Putting it all together, the constant "e" raised to the power of the imaginary "i" multiplied by pi equals -1. Like "e," it seems to suddenly arise in a huge number of math and physics formulas. Pi, the ratio of a circle's circumference to its diameter, is one of the best-loved and most interesting numbers in math. The letter "i" is therefore used as a sort of stand-in to mark places where this was done. But in math, there are many situations where one is forced to take the square root of a negative. It is thus called because, in reality, there is no number which can be multiplied by itself to produce a negative number (and so negative numbers have no real square roots). Next, "i" represents the so-called "imaginary number": the square root of negative 1.
Indeed, the constant "e" pervades math, appearing seemingly from nowhere in a vast number of important equations.
In math, the number exhibits some very surprising properties, such as - to use math terminology - being equal to the sum of the inverse of all factorials from 0 to infinity.
Discovered in the context of continuously compounded interest, it governs the rate of exponential growth, from that of insect populations to the accumulation of interest to radioactive decay. Stanford mathematician Keith Devlin wrote these words about the equation to the left in a 2002 essay called "The Most Beautiful Equation." But why is Euler's formula so breath-taking? And what does it even mean?įirst, the letter "e" represents an irrational number (with unending digits) that begins 2.71828.
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"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence." Wall MathĮuler's Equation (Image credit: public domain) Incredibly, the topologist Bernard Morin, a key developer of the complex method of sphere eversion shown here, was blind. Watch the video above to see how it's done. But in fact, "sphere eversion," as it's called, is possible. Topologists long wondered: Is a sphere homeomorphic with the inside-out version of itself? In other words, can you turn a sphere inside out? At first it seems impossible, because you aren't allowed to poke a hole in the sphere and pull out the inside. On the other hand, Moebius bands - loops with a single twist in them - are not homeomorphic with twist-free loops (cylinders), because you can't take the twist out of a Moebius band without cutting it, flipping over one of the edges, and reattaching. Thus, a classic math joke is to say that topologists can't tell their doughnuts from their coffee cups. If you turn it upright, widen one side and indent the top of that side, you then have a cylindrical object with a handle. In an important field of mathematics called topology, two objects are considered to be equivalent, or "homeomorphic," if one can be morphed into the other by simply twisting and stretching its surface they are different if you have to cut or crease the surface of one to reshape it into the form of the other.Ĭonsider, for example, a torus - the dougnut-shape object shown in the intro slide.
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