The Math With Bad Drawing blog has a nice little post explaining very lucidly who the notion of exponents of repeated multiplication was generalized in a way that is pretty much unique to allow for exponents that have values other than whole numbers.
This fact, typically first taught in middle school or high school algebra, has been well known for a long time. Euclid toyed with the idea a little. Ancient Greek scientist Achimedes first generalized the concept and proved the law of exponents. A fairly efficient form of exponential notation was invented by Nicolas Chuquet in 1484. More than three hundred years ago René Descartes established the modern superscript notation for exponents in the late 1600s around the same time that Newton's law of gravity and motion were invented and around the same time that Newton and Leibniz invented calculus (the modern notation used in undergraduate calculus follows the practice of Leibniz and not Newton's much more awkward notation).
There has been one notable elaboration of a similar concept in mathematics since then, called the fractal dimension which was first formally defined using that name by the late Benoit Mandelbrot in 1967 and entered the upper level college mathematics curriculum in the late 1980s and early 1990s, around the time I was an undergraduate math major. This concept was also invented in Newton's day, but then consigned to the dustbin of history as a curiosity until the late 1800s when several mathematicians developed it some more, and then remained out of sight until Mandelbrot, more or less single handedly repopularized the concept in a way that actually stuck and found practical applications.
The fractal dimension generalizes the notion of a dimension in a manner similar to the way that the law of exponents generalizes the notion of repeated multiplication by relating change in detail to change in scale. For example, the smaller the ruler you use to measure a shoreline, the longer the shore gets in ruler lengths, because the ragged pattern of a shoreline has a high fractal dimension, while a smooth shoreline would have a low fractal dimension and doesn't change in length at all based upon the length of the ruler used to measure it.
I probably wouldn't ordinarily have found any of the blog post on exponents notable at all. But, earlier just this week, I had been thinking about the precise issue of how the generalized notion of an exponent is so much more subtle than the naive repeated multiplication definition, in the context of thinking about Euler's formula and the Euler's number "e", which is equal to approximately 2.71828 and is a transcendental number that cannot be produced from the ratio of any two integers (something called a rational number). It felt remarkable to see in illustrated print found at random on the Internet, almost exactly the same line of thought.
I guess I still belong to the math tribe, even though I'm a lawyer now.
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