On Sat, 2 Sep 2006 aliensrcooluk at yahoo.co.uk wrote: I think that you would enjoy: Schaum's Outline Series : Essential Computer Mathematics by Seymour Lipschutz ISBN:0-07-037990-4 It's a much more comfortable casual read than Rosen, etc. Do you understand 2's complement notation? IEEE floating point has 1 bit for sign (an actual sign bit!, rather than 2's complement), 8 bits for exponent (power of TWO to be multiplied to the mantissa, stored as a "biased" number, NOT sign bit, nor 2's complement) 23 bits for 24 bit mantissa (when normalized, the high order bit is known, and doesn't need to be stored) (Lipschutz has a minor error on that!) with practice, it gets easier, like any other binary conversion 3.875 3 7/8 If you have CARPENTRY experience, then making .875 out of one half, plus one quarter, plus one eighth becomes more obvious There isn't a fixed standard for naming them. Some people use 0 through 7 (or whatever) for the bits to the left of the binary point, and use NEGATIVE numbers for the bits to the right, which keeps the bit "name" matching the power of two. But, not everybody likes to do it that way. -- Grumpy Ol' Fred cisin at xenosoft.com

As I think someone else mentioned, IEEE there stands for the (US) Institute of Electrical and Electronics Engineers, the standards body responsible for the format. An IEEE floating-point number consists of three parts: a sign bit, an exponent field, and a mantissa field, normally stored in that order. I know of two common formats, 32-bit format and 64-bit format; I don't know whether the standard defines any others, but it's a fairly easy format to extend to other sizes. The sign bit is always one bit, and is 0 for positive and 1 for negative. (It's a sign-magnitude format; unlike the 2's-complement that's now nigh universal for integers, negating a number here involves nothing but flipping the sign bit.) The exponent field is variable size. For 32-bit it's 8 bits; for 64-bit, 11. This exponent field is what's called `biased'. A number N>0 can always be written as N = M * (2^E) for M in [1,2) and some integer E. IEEE format stores not E but E+K, where K depends on the size of the exponent field, such that E=0 corresponds to the exponent field having high bit 0 and its other bits all 1. For 32-bit, for example, this means that E=0 corresponds to an exponent field of 01111111, or 127 decimal; this is sometimes called "excess-127". An exponent field of all 0 bits or all 1 bits is special; I'll mention this more below. If a number calls for an E value that's out of range, that number is not representible (except for certain very small numbers; see below). The mantissa field is also variable size, and occupies the balance of the bits. Because M is >=1 and <2, it is of the form 1.xxxx... when written in binary. The high bit thus carries no information; they get one more bit of precision by not actually storing it. For example, the number 3 corresponds to E=1 and M=1.1 (that M value being in binary); the M value stored in the number is 100000...000, the 1 bit before the binary point being the non-stored bit (often called the hidden bit). Because the hidden bit is hidden, it is not available to distinguish between a nonzero number that's a power of two and a number that's zero. I mentioned above that an exponent field of all 0 bits was special. One of the things it's used for is representing zero: zero is represented with all its bits - sign, exponent, and mantissa - 0. This leaves a number of possible conditions unspecified: any bit pattern with all exponent bits zero but some other bits set has no defined meaning according to the above. I'm a little hazy on exactly what's what for them. Part of this is because I also know another floating-point format - VAX format - well, and it is very similar to the above, differing only in its treatment of these "exceptional" cases (and its lack of the all-1-bit-exponent special case). It's been too long since I looked at either; the details are blurring together in my memory. In VAX format, I think any bit pattern with sign and exponent bits zero represents zero; any bit pattern with sign bit 1 and exponent bits 0 is a "reserved operand", which the floating point unit raises an exception when asked to do anything with. In IEEE format, there are denormalized numbers, which are numbers for which the non-stored ("hidden") bit is 0, not 1; I *think* these are all bit patterns with exponent bits zero, and they correspond to the above for values of E too small to represent with nonzero bits in the exponent field. These account for all the other bit patterns with exponent bits all 0. This leaves exponent fields of all 1s. IEEE format has other values, notably infinities and NaNs; all-1 exponent fields represent them. Positive infinity is exponent field all 1, sign bit 0, mantissa all 0; negative infinity is the same but with sign bit 1. I infer that NaNs are any pattern with exponent bits 1 and mantissa bits not all 0. (There is a distinction between "quiet NaNs" and "signaling NaNs"; while I don't really know, I suspect this is done with the sign bit. A signaling NaN is like a VAX reserved operand; a quiet NaN is similar, except that it simply results in another NaN when you do something for which its exact value would matter. It disappears when you do things like multiply it by zero, though.) Infinity is used to represent values that are actually infinite, such as dividing any nonzero number by zero, and values that, while not theoretically deserving to be called infinite, overflow the available number range, such as 1e200*1e200. There is a negative infinity as well; this is a conventional number line, not a projective line with only one point at infinity. "NaN" stands for "Not a Number" and represents something that does not exist on the real line at all; they normally indicate uninitialized data that happens to contain such a bit pattern, but can be generated by some operations, such as 0/0, sqrt(-1), or adding infinities of opposite signs. This concludes today's class on floating point. :-) /~\ The ASCII der Mouse \ / Ribbon Campaign X Against HTML mouse at rodents.montreal.qc.ca / \ Email! 7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B