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11

You're mixing the actual number with its representations. Yes, in base $n$, the string $10$ represents exactly $n$. But now forego of the decimal representation of $10$, and think about it as "how many digits a healthy human being has on both their hands". This is your $n$, now. Let's for the sake of simplicity call this number "ten". Now we are counting ...


10

This is more like a comment but it's too long, so I'm putting it as an answer instead, please accept my apology. In base $10$, the "symbol" $78152_{10}$ represents the number $78152_{10}=7\cdot 10^4 + 8\cdot 10^3 + 1\cdot 10^2 + 5\cdot 10^1 + 2\cdot 10^0$. In base $n$, the "symbol" $78152_n$ represents the number $78152_n=7\cdot n^4 + 8\cdot n^3 + ...


2

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.)


2

Let's approximately model the digits of a number in positional notation with coprime bases as independent random variables, and likewise for the number's residues with respect to powers of different primes, with mutual conditional independence. (I think you could make this more rigorous by uniformly randomly selecting a number between $1$ and $N$ and taking ...


2

When I work with numbers in base 16 or base 2, the string "10" is not called "ten", but "one zero". We generally don't have names for specific integers that reflect some other base, so although a full 16 bit value is ffff and then 1 0000 is significant to me, I don't have a spoken name for it that corresponds to thousand. It's only known as "sixty ...


2

The name refers to the way we choose to group our items. The way we've evolved, we found that nine counting symbols and a symbol for nothing suffice for our preferred base. We're able to recycle the glyphs $1$ and $0$ to denote our grouping, which is given by the combined symbol $10$. To illustrate why it's all about the grouping, and not about the final ...


1

Try long division. For example, to calculate $1 \div 7$, do the following. (Note that $7_{10} = 13_4$.) $$ \require{enclose} \begin{array}{r} 0.02102\ldots \\[-3pt] 13 \enclose{longdiv}{1.00000\ldots} \\[-3pt] \underline{32}\phantom{0000000} \\[-3pt] 20\phantom{000000} \\[-3pt] ...


1

$\frac18<0.21 <\frac28\\ 0.21-\frac18 = 0.085\\ 0.085*64 = 5.4\\ 0.085 - \frac5{64} = 0.006875\\ 0.006875 * 8^3 = 3.5\\ 0.006875 - \frac 3{8^3} = 0.00101625\\ $ etc. $0.153$


1

Your figure out fractions in another base just as you figure them out in base 10. To figure out $a/b$ in base 8 you divide be into $a$ and take the remainder, multiply it by the base and divide $b$ into it and repeat. Example $3/5$ in base $8$ is... $5$ goes into $3$, $0$ times with $3$ remainder. So $0.rem3$. So we multiply the $3$ by $8$ to get $24$. ...


1

$$0.412\overline{8754}=\frac{4128754-412}{9999000}$$ 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 $000$.


1

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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