The examples of floating-point numbers shown on the previous page illustrated the most common general type of floating-point format, the one shown in the first line of the diagram below: The format shown in the first line begins with a single sign bit, which is 0 if the number is positive, and 1 if the number is negative. If the exponent is eight bits long, as shown in the diagram, it is in excess-128 notation, so that the smallest exponent value, 00000000, stands for -128, and the largest exponent value, 11111111, stands for 127.
The second line in the diagram illustrates the kind of floating-point format used on computers such as the PDP-8 and the RECOMP II.
Here, a floating-point number is simply represented by two signed binary numbers, the first, being the exponent, treated as an integer, and the second, being the mantissa, treated as a fraction, both represented in the ordinary format for signed fixed-point numbers used on the computer.
The third line of the diagram illustrates a kind of format which, with a number of variations, was found on most computers with a 24-bit word length.
Computers with a 48-bit word length, on the other hand, typically had hardware floating-point, and used a floating-point format of the type given in the first line.
A Group III format tends to be used where the word length is too long to use a full word for the exponent, and hardware multiplication features make it useful to align the beginning of the mantissa with the beginning of a word.
The Group II format is most popular with machines with small word lengths and limited hardware arithmetic, but it was also used, as in the case of the Maniac II and the Philco 2000, with architectures that started out with hardware floating-point as well.
Also, in some cases, the floating point formats of different sizes for some machines belonged to different groups.
When this happened, all the formats for any one architecture were placed within the discussion of one of the groups of formats included.
Why did these computers use such an unusual floating-point format?
Typically, although these computers did not have hardware floating-point support, the way bigger computers with a 32-bit, 36-bit, 48-bit, or 64-bit word length did, they did come standard with hardware integer multiplication, unlike smaller computers with an 18-bit, 16-bit, or 12-bit word length.
In order to support floating-point arithmetic, the format of double-precision fixed-point numbers on most of these computers omitted the first bit of the second word of the number from the number itself, sometimes treating it as a second copy of the sign, so that fixed-point numbers could be treated as having the binary point on the right, making them integers, or on the left, after the sign, making them fractions on the interval [0,1), without having to adjust them by shifting them one place to the left after a multiplication.