Why create an irrational numbers search engine? Why not! Built in just for fun, the original implementation only offered digits for Pi and ran on a makeshift server in my basement. The hardware has since been continually upgraded and the application tuned for performance. Usage has steadily increased over time, with , searches processed in the 5 years between and Usage spiked dramatically in with , searches, and tapered off slightly in with a total of , As of June , the search engine is processing an average of 47K queries per month!
Between and , 14, unique visitors from countries accessed the search engine, including numerous corporations, universities and schools across the globe. I have received a range of user feedback, citing uses spanning research from the mainstream to the fringes of credibility to Geocachers who have used the site to embed clues within the digits of Pi.
Euler-Mascheroni Constant Digits -- from Wolfram MathWorld
It has been interesting and rewarding to see the variety of 'uses' for the search engine! Searches during this period averaged 6. This is a hole. Worse: there are infinitely many irrational numbers, they are even more abundant than rational ones, therefore the power function might be shock full of holes, making it un-smooth. One way around this hole is using a rational approximation that "converts" a irrational exponent into a rational one, like this:.
This technique seems to "solve" the problem, but two issues remain. The "small" issue: we would have to handle huge numbers, with millions of digits. The big issue is we haven't proven that a power with irrational exponent really exists. There are other problems. But, if the exponent is irrational, we simply can't determine whether a or b are even or odd, so we don't know whether the result is real or complex. This fact invalidates the approximation technique.
To fill these holes, we must redefine the power function. The teachings from the primary school will suddently look useless.
I hope you can handle that :. It looks simple.
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Many mathematicians tried to find the area beneath this curve, and didn't succeed. This problem was finally solved when calculus was invented by Newton and Leibniz. Before calculus, one thing was "known" actually suspected, without proof : the area grows logarithmically as x grows.
e Digits 2.71828...
Due to this, the function of this area was baptized "ln". Then ln must be, for all intents and purposes, a logarithmic function.
It is a case of "duck typing" in math: if the function squawks like a duck and walks like a duck, it is a duck. Due to the many special properties of ln, it is nicknamed "natural logarithm". Before calculus, the base of ln was already suspected to be around 2. By choosing x carefully, we can make y to be any real number. Well, we have found one case where an exponent can be irrational: when the base is the e number. Now we use this finding to redefine the power function in a way that it can accept any real number as exponent.
The properties of logarithms can help us in this. For example:. The definition of power we learn at primary school is a corner case, valid only for rational exponents. Ok, maybe this is a bit harsh.
Algorithm used for computation
The "primary school" definition is still the ultimate when we deal with integer numbers exclusively. Modern cryptography and many other tricks in discrete math are based on powers of integers. But, when dealing with real numbers, the "new" definition is the valid one. The definition above is valid even for negative bases, even though this forces us to deal with complex numbers in intermediate calculations. Note the capital-letter L in "Ln".
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A negative number has infinitively many logarithms, all of them complex. One of them, exactly one, is considered the "principal". The capital-letter Ln means that we get this one. The new definition works equally well for complex bases and exponents, which gives birth to the intriguing Euler's identity whose exponent is imaginary and irrational:.