When I type into wolfram the query, exp(1) in base 100, the answer comes back "2.71:82:81...."

What does the colon (:) mean?

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    $\begingroup$ I would guess that the base-100 digits are the pairs separated by colons. For example, the first base-100 digit past the radix point is 71. $\endgroup$ – MPW Jul 17 '14 at 21:31
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    $\begingroup$ to expand on what MPW said, you'd need 100 different characters to not need colons $\endgroup$ – Jam Jul 17 '14 at 21:33

You are asking for a number to be expressed in base $100$. That means the first "place" after the "decimal" point is in units of $1/100$, the second "place" after the "decimal" point is in units of $(1/100)^2$, and so on. Each "place" is populated by a whole number ("digit") from 0 to 99. Because we do not have distinct symbols for each of those possible digits, each one of those base-100 "digits" is actually written using a pair of base-10 digits. This makes reading the number hard, so it is helpful to use a delimiter to keep the place values separated for readability's sake.

Strictly speaking, one could leave the delimiter out, since each "place" always is populated by a two-digit "digit", so one could simply group the "digits" in pairs.

The particular number you are looking at is $$2 + 71(1/100) + 82(1/100)^2 + 81(1/100)^3 + \dots$$

Note that the use of a colon to separate "places" when the "places" hold two-digit numbers is standard in time-keeping, which is (at least partly) in base 60: We write 15 minutes and 8 seconds as 15:08, and each "place" can hold any value from 0 to 59.

The irony here is that the "base 100" representation of any number is precisely the same as its "base 10" representation of the number, if you just eliminate the colons and interpret each pair of digits as two distinct place values.

  • $\begingroup$ thank you both for your answers. mweiss, your explanation was particularly thorough and clear. I fully understand the notation now. Michigan State is lucky to have you as a teacher. A quick followup question if I might: writing 1000 in base 3 gives 1x3^6 + 1x3^5 + ... however, isn't it strange to place 3 to the 6th power or 5th or 4th when the base is 3? 6 in base 3 is written 20, so wouldn't we more appropriately start by writing 1000 as 1x3^(20) +...? $\endgroup$ – steve Jul 18 '14 at 22:20
  • $\begingroup$ When we write 1000 in base 3, we write it as $1101001$. We interpret this as $1 \cdot 3^6 + 1 \cdot 3^5 +1 \cdot 3^3 + 1$, but that is actually a base 10 expression -- which you can tell because it contains not just the symbol 6, but also a 5 and a 3, none of which are legal symbols in base 3. If we wanted to write the same expanded form using only base 3 symbols, we would write it as $1 \cdot 10^{20} + 1 \cdot 10^{12} + 1 \cdot 10^{10} + 1$. $\endgroup$ – mweiss Jul 18 '14 at 23:13

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