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Is there a standard or common way to write scientific notation in different bases that doesn't require repeating the base in both the coefficient and the exponent base?

For example, this notation is correct, but is quite verbose:

  • Implicit base 10: $100 \cdot \tau = 6.283 \times 10^2$
  • Explicit base 10: $100 \cdot \tau = 6.283_{10} \times 10_{10}^2$
  • Explicit base 16: $100 \cdot \tau = 2.745_{16} \times 10_{16}^2$

Most calculators & programming languages shorten this to an "exponent" notation such as:

6.283e2

Some rare ones even support a base, such as:

16#2.745#e2#

Unfortunately, these are not very nice notations outside of specific domains.

Ideally, I'm looking for an operator notation that concisely does the function:

  • $f(coefficient_{b},exponent) = coefficient_{b} \times b^{exponent}$

If there is no such standard or common notation, I'd be happy to hear any suggestions.

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  • $\begingroup$ How about "$\tau = 2.745 \times 10^2$ in base $16$" (or "τ = 2.745e2 in base 16", whichever you prefer)? $\endgroup$
    – ShreevatsaR
    Jun 5, 2011 at 22:42
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    $\begingroup$ what's a bit ugly is that you're still using base 10 to denote the base you're using. Maybe, in a way, it's more natural to denote your base by the highest number below the radix, instead of the radix itself, so you'd have base 9 or base F instead of base 10 or base 16 in 'metabase' 10. $\endgroup$
    – gfes
    Jun 5, 2011 at 23:03
  • $\begingroup$ @gfes I think your (base-1) notation has some appeal, but using 10 as the "metabase" seems to be the only way I've ever seen it done, and seems to be universally understood. "Base F" immediately makes one think of hexadecimal, but I think if you wrote "base 9" it would be confusing! $\endgroup$
    – wjl
    Jun 5, 2011 at 23:33
  • $\begingroup$ True. Although when working in base 10, probably anyone would omit the base 9 / base 10 altogether anyway. Also, I thought of a neater, but less practical way, namely writing the alternative expansion for the radix. I.e. 9.9999999999 etc. or F.FFFFFFFFF etc. $\endgroup$
    – gfes
    Jun 5, 2011 at 23:37
  • $\begingroup$ @gfes The alternate expansion is quite clever. It would work especially well with an overbar to indicate a the repeating fractional part. $\endgroup$
    – wjl
    Jun 5, 2011 at 23:43

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Unfortunately, $2.546_{16}\times 10_{16}^{23}$ is cumbersome, and it isn't immediately clear if the exponent is being written in base $10$ or base $16$, and $2.546_{16}\times 16^{23}$ has a strange mixing of base $10$ and base $16$ elements. For programming things, there are explicit prefixes you can put on a number to denote whether it is base $2,8,10$ or $16$, but that isn't general, and it still doesn't really address any of the previous problems.

Maybe a notation like $(2.546\times b^{23})_{16}$ would work to indicate that everything is supposed to be taken base $16$, with $b$ being implicit for the base (as writing either 10 or the base 10 expression of the base is awkward). However, I'm not fully enamored with this notation either.

What is your application of this? In what circumstances is some sort of scientific notation required but base $10$ is undesirable?

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  • $\begingroup$ I actually like the parenthesized notation better than anything else I've seen or thought of so far. As far as the application, this is more of a general question that has bugged me for a long time. However, this comes up a lot for me specifically because I often write up detailed mathematical descriptions of the behavior of low-level, digitial electrical engineering applications. For instance, one place a concise notation would be obviously handy is when writing things about DSP algorithms using IEEE floating point in hexadecimal. $\endgroup$
    – wjl
    Jun 5, 2011 at 23:38
  • $\begingroup$ Um, one possibility is also to use the shift operators a la C/C++. So $(2.546 \ll 23)_{16}$? $\endgroup$
    – Willie Wong
    Jul 5, 2011 at 23:47

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