I'm confused that MSN states a Single can store up to 39 significant figures.
Can you return to that article and examine it more carefully? A SINGLE can store a number up to 10^38 with precision to 7 full digits and that 8th digit is a "rarely" case. Any "significant digits" discussion becomes trickier because the position of the decimal point becomes an issue in that discussion. How many significant digits are in the number
1.0000000000000? That depends on how you measured it. Scientists in a lab with no computer presence whatsoever argued over "numeric significance" 100 years ago.
To answer your question, you need to understand the format and storage concepts. To start with, you have a bucket of bits for each data type. For a SINGLE variable, the bucket holds 32 bits. The DOUBLE bucket holds 64 bits. There is a format supported by the Intel "Longhorn" series of processors that holds 128 bits, but Access doesn't support that one so we can skip it for this forum. Since the hardware data bus width is your hard limit, you can only transport those buckets of bits in fixed-size gulps. So a SINGLE is ALWAYS 32 bits and a DOUBLE is ALWAYS 64 bits.
You have to allocate bits in three groups. These groups are the sign, the exponent, and the mantissa; some older authorities use "fraction" instead of "mantissa" but ever since the standard came out, it is "mantissa". The organization IEEE (you can look that up online) established "IEEE 754" as a standard for SINGLE and DOUBLE. That aforementioned 128-bit number has a different standard because it came much later. Here is a link from AMD that explains the IEEE 754 standards for you.
In brief, both SINGLE and DOUBLE allocate 1 bit to the sign. The convention is that if sign bit is SET/TRUE/ON, the number if negative. That leaves 31 bits for SINGLE and 63 bits for DOUBLE.
The standard is for scientific numbers, so we are talking about a format that,
in decimal, has some number of digits and a decimal point plus a separate exponent representing a power of 10. So (for chemists, you knew this) Avogadro's number is
6.023x10^23, the number of molecules in one "mole" of an elemental substance. If you have the Atomic Weight (in grams) of a substance, you have one mole.
However, for computers, you aren't working in decimal, you are working in binary. All exponents are powers of 2, not of 10. A SINGLE number allocates 8 bits to the exponent. A DOUBLE allocates 11 bits. IEEE 754 also states that the exponents are stored in "excess" notation. For SINGLE, that means that the exponent appears as an unsigned integer but to use it, the hardware subtracts 128 - and thus you hear about "excess 128" notation. For DOUBLE, that is "excess 1024" notation, same idea but different number of bits.
What is left is the mantissa. A difference between "traditional decimal" and "computer binary" mantissas is the location of the fractional point. In decimal scientific notation, a decimal scientific number is
d.dddd...dddx10^eeee - which is to say scaling of the exponents leaves one digit to the left of the decimal point and the remainder of fractional digits to the right, with a couple of exceptions. A binary scientific fraction has NO digits to the left of the fractional point. ALL digits are considered as fractions. That is, the number is
.bbbbbbb...bbbbbx2^(eeee-"excess") - more or less.
When I mentioned that scaling moves the fractional point for decimal numbers to leave one digit to the left, that process of adjusting the fractional point is called "normalization." An assumption is made that for a binary fraction, the first bit is always 1 unless the number is "true zero" (all bits 0, whether 32-bit or 64 bit format) - and therefore IEEE 754 includes a feature called "hidden-bit" normalization. Since you know that 1st bit is always 1, you don't need to show it. Which means that your mantissa for SINGLE = 32 bits - 1 sign - 8 exponent + 1 hidden bit = 24 bits of mantissa. For DOUBLE, 64 bits - 1 sign - 11 exponent + 1 hidden bit = 53 bits of mantissa.
Here's a rule of thumb to figure out how many decimal digits you can expect. Remember that 10^3 is close to 2^10 (1000 ~ 1024). So every 10 bits of mantissa is about 3 decimal digits. Single = 24 bits, so that's 10 bits + 10 bits + 4 bits, or 1000x1000x16 - and you can express (roughly) 16,000,000 with that mantissa - and that is 7 decimal digits. Double = 53 bits = 8,000,000,000,000,000 so you can express 15 decimal digits and squeak out a 16th in rare cases (when the first digit is 4 or less.)
Side effects of this notation scheme allow greater range for negatives than for positives. This hits hardest for the exponents. The side effect occurs because signed binary numbers have to include 0, which is not a negative number. So a short binary integer ranges from +127 through 0 to -128 - and that same difference permeates number representation on computers to this day. SINGLE and DOUBLE have one extra step for negative exponents representing numbers less than 1. A SINGLE ranges from 10^38 to 10^-39, roughly. Note also that when your problem is numbers with larger exponents, DOUBLE has 3 extra bits, so you might in some cases want to use DOUBLE, not because of the extra precision but because of the extra exponent range.
@dalski, this should tell you everything you needed to know (and maybe just a little bit more) about a computer's precision of numbers. Remember that rule of 2^10~10^3 because it applies to EVERYTHING in binary computing.
