Disclaimer - I'm not an information theorist, just a code monkey who works primarily in C and C++ (and thus, with fixed-width types), and my answer is going to be from that particular perspective.

It takes *on average* 3.2 bits to represent a single decimal digit - 0 through 7 can be represented in 3 bits, while 8 and 9 require 4. `(8*3 + 2*4)/10 == 3.2`

^{1}.

This is less useful than it sounds. For one thing, you obviously don't have fractions of a bit. For another, if you're using native integer types (i.e., not BCD or BigInt), you're not storing values as a sequence of decimal digits (or their binary equivalents). An 8 bit type can store some values that take up to 3 decimal digits, but you can't represent all 3-decimal-digit values in 8 bits - the range is `[0..255]`

. You cannot represent the values `[256..999]`

in only 8 bits.

When we're talking about *values*, we'll use decimal if the application expects it (e.g., a digital banking application). When we're talking about *bits*, we'll usually use hex or binary (I almost never use octal since I work on systems that use 8-bit bytes and 32-bit words, which aren't divisible by 3).

Values expressed in decimal don't map cleanly on to binary sequences. Take the decimal value `255`

. The binary equivalents of each digit would be `010`

, `101`

, `101`

. Yet, the binary representation of the value `255`

is `11111111`

. There's simply no correspondence between *any* of the decimal digits in the value to the binary sequence. But there is a direct correspondence with hex digits - `F == 1111`

, so that value can be represented as `FF`

in hex.

If you're on a system where 9-bit bytes and 36-bit words are the norm, then octal makes more sense since bits group naturally into threes.

^{
Actually, the average per digit is smaller since 0 and 1 only require a single bit, while 2 and 3 only require 2 bits. But, in practice, we consider 0 through 7 to take 3 bits. Just makes life easier in a lot of ways.
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`d`

, it covers one decimal digit, the range of`0..9`

.`3*d`

bits mean three decimal digits and allow you to represent integers from the range`0..999`

. Whole ten bits (think binary now) give a range of`0..1023`

. 999 is quite close to 1023, yet a little less. So you may expect`d`

should be little less than 10/3.10more comments