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11

No -- according to the Wikipedia article, axiomatizations of arithmetic were proposed by Peirce and Dedekind in the years leading up to Peano's publication of his version in 1889. Even so, what Peano published in 1889 was not quite the thing we call "Peano Arithmetic" today. The 1889 axioms were, in today's terminology, a second-order system where the ...


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One thing that we must be very clear about: if you have a certain number, it is the same number no matter what base you write it in. If you have $x$ and divide it by $b$ using integer arithmetic only, with integer quotient $q$ and remainder $r$, you will have the same four numbers no matter whether $x$, $b$, $q$, and $r$ are written in base ten, in base ...


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Simplifying this equation is really easy if you use a cool factoring trick. $$xy=2^2 3^4 5^7(x+y)$$ Distribute the $2^2 3^4 5^7$. $$xy=2^2 3^4 5^7x+2^2 3^4 5^7y$$ Subtract both sides by $2^2 3^4 5^7x+2^2 3^4 5^7y$. $$xy-2^2 3^4 5^7x-2^2 3^4 5^7y=0$$ Now, add the product of the coefficients of $x$ and $y$, which is $(2^2 3^4 5^7)^2=2^4 3^8 5^{14}$, to ...


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If the $\ldots$ means that you go on to infinity on the left, then it's called the 10-adic representation (see https://en.wikipedia.org/wiki/P-adic_number). If you truncate to a fixed finite number of digits (say $n$) you call it an $n$ digit 10s complement representation. (Google will find you lots of links.)


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Presumably you know how to convert integers to decimal by dividing by two and recording the remainders then reversing the order of the remainders. The same process in reverse will work for values less than 1. So multiple by 2 then record if the value is less than or more than one. Repeat with any fractional part. Example with $\frac{3}{17}$: ...


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The Knuth arrow notation is for expressing primitive recursive functions at a specific level of the hierarchy of such functions. Level is the depth of nested FOR loops needed to compute the function. The most frequent (but still quite uncommon) use of the notation would be to graphically display how inefficient a particular upper bound argument is in a ...


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An unfortunate truth about converting a number from base $a$ to base $b$ is that unless there are integers $m > 0$ and $n > 0$ such that $a^m = b^n$, you need to use an arbitrarily large amount of memory in order to convert arbitrarily large numbers from base $a$ to base $b$. Suppose there are no such numbers $m$ and $n$. There are various ways this ...


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Example: Convert $30$ (base ten) to base $4$: $30=4(7)+2+4(4(1)+3)+2=4^2(1)+4(3)+2$. So the answer is $132_{\rm{four}}$. But note that the first remainder ($2$) is the least significant digit--i.e., it goes in the ones place. And basically we are acquiring digits right to left.


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Dedekind had a contemporaneous and slightly earlier-published essay on "what are numbers" in which he gave the categorical definition, before categories existed. His definition of a natural number object is a set $N$ with a self-similarity transformation (an injective endomorphism) satisfying properties that make the transformation correspond to the ...


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$$0.412\overline{8754}=\frac{4128754-412}{9999000}$$ Edit: the numerator is the difference of the number build by the preperiod followed by the period, in our case $4128754$, and the preperiod, here $412$. For the denominator, write down as much nines as the period is long, here $9999$, followed by as much zeros as the preperiod is long, in our case $000$.


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Another description of the Mayan numbering system, including important historical facts explaining how limited our knowledge is, is at the site http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Mayan_mathematics.html. We have only a few surviving documents from the Mayan civilization. We have examples of large numbers written in a not-quite-base-$20$ ...



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