equation: Today, all of our semiconductors (transistors, integrated circuits, Intel CPU chips, etc.) depend on the science of quantum mechanics that wouldn’t have been possible to understand without Schrödinger’s equation. It also paved the way for nuclear power, microchips, and electron microscopes. Most people have heard this guy's name in connection with a famous thought experiment involving a cat that might or might not be alive. I prefer living cats, myself. The point is, though, that if you're reading this, you're doing so on a device that relies on quantum mechanical processes. Well. I don't know. Maybe you printed it out from such a device. Still, it's clear that we don't have to understand how something works to use it effectively. 10. Fourier Transform: The Fourier Transform defines the mathematics that allows us to put many different signals onto one wire, or one radio signal, and to then extract each individual signal at the other end. Ever wonder how a fiber optic bundle can transmit so much data with little to no loss? That's why. Well, that and the near-magical property of complete, lossless reflection off of the boundary of a medium such as glass (or water for that matter) when the angle of reflection is in the right range. Like I said, a lot of this stuff is beyond me. There are plenty of resources for anyone who wants to delve deeper into these concepts, and I've done so myself. Also, this list is only a starting point; there are other important equations that advanced our understanding of science and the universe. But you don't need to understand the jargon to feel a sense of wonder at how we products of the universe have worked at understanding said universe, or how far we still have to go. Pure mathematics is, in its way, the poetry of logical ideas. --Albert Einstein |
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